Universidade Federal de Santa Catarina Curso de Pós-Graduação em Matemática Pura e Aplicada

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  

❊st✉❞♦ ❛ss✐♥tót✐❝♦ ♣❛r❛ ✉♠

♠♦❞❡❧♦ ❞❡ ❡✈♦❧✉çã♦ ❝♦♠

♦♣❡r❛❞♦r❡s ❢r❛❝✐♦♥ár✐♦s ❡

❝♦❡✜❝✐❡♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❞♦

t❡♠♣♦

  

❏éss✐❦❛ ❘✐❜❡✐r♦

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❧❡✈❡rs♦♥ ❘♦❜❡rt♦ ❞❛ ▲✉③

  ❋❧♦r✐❛♥ó♣♦❧✐s ❙❡t❡♠❜r♦ ❞❡ ✷✵✶✻

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  ❊st✉❞♦ ❛ss✐♥tót✐❝♦ ♣❛r❛ ✉♠ ♠♦❞❡❧♦ ❞❡ ❡✈♦❧✉çã♦ ❝♦♠ ♦♣❡r❛❞♦r❡s ❢r❛❝✐♦♥ár✐♦s ❡ ❝♦❡✜❝✐❡♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❞♦ t❡♠♣♦

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐✲ ê♥❝✐❛s ❋ís✐❝❛s ❡ ▼❛t❡♠át✐❝❛s✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛✲ t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ❆♥á❧✐s❡✳

  ❏éss✐❦❛ ❘✐❜❡✐r♦ ❋❧♦r✐❛♥ó♣♦❧✐s

  ❙❡t❡♠❜r♦ ❞❡ ✷✵✶✻

  

Ficha de identificação da obra elaborada pelo autor,

através do Programa de Geração Automática da Biblioteca Universitária da UFSC.

Florianópolis, SC, 2016. / Jéssika Ribeiro ; orientador, Cleverson Roberto da Luz -

operadores fracionários e coeficiente dependendo do tempo

Estudo assintótico para um modelo de evolução com

Ribeiro, Jéssika

Programa de Pós-Graduação em Matemática Pura e Aplicada.

Dissertação (mestrado) - Universidade Federal de Santa

165 p. Inclui referências Catarina, Centro de Ciências Físicas e Matemáticas.

Graduação em Matemática Pura e Aplicada. III. Título.

Universidade Federal de Santa Catarina. Programa de Pós

Diferenciais Parciais. I. Luz, Cleverson Roberto da. II.

Propriedades Assintóticas da Solução. 5. Equações Dissipação com Coeficiente Dependendo do Tempo. 4.

  

❊st✉❞♦ ❛ss✐♥tót✐❝♦ ♣❛r❛ ✉♠ ♠♦❞❡❧♦ ❞❡

❡✈♦❧✉çã♦ ❝♦♠ ♦♣❡r❛❞♦r❡s ❢r❛❝✐♦♥ár✐♦s ❡

❝♦❡✜❝✐❡♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❞♦ t❡♠♣♦

  ♣♦r

  ✶

  ❏éss✐❦❛ ❘✐❜❡✐r♦ ❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠

  ▼❛t❡♠át✐❝❛✱ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ❆♥á❧✐s❡✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛ ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡

  ❆♣❧✐❝❛❞❛✳ Pr♦❢✳ ❉r✳ ❘✉② ❈♦✐♠❜r❛ ❈❤❛rã♦

  ❈♦♦r❞❡♥❛❞♦r ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛

  Pr♦❢✳ ❉r✳ ❈❧❡✈❡rs♦♥ ❘♦❜❡rt♦ ❞❛ ▲✉③ ✭❖r✐❡♥t❛❞♦r ✲ ❯❋❙❈✮

  Pr♦❢✳ ❉r✳ ❘✉② ❈♦✐♠❜r❛ ❈❤❛rã♦ ✭❯❋❙❈✮

  Pr♦❢✳ ❉r✳ P❛✉❧♦ ▼❡♥❞❡s ❞❡ ❈❛r✈❛❧❤♦ ◆❡t♦ ✭❯❋❙❈✮

  Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❘❡♠♣❡❧ ❊❜❡rt ✭❯❙P✮

  ❋❧♦r✐❛♥ó♣♦❧✐s✱ s❡t❡♠❜r♦ ❞❡ ✷✵✶✻✳

  ✶ ❇♦❧s✐st❛ ❞❛ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✲ ❈❆P❊❙

  ❖ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛ é ♦ ♠❛✐s ✐♥❞✐❝❛❞♦ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛s ❢❛❝✉❧❞❛❞❡s✱ ❢♦rt❛❧❡❝❡r ♦ r❛❝✐♦❝í♥✐♦ ❡ ✐❧✉♠✐♥❛r ♦ ❡s♣ír✐t♦✳

  ✭❙ó❝r❛t❡s✱ ❋✐❧ós♦❢♦ ●r❡❣♦✮

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡❝❡r é ✉♠❛ ❛rt❡✱ é ✉♠ ❣❡st♦ ❞❡ ❛♠♦r✱ é s❡❧❛r ✉♠❛ ✉♥✐ã♦✱ é ❛❞♠✐t✐r q✉❡ ❤♦✉✈❡ ✉♠ ♠♦♠❡♥t♦ ❡♠ q✉❡ s❡ ♣r❡❝✐s♦✉ ❞❡ ❛❧❣✉é♠❀ é r❡❝♦♥❤❡❝❡r q✉❡ ♦ ❤♦♠❡♠ ❥❛♠❛✐s ♣♦❞❡rá ❧♦❣r❛r ♣❛r❛ s✐ ♦ ❞♦♠ ❞❡ s❡r ❛✉t♦✲s✉✜❝✐❡♥t❡✳ ◆✐♥❣✉é♠ s❡ ❢❛③ s♦③✐♥❤♦✿ s❡♠♣r❡ é ♣r❡❝✐s♦ ✉♠ ♦❧❤❛r ❞❡ ❛♣♦✐♦✱ ✉♠❛ ♣❛❧❛✈r❛ ❞❡ ✐♥❝❡♥t✐✈♦✱ ✉♠ ❣❡st♦ ❞❡ ❝♦♠♣r❡❡♥sã♦✱ ✉♠❛ ❛t✐t✉❞❡ ❞❡ ❛♠♦r✳ ❆ t♦❞♦s ✈♦❝ês✱ q✉❡ ❝♦♠♣❛rt✐❧❤❛r❛♠ ❞♦ ♠❡✉ ✐❞❡❛❧✱ ❞❡❞✐❝♦ ❡ss❛ ✈✐tór✐❛ ❝♦♠ ❛ ♠❛✐s ♣r♦❢✉♥❞❛ ❣r❛t✐❞ã♦ ❡ r❡s♣❡✐t♦✳

  Pr✐♠❡✐r❛♠❡♥t❡ q✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s✱ ♣♦r ♠❡ ❝♦♥❝❡❞❡r ❛ ❣r❛ç❛ ❞❡ ✈✐✈❡r✱ s❛ú❞❡ ♣❛r❛ q✉❡ ♣✉❞❡ss❡ ♣❡rs❡❣✉✐r ♠❡✉s s♦♥❤♦s ❡ ♣♦r t❡r ❝♦❧♦❝❛❞♦ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ♣❡ss♦❛s ♠❛r❛✈✐❧❤♦s❛s q✉❡ ♠❡ ❛❥✉❞❛♠✱ ♠❡ ✐♥s♣✐r❛♠ ❡ ♠❡ ❡♥❝♦r❛❥❛♠✳

  ❋❛❧❛♥❞♦ ❡♠ ♣❡ss♦❛s ♠❛r❛✈✐❧❤♦s❛s✱ q✉❡r♦ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ♣❛✐s✱ ❊❧❡♥❛ ❡ ◆✐❧s♦♥✱ ♣❡❧♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❛♦s ❡st✉❞♦s✱ ♣❡❧❛ ❡❞✉❝❛çã♦✱ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛❥✉❞❛ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s ❡ ♣♦r ✜❝❛r❡♠ ♠❛✐s ❢❡❧✐③❡s q✉❡ ❡✉ ♣♦r ♠✐♥❤❛s ❝♦♥q✉✐st❛s✳ ❱♦❝ês ♣r❡♣❛r❛r❛♠

  ❝✉✐❞❛❞♦s❛♠❡♥t❡ ♠❡✉ ❝❛♠✐♥❤♦ ♣❛r❛ q✉❡ ♦ ❞✐❛ ❞❡ ❤♦❥❡ ❢♦ss❡ ♣♦ssí✈❡❧✳ ❆♠♦ ✈♦❝ês ❝♦♠ t♦❞♦ ♠❡✉ ❝♦r❛çã♦✦

  ➚ ♠✐♥❤❛ ♠❡❧❤♦r ❛♠✐❣❛✱ q✉❡ ❝♦♠♣❛rt✐❧❤❛ s♦❜r❡♥♦♠❡ ❝♦♠✐❣♦✱ ◆❛✐s❡✱ q✉❡ ❝♦♥❤❡❝❡ ♠✐♥❤❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ♠❡ ❛♣♦✐❛✱ ♣❡❧♦ ❛♠♦r ❡ ❛t❡♥çã♦✳ ❖❜r✐✲ ❣❛❞❛ ♣♦r s❡r ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r✐♥❤❛ ❡ ❣✉❛r❞✐ã ❞❛s ♠✐♥❤❛s ♠❡❧❤♦r❡s r❡✲ ❝♦r❞❛çõ❡s✳ ❚❡♥❤♦ ♦r❣✉❧❤♦ ❡♠ t❡r ✈♦❝ê ❝♦♠♦ ✐r♠ã ❡ ❡st❛r❡✐ s❡♠♣r❡ ❛♦ s❡✉ ❧❛❞♦✦ ❚❡ ❛♠♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡✳

  ❆♦s ♠❡✉s ❛✈ós✱ ♣❡❧♦ ❝♦♥st❛♥t❡ ✐♥❝❡♥t✐✈♦ ❛♦s ❡st✉❞♦s✱ ♣❡❧♦s ❡♥s✐✲ ♥❛♠❡♥t♦s✱ ♣❡❧❛ ❛t❡♥çã♦ ❡ ❛❥✉❞❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❛✈ó ❊❧✐③✐❛ ✭✐♥ ♠❡♠♦r✐❛♥✮✱ ♣❡❧❛s t❛r❞❡s ❡♠ q✉❡ ♠❡ ❢❛③✐❛ ❝♦♠♣❛♥❤✐❛ ❡♥q✉❛♥t♦ ❡✉ ❡s✲ t✉❞❛✈❛ ❡ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ♣❛ss❛❞♦s ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s ♣♦r s❡✉s ❣❡st♦s✳

  ❆♦ ♠❡✉ ♥❛♠♦r❛❞♦ ❡ ❛♠✐❣♦ ❋❛❜✐♦ ❈❛s✉❧❛✱ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛✱ ♣♦r ♠❡ ♦✉✈✐r✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♠♣r❡❡♥sã♦✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡✱ ❝❧❛r♦✱ ♥❛ ❝♦♥❞✐çã♦ ❞❡ ✉♠ ❜♦♠ ♠❛t❡♠át✐❝♦✱ ♣♦r ♠❡ ❛❥✉❞❛r t❛♠❜é♠ ♥♦s ❝á❧❝✉❧♦s✳ ❚❡ ❛❞♠✐r♦ ❡ t❡ ❛♠♦✳

  ❆♦s ♠❡✉s ❛♠✐❣♦s✱ P✐❡rr②✱ ❋r❛♥❝✐s❝❛✱ ◆✐❝♦✱ ❉✳ ◆❡r❝✐✱ ❙❤❡②❧❛✱ ❖r✐✲ ❛♥❛✱ ❈❛r❧♦s ❡ ■♥❣r✐❞ ❡ ❉♦♠✐♥❣♦s q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t❛r❛♠ ❡♠ ♠✐♠✱ ♠❡ ❛♣♦✐❛r❛♠✱ ❛❧❣✉♥s ♠❡ ❞❡r❛♠ s✉♣♦rt❡ ♠❛t❡♠át✐❝♦✱ ✜③❡r❛♠ ♣❛rt❡ ❞❛s ♠✐✲ ♥❤❛s ♠❡❧❤♦r❡s ❤♦r❛s ❞❡ ❧❛③❡r ❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❝❛rr❡❣❛r ♦ ♣❡s♦ ❞❡ ❞✐❛s ❞✐❢í❝❡✐s✱ ♥ã♦ ♠❡❞✐♥❞♦ ❡s❢♦rç♦s ♣❛r❛ ✐ss♦✳

  ❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ♠❡str❛❞♦ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛ ♥♦s ❡st✉❞♦s✱ ♣❡❧❛ ❛❥✉❞❛✱ ♣❡❧♦s ✐♥t❡r✈❛❧♦s ❞❡ ❛✉❧❛ r❡❣❛❞♦s ❛ ❝❛❢és✱ ❜♦❛s r✐s❛❞❛s ❡ ❜r❛✐♥s✲ t♦r♠✐♥❣s s♦❜r❡ ♦s ❡①❡r❝í❝✐♦s✳ ❆♦ Pr♦❢✳ ❈❧❡✈❡rs♦♥✱ ♣♦r t❡r ❛❝❡✐t❛❞♦ s❡r ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣❡❧❛ s✉❛

  ♣r❡st❛t✐✈✐❞❛❞❡✱ ❝♦♠♣❡tê♥❝✐❛✱ ♦r❣❛♥✐③❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ ✭❛té ♠❡s♠♦ ♥♦s ✜♥❛✐s ❞❡ s❡♠❛♥❛✮ ❝♦♠✐❣♦ ❡ ❝♦♠ ♦ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❙❡♠ ✈♦❝ê ♥❛❞❛ ❞✐ss♦ s❡r✐❛ ♣♦ssí✈❡❧✳ ▼❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✳

  ❆s ♣❡ss♦❛s q✉❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❛ ❡st✉❞❛r ♠❛t❡♠át✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❚✐❛ ▼❛r✐✱ Pr♦❢✳ ❙✐♠♦♥❡✱ Pr♦❢✳ ❱✐❧♠❛r✱ Pr♦❢✳ ❖s❝❛r ❏❛♥❡s❝❤ ❡ Pr♦❢✳ ●✐✉❧✐❛♥♦ ❇♦❛✈❛✳

  ❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ss❛ ❥♦r♥❛❞❛✱ ♣❡❧❛s ❛✉❧❛s✱ ♣❡❧❛s ♠♦♥✐t♦r✐❛s✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❞✐s♣♦s✐çã♦ ❡♠ ❛❥✉❞❛r ❡ t❛♠❜é♠ ❛♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ♣♦r t♦❞❛ ❛ss✐stê♥❝✐❛✳

  ➪ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ♥❡ss❡ ú❧t✐♠♦ ❛♥♦✳

  ❘❡s✉♠♦

  ◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ σ

  ✲❡✈♦❧✉çã♦✱ ❝♦♠ ♦ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉r❛❧ ❞❡♣❡♥❞❡♥❞♦ ❞♦ t❡♠♣♦✱ ❜❛s❡❛❞♦ ♥❛s ✐❞❡✐❛s ❞❡ ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪ ❡ ❉✬❆❜❜✐❝❝♦✲❈❤❛rã♦✲❞❛ ▲✉③ ❬✺❪✳ P❛r❛ ❡♥❝♦♥tr❛r t❛①❛s ❡①♣❧í❝✐t❛s ❞❡ ❞❡✲ ❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❞♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✱ ❞✐✈✐❞✐♠♦s ♦ ❡s♣❛ç♦

  n

  ❞❡ ❋♦✉r✐❡r✱ R ✱ ❡♠ ❞✉❛s r❡❣✐õ❡s✿ ❛❧t❛ ❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✳ ❯t✐❧✐③❛♠♦s

  ξ

  ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s ♣❛r❛ ❝❛❞❛ r❡❣✐ã♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛s♦ ❝♦♠ ❛♠♦r✲ t❡❝✐♠❡♥t♦ ✧❡✛❡❝t✐✈❡✧ ❞❛❞♦ ♣❡❧❛ ❝♦♥❞✐çã♦ 2δ < σ(1 + α)✳ ❆♣❧✐❝❛♠♦s ♦ ♠ét♦❞♦ ❞❡ ❞✐❛❣♦♥❛❧✐③❛çã♦ ✉s❛❞♦ ♣♦r ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪ ♣❛r❛ ♦❜t❡r ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✳ ◆❛ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ✉t✐✲ ❧✐③❛♠♦s ♦ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❘✳ ❈✳ ❈❤❛rã♦✱ ❈✳ ❘ ❞❛ ▲✉③ ❡ ❘✳ ■❦❡❤❛t❛ ❡♠ ❬✸❪ ❡ ❬✹❪✳

  ❆❜str❛❝t

  ■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❢♦r ❛ σ✲❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ t✐♠❡✲❞❡♣❡♥❞❡♥t str✉❝t✉r❛❧ ❞❛♠♣✐♥❣✱ ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛s ♦❢ ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪ ❛♥❞ ❉✬❆❜❜✐❝❝♦✲❈❤❛rã♦✲❞❛ ▲✉③ ❬✺❪✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❡①♣❧✐❝✐t ❞❡❝❛② r❛t❡s ❢♦r t❤❡ ❡♥❡r❣② ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❈❛✉✲

  n

  ❝❤② ♣r♦❜❧❡♠✱ ✇❡ s♣❧✐t t❤❡ ❋♦✉r✐❡r s♣❛❝❡✱ R ✱ ✐♥ t✇♦ r❡❣✐♦♥s✿ ❤✐❣❤

  ξ

  ❛♥❞ ❧♦✇ ❢r❡q✉❡♥❝②✳ ❲❡ ✉s❡ ❞✐✛❡r❡♥t ♠❡t❤♦❞s ✐♥ ❡❛❝❤ r❡❣✐♦♥ t♦ ❣❡t ♦✉r ❡st✐♠❛t❡s✱❝♦♥s✐❞❡r✐♥❣ t❤❡ ❡✛❡❝t✐✈❡ ❞❛♠♣✐♥❣ ❝❛s❡✱ ❣✐✈❡♥ ❜② ❝♦♥❞✐✲ t✐♦♥ 2δ < σ(1 + α)✳ ❲❡ ❛♣♣❧② t❤❡ ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ♠❡t❤♦❞ ✉s❡❞ ❜② ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪ t♦ ♦❜t❛✐♥ ❡st✐♠❛t❡s ✐♥ t❤❡ ❧♦✇ ❢r❡q✉❡♥❝② r❡❣✐♦♥ ♦❢ t❤❡ ❋♦✉r✐❡r s♣❛❝❡✳ ■♥ t❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝② r❡❣✐♦♥ ✇❡ ✉s❡ t❤❡ ♠❡t❤♦❞ ❞❡✈❡❧♦♣❡❞ ❜② ❘✳ ❈✳ ❈❤❛rã♦✱ ❈✳ ❘ ❞❛ ▲✉③ ❛♥❞ ❘✳ ■❦❡❤❛t❛ ✐♥ ❬✸❪ ❛♥❞ ❬✹❪✳

  ❙✉♠ár✐♦

  ■♥tr♦❞✉çã♦ ①✐✈

  ✶ ◆♦t❛çõ❡s ❡ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✽ ✶✳✶ ◆♦t❛çõ❡s ❡ Pr✐♠❡✐r♦s ❈♦♥❝❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

  p

  (Ω) ✶✳✸ ❊s♣❛ç♦s L ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✹ ❊s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③ ❡ ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✺ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✻ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✼ ❉❡s✐❣✉❛❧❞❛❞❡s ■♠♣♦rt❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✽ ❚❡♦r❡♠❛ ❞❛ ❉✐✈❡r❣ê♥❝✐❛ ❡ ❋ór♠✉❧❛s ❞❡ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✾ ▲❡♠❛ ❞❡ ▼❛rt✐♥❡③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✶✵ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧ ✭❱❡rsã♦ ■♥t❡❣r❛❧✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

  ✷ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛ ❇❛✐①❛ ❋r❡q✉ê♥❝✐❛ ✸✶ ✷✳✶ ❩♦♥❛ Ps❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

  ✷✳✷ ❩♦♥❛ ❊❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛ ❆❧t❛ ❋r❡q✉ê♥❝✐❛ ✼✶ ✹ ❘❡s✉❧t❛❞♦ Pr✐♥❝✐♣❛❧ ✼✽ ❆ ❘❡s✉❧t❛❞♦s ❆❞✐❝✐♦♥❛✐s ✶✵✷ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✹✼

  ■♥tr♦❞✉çã♦

  ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ♣r♦♣r✐❡❞❛❞❡s ❞❛ s♦✲

  n

  ❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ σ✲❡✈♦❧✉çã♦ ❡♠ R ✱ ❝♦♠ ♦ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉r❛❧ ❞❡♣❡♥❞❡♥❞♦ ❞♦ t❡♠♣♦✱ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡s❝r✐t♦ ♣♦r ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪ ♥❛ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❡ ♦ ♠ét♦❞♦ ❞❛ ❡♥❡r❣✐❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❋♦✉r✐❡r ❝♦♠ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❛❞❡q✉❛❞♦s✱ ✉s❛❞♦ ♣♦r ❉✬❆❜❜✐❝❝♦✲❈❤❛rã♦✲❞❛ ▲✉③ ❬✺❪✱ ♥❛ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✳

  Pr❡❝✐s❛♠❡♥t❡✱ q✉❡r❡♠♦s ♦❜t❡r t❛①❛s ❡①♣❧í❝✐t❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛

  2

  ❛ ♥♦r♠❛ L ❞❛ s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ ♠♦❞❡❧♦✿

  σ δ n

  u (t, x) + ( u(t, x) + 2b(t)( u (t, x) = 0, t > 0, x

  tt t ✭✶✮

  −∆) −∆) ∈ R

  n

  u(0, x) = u (x), u t (0, x) = u

  1 (x), x

  ∈ R

  α

  ♦♥❞❡ σ > 0✱ δ ∈ (0, σ)✱ 2δ < σ(1 + α) ❡ b(t) = µ(1 + t) ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❝♦♠ µ > 0 ❡ 0 < α < 1.

  δ

  u ❖ t❡r♠♦ 2b(t)(−∆) t r❡♣r❡s❡♥t❛ ♦ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉r❛❧ ❞❡✲

  δ

  ♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ❡ ♦ ♦♣❡r❛❞♦r (−∆) é ❞❡✜♥✐❞♦ ♣♦r

  δ 2δ 2δ n

  ( v(x) := F ( (R ) ✱ ♦♥❞❡ F ❞❡✲

  −∆) |ξ| bv(ξ))(x), ♣❛r❛ t♦❞♦ v ∈ H ♥♦t❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❝♦♠ r❡s♣❡✐t♦ ❛ ✈❛r✐á✈❡❧ x ❡ r❡♣r❡s❡♥t❛✲ ♠♦s ♣♦r bv = F(v).

  ◆❛t✉r❛❧♠❡♥t❡✱ ♦ ♣❛♣❡❧ ❞❡s❡♠♣❡♥❤❛❞♦ ♣❡❧♦ t❡r♠♦ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ✈❛r✐❛ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❞♦ ❝♦❡✜❝✐❡♥t❡ b(t) ❡ ❞♦s ♣❛r❛♠êtr♦s σ ❡ δ✳ ❉❡✲ ✈✐❞♦ ❛ ❡st❡ ❢❛t♦✱ sã♦ ♥❡❝❡ssár✐❛s ❞✐❢❡r❡♥t❡s ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ ❡st✉❞❛r ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ss❡s t❡r♠♦s ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ♥♦ ♣❡r✜❧ ❛ss✐♥tót✐❝♦ ❞❛ s♦❧✉çã♦✱ q✉❛♥❞♦ t ✈❛✐ ♣❛r❛ ✐♥✜♥✐t♦✳ ❖ ♠♦❞❡❧♦ ❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ♦ ♥♦ss♦ tr❛❜❛❧❤♦ é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

  α

  b(t) = µ(1 + t) ♣❛r❛ ❛❧❣✉♠ µ > 0 ❡ α ∈ (0, 1). ❚❡♠♦s q✉❡ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❡s♣❡❝í✜❝❛ ❞❡ ❝♦❡✜❝✐❡♥t❡s b(t) q✉❡ ❞❡✲

  ♣❡♥❞❡♠ ❞♦s ❡①♣♦❡♥t❡s ❢r❛❝✐♦♥ár✐♦s σ ❡ δ✱ ♦ ♣❡r✜❧ ❛ss✐♥tót✐❝♦ ❞❛ s♦❧✉çã♦ ❞❡ ✭✶✮✱ q✉❛♥❞♦ t → ∞✱ é ♦ ♠❡s♠♦ ❞♦ ♣r♦❜❧❡♠❛

  1

  σ−δ

  v (t, x) + ( v(t, x) = 0, v(0, x) = v (x),

  t

  ✭✷✮ −∆)

  2b(t) , u , b(t), σ

  ♣❛r❛ ✉♠❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞❡ v ✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ u 1 ❡ δ.

  ❖ ♣r♦❜❧❡♠❛ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞❡ ♣r♦❜❧❡♠❛ ❞❡ ❞✐❢✉sã♦ ❛♥ô♠❛❧❛✱ q✉❡ s✐❣♥✐✜❝❛ q✉❡✱ ❡♠ ❣❡r❛❧✱ σ − δ 6= 1✱ ❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r✱ q✉❛♥❞♦ σ − δ = 1.

  ❈♦♥s✐❞❡r❛♥❞♦ ♦✉tr❛ ❝❧❛ss❡ ❞❡ ❝♦❡✜❝✐❡♥t❡s b(t) ♦ ♣❡r✜❧ ❛ss✐♥tót✐❝♦ ❞❡ ✭✶✮ ♥ã♦ s❡ ❝♦♠♣♦rt❛ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ♦ ❞❡ ✭✷✮✳ ❊♠ ❉✬❆❜❜✐❝❝♦✲ ❊❜❡rt ❬✻❪✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❝♦♠♣❧❡t❛ ❞✐❢❡r❡♥❝✐❛♥❞♦ ❡st❡s ❞♦✐s ❝❛s♦s✳ ❖ ❝❛s♦ ✧❡✛❡❝t✐✈❡✧ é q✉❛♥❞♦ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❞✐❢✉sã♦ ❛♥ô♠❛❧❛ ♦❝♦rr❡ ❡ ❛ ❝♦♥❞✐çã♦ 2δ < σ(1 + α) é s❛t✐s❢❡✐t❛✳ ❖ ❝❛s♦ ✧♥♦♥✲❡✛❡❝t✐✈❡✧ é q✉❛♥❞♦ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❞✐❢✉sã♦ ❛♥ô♠❛❧❛ ♥ã♦ ♦❝♦rr❡✳ ◆❡ss❡ ❝❛s♦ ♦ ♣❡r✜❧ ❞❛ s♦❧✉çã♦ ❞❡ ✭✶✮ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛♦ ♣❡r✜❧ ❞❛ ❢✉♥çã♦ r❡s✉❧t❛❞♦

  R t

  δ

  b(τ ) dτ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ♦♣❡r❛❞♦r ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧ exp

  −(−∆) à s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❞❛ ❡q✉❛çã♦ ❞❡ ❡✈♦❧✉çã♦ ❧✐✈r❡ u

  • tt σ

  ( u = 0 ✱ ❝♦♠ ❞❛❞♦s ✐♥✐❝✐❛✐s ❛❞❡q✉❛❞♦s✳ ◆❡ss❡ ❝❛s♦ ❝♦♥s✐❞❡r❛✲s❡ ❛

  −∆) ❝♦♥❞✐çã♦ 2δ > σ(1 + α).

  ❘❡s✉❧t❛❞♦s s✐♠✐❧❛r❡s ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♣❛r❛ −1 ≤ α < 0✳ P❛r❛ α < −1 t❡♠♦s ✧s❝❛tt❡r✐♥❣✧✱ ❡♥q✉❛♥t♦ ♣❛r❛ α > 1 t❡♠♦s ✧♦✈❡r❞❛♠♣✐♥❣✧✳

  ❆ ❝❧❛ss✐✜❝❛çã♦ ❛♣r❡s❡♥t❛❞❛ s❡❣✉❡ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❡♠ ❏✳ ❲✐rt❤ ❬✶✼❪✳ ◆❡ss❡ ♣❛♣❡r✱ ❛ss✉♠✐♥❞♦ ✉♠ ❝♦♥tr♦❧❡ ❛❞❡q✉❛❞♦ ♥❛ ♦s❝✐❧❛çã♦ ❞❡ b(t)✱ ❡❧❡ ♣r♦♣ôs ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ ❡①t❡r✐♦r ✭σ = 1 ❡ δ = 0✮✱ ✐st♦ é u (t, x) (t, x) = 0 (u, u )(0, x) = (u , u )(x),

  tt t t

  1

  − ∆u(t, x) + 2b(t)u ♣r♦✈❛♥❞♦ q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ss❛ ❡q✉❛çã♦ t❡♠ ♠✉❞❛♥✲ ç❛s ♣r♦❢✉♥❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ b(t)✳

  ❘❡❝❡♥t❡♠❡♥t❡ ❡st❡ tó♣✐❝♦ ✈❡♠ s❡♥❞♦ ❢♦rt❡♠❡♥t❡ ❡st✉❞❛❞♦ ♣♦r ✈ár✐♦s ❛✉t♦r❡s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥♦ q✉❡ s❡ r❡❢❡r❡ ❛ ♠♦❞❡❧♦s ❝♦♠ ❞✐ss✐♣❛çã♦ ❡s✲ tr✉t✉r❛❧ ❡ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦✳ ❆❜❛✐①♦ ❞❡st❛❝❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✳

  ▼✳ ❘❡✐ss✐❣ ❬✶✺❪ ❡①♣❧✐❝❛ ❛ t❡♦r✐❛ ❞❛s t❛①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ L p q ♣❛r❛ −L

  ♠♦❞❡❧♦s ❞❛ ♦♥❞❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s q✉❡ ❞❡♣❡♥❞❡♠ ❞♦ t❡♠♣♦✱ ❡①♣❧✐❝❛♥❞♦ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ ♦❝✐❧❛çõ❡s ♥♦s ❝♦❡✜❝✐❡♥t❡s ✉s❛♥❞♦ ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ♣r❡✲ ❝✐s❛ ❡ ❞❡st❛❝❛♥❞♦ ❝♦♠♦ ❛ ♠❛ss❛ ❡ ♦s t❡r♠♦s ❞❡ ❞✐ss✐♣❛çã♦ ✐♥✢✉❡♥❝✐❛♠ ❡ss❛s t❛①❛s✳

  ❊♠ ❬✶✻❪✱ ▼✳ ❘❡✐ss✐❣ ❡ ❳✳ ▲✉ ❡st✉❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♠♦❞❡❧♦s ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉r❛❧ ❡♥tr❡ ❛ ❡q✉❛çã♦ ❝❧áss✐❝❛ ❞❛ ♦♥❞❛ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ ❡ ♦ ♠♦❞❡❧♦ ❞❛ ♦♥❞❛ ❝♦♠ ❞✐ss✐♣❛çã♦ ✈✐s❝♦❡❧ást✐❝❛✳ ◆♦ ♣❛♣❡r ❡❧❡s ❡♥❝♦♥tr❛r❛♠ t❛①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❞❛ s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②

  σ

  u (t, x) u (t, x) = 0

  tt t

  − ∆u(t, x) + b(t)(−∆) u(0, x) = u (x), u (0, x) = u (x),

  t

  1 −δ

  ♦♥❞❡ σ ∈ (0, 1]✱ δ ∈ [0, 1] ❡ b(t) = µ(1 + t) ❝♦♠ µ > 0✳ ❆q✉✐ −∆ é ♦

  σ

  u t ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡✱ (−∆) ♦ t❡r♠♦ ❞❡ ❞✐ss✐♣❛çã♦ ❡ b(t) ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡❝r❡s❝❡♥t❡ q✉❡ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦✳

  ▼✳ ❑✳ ▼❡③❛❞❡❦ ❬✶✹❪ ✐♥✈❡st✐❣♦✉ ❡ ❛♣r❡s❡♥t♦✉ ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛ ❡♥❡r✲ ❣✐❛ ❞❡ ♦r❞❡♠ ♠❛✐s ❛❧t❛s ❛ss♦❝✐❛❞❛ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❈❛✉❝❤② ♣❛r❛ ♠♦❞❡❧♦s σ✲❡✈♦❧✉çã♦ ❝♦♠ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉r❛❧✳ ❖ ♣r♦❜❧❡♠❛ ❡st✉❞❛❞♦ ❢♦✐

  σ δ

  u (t, x) + ( u(t, x) + b(t)( u (t, x) = 0

  tt t

  −∆) −∆) u(0, x) = u (x), u t (0, x) = u

  1 (x)

  ♦♥❞❡ σ > 1✱ δ ∈ (0, σ] ❡ b(t) é ✉♠❛ ❢✉♥çã♦ ♠♦♥ót♦♥❛ ♣♦s✐t✐✈❛✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ❝♦♠ ❞❡t❛❧❤❡s ♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❛

  ❡q✉❛çã♦ ❞❡ ❡✈♦❧✉çã♦ ❞❛❞❛ ❡♠ ✭✶✮✳ ◆❛ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✱ ✉t✐❧✐③❛♠♦s ♦ ♠ét♦❞♦ ❞❡s❝r✐t♦ ♣♦r ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪✱ ❡✱ ♥♦ ❝♦♥t❡①t♦ ❞❛ ❛❧t❛ ❢r❡q✉ê❝✐❛✱ ✉s❛♠♦s ♦ ♠ét♦❞♦ ❞❛ ❡♥❡r❣✐❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❋♦✉r✐❡r ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧✱ q✉❡ ❛ss✉♠❡ ❛♣❡♥❛s q✉❡ ♦ t❡r♠♦ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉✲ r❛❧ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ b(t)✱ é ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ♠♦♥ót♦♥❛ ❝r❡s✲ ❝❡♥t❡ ❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣♦r ♣❛rt❡s✱ ♦❜t✐❞♦ ♣♦r ❉✬❆❜❜✐❝♦✲❈❤❛rã♦✲❞❛ ▲✉③ ❬✺❪✳ ◆❛ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✱ ❞❡✈✐❞♦ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ m(t, ξ) ✱

  N (t, ξ)

  ✉s❛♠♦s ❛ ❝✉r✈❛ Γ ✱ ♣❛r❛ ❞✐✈✐❞✐r ❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❡♠ ❞✉❛s ③♦♥❛s✿ ❡❧í♣t✐❝❛ ❡ ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧ ❡✱ ❡♠ ❛♠❜❛s s❡♣❛r❛❞❛♠❡♥t❡✱ ❡♥❝♦♥tr❛♠♦s ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛ s♦❧✉çã♦✱ ♣r♦✈❛♥❞♦ q✉❡ ❡st❛ ❞❡❝❛✐ ♣♦✲ ❧✐♥♦♠✐❛❧♠❡♥t❡✳ ◆❛ r❡❣✐ã♦ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ♣r♦✈❛♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ♥♦ ❡s♣❛ç♦ ❞❡ ❋♦✉r✐❡r ❞❡❝❛✐ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡✳

  ❖ tr❛❜❛❧❤♦ ❡♥❝♦♥tr❛✲s❡ ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳ ❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❢♦✐ ❡s❝r✐t♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ t♦r♥❛r ♦ tr❛❜❛❧❤♦ ♦ ♠❛✐s ❛✉t♦✲ ❝♦♥t✐❞♦ ♣♦ssí✈❡❧✱ ❡ ♣♦r ✐ss♦✱ ♥❡❧❡ ❛♣r❡s❡♥t❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♥♦t❛çõ❡s✱

  p

  ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s s♦❜r❡ ❊s♣❛ç♦s L ✱ ❚❡♦r✐❛ ❞❡ ❉✐s✲ tr✐❜✉✐çõ❡s✱ ❊s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③ ❡ ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s✱ ❚r❛♥s❢♦r✲ ♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❡ ❛❧❣✉♥s ❧❡♠❛s q✉❡ ✉s❛r❡♠♦s ♥♦ ❞❡❝♦rr❡r ❞♦ t❡①t♦✳

  ◆♦ ❈❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s ♦ ♣r♦❜❧❡♠❛ ♥❛ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❡ ❡♥✲ ❝♦♥tr❛♠♦s ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❜❛s❡❛❞♦ ♥♦ ♠ét♦❞♦ ❞❡ ❞✐❛❣♦♥❛✲ ❧✐③❛çã♦ ✉t✐❧✐③❛❞♦ ❡♠ ❬✻❪✳ ▼♦t✐✈❛❞♦s ♣❡❧♦s ❞✐❢❡r❡♥t❡s ❝♦♠♣♦rt❛♠❡♥t♦s ❞❡ m(t, ξ)✱ ❛♣r❡s❡♥t❛❞♦s ♥♦ t❡①t♦✱ ✐♥tr♦❞✉③✐♠♦s ❞✉❛s ③♦♥❛s ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✿ ❩♦♥❛ ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧✱ ❞❡♥♦t❛❞❛ ♣♦r Z pd ❡ ❡❧í♣t✐❝❛ ❞❡♥♦✲

  low

  t❛❞❛ ♣♦r Z ✳

  ell

  ◆♦ ❈❛♣ít✉❧♦ ✸ ❡♥❝♦♥tr❛♠♦s t❛①❛ ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ♥❛ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❛ ❡♥❡r❣✐❛ ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧✱ ❛♣r❡s❡♥✲ t❛❞♦ ❡♠ ❬✺❪✳ ❯t✐❧✐③❛♠♦s ♦ ♠ét♦❞♦ ❞♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❡ ❡♥❝♦♥tr❛♠♦s ❛❧❣✉♠❛s ❡st✐♠❛t✐✈❛s ❡ ♣❛r❛ ❝♦♥❝❧✉✐r ❛♣❧✐❝❛♠♦s ♦ ▲❡♠❛ ❞❡ ▼❛rt✐♥❡③ ❬✶✶❪ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ❡♠ q✉❡stã♦✳ ❯t✐❧✐③❛♠♦s ❡st❡ ♠ét♦❞♦ ♣❛r❛ ❛♣r❡s❡♥t❛r ❛♦ ❧❡✐t♦r ✉♠ ♠ét♦❞♦ ❞✐❢❡r❡♥t❡ ❛♦ ❛♣r❡s❡♥✲ t❛❞♦ ♥❛ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✱ ✈✐st♦ q✉❡✱ s❡❣✉✐♥❞♦ ♦ ♠ét♦❞♦ ✉t✐❧✐③❛❞♦ ❡♠ ❬✻❪✱ ♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❛ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ s❡❣✉✐❛♠ ❞❡ ♠♦❞♦ s✐♠✐❧❛r ❛♦s ❛♣r❡s❡♥t❛❞♦s ♥❛ ❜❛✐①❛✳

  P♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✹ ♣r♦✈❛♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞♦ tr❛❜❛❧❤♦ ✉s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♣r♦✈❛♠♦s q✉❡ ❛ s♦❧✉çã♦ ❞♦ ♠♦❞❡❧♦ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ♣❛r❛

  2δ

  ¯ θ > :

  1+α k 2 2 n p n p n −k 01 −k 11

  u(t) + Ct

  L (R ) L (R )

  1 L (R )

  k(−∆) k ≤ Ct ku k ku k 1−α k n k−σ n −C (1+t)

  1 e

  ku k H (R ) ku k H (R ) ❝♦♠ ¯θ✱ k

  01 ❡ k 11 ❞❡✜♥✐❞♦s ♥♦ ❚❡♦r❡♠❛ ✹✳✷✳ ❚❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s r❡✲ 2δ 2δ

  s✉❧t❛❞♦s ♣❛r❛ θ < ❡ θ = ✳

  1+α 1+α

  ❈♦♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ♥ã♦ ❢♦r❛♠ ✉s❛❞♦s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞♦ tr❛❜❛❧❤♦ ♠♦♥t❛♠♦s ✉♠ ❛♣ê♥❞✐❝❡✳ ❖s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❛♣ê♥❞✐❝❡ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♣r♦✈❛r r❡s✉❧t❛❞♦s ❞❡ ♣❡r✜❧ ❛ss✐♥tót✐❝♦✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❛ ♦t✐♠❛❧✐❞❛❞❡ ❞❛s ❡st✐♠❛t✐✈❛s ❧✐♥❡❛r❡s✱ ✐st♦ é✱ ♣❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ✷ ❞❛❞♦ ❡♠ ❬✻❪✳

  ❈❛♣ít✉❧♦ ✶ ◆♦t❛çõ❡s ❡ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s

  ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✱ ❛ ✜♠ ❞❡ ❛✉①✐❧✐❛r ❛ ♣r♦✈❛ ❞♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧✳ ◆ã♦ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❞❡♠♦♥str❛çõ❡s✱ ♣♦r s❡r❡♠ ❝♦♥s✐❞❡✲ r❛❞♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s✳ ❈♦♥t✉❞♦✱ ✈❛♠♦s ♣r♦❝✉r❛r r❡❢❡r❡♥❝✐á✲❧♦s ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r✱ ❝❛s♦ t❡♥❤❛ ✐♥t❡r❡ss❡✱ t❡♥❤❛ ❝♦♥t❛❞♦ ❝♦♠ s✉❛s r❡s♣❡❝✲ t✐✈❛s ♣r♦✈❛s✱ ❢❛❝✐❧✐t❛♥❞♦ ❛ ❧❡✐t✉r❛ ❡ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ r❡s✉❧t❛❞♦ ♥❛ s✉❛ t♦t❛❧✐❞❛❞❡✳

  ✶✳✶ ◆♦t❛çõ❡s ❡ Pr✐♠❡✐r♦s ❈♦♥❝❡✐t♦s

  ✶✳ K ✐♥❞✐❝❛ ♦ ❝♦r♣♦ R ♦✉ C❀

  n

  , x , x , ) ✷✳ x = (x

  1

  2 3 n ♣♦♥t♦ ♥♦ ❡s♣❛ç♦ R ❀

  · · · , x

  n

  ✸✳ | · | ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ ❡♠ R ❀

  n

  • 1

  2 n ♣❛r❛ α = (α 1 n

  · · · + α · · · , α ∈ N ∈ N❀

  • α , ) , n ✹✳ |α| = α

  2 n n

  (R ) ✺✳ L é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s u : R → R✱ ♠❡♥s✉rá✈❡✐s t❛✐s q✉❡

  Z

  2

  |u(x)| ∞ ; dx < + R n

  1/2

  Z

  2 n

2 n

  2

  (R ) = dx ✻✳ ❙❡ u ∈ L ❡♥tã♦ kuk L (R ) ❞❡✜♥❡ R n |u(x)|

  ♥♦r♠❛❀ ∂u

  = ✼✳ u t ❞❡r✐✈❛❞❛ ❞❡ u ❡♠ r❡❧❛çã♦ ❛ t❀

  ∂t

  2

  ∂ u =

  ✽✳ u tt s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞❡ u ❡♠ r❡❧❛çã♦ ❛ t❀

  2

  ∂t

  |α|

  ∂ u

  α n

  u = , α = (α , ) ✾✳ D

  1 n ❀ α 1 α · · · , α ∈ N n

  ∂x ... ∂x n

  1

  ∂u ∂u ∂u ∂u , , ,

  ✶✵✳ ▽ u = grad u = · · · , r❡♣r❡s❡♥t❛ ♦ ❣r❛❞✐✲ ∂x ∂x ∂x ∂x

  1

  2 3 n

  ❡♥t❡ ❞❛ ❢✉♥çã♦ u❀

  n

  X ∂u

  i

  , u , u , ..., u ) ✶✶✳ ❙❡ u = (u

  1

  2 3 n ❡♥tã♦ div u = r❡♣r❡s❡♥t❛ ♦

  ∂x

  i i=1

  ❞✐✈❡r❣❡♥t❡ ❞❛ ❢✉♥çã♦ u❀

  n

  ∂ u ✶✷✳ △ u = r❡♣r❡s❡♥t❛ ♦ ❧❛♣❧❛❝✐❛♥♦ ❞❛ ❢✉♥çã♦ u❀

  2

  ∂x

  i i=1 n

  , ξ , ξ , ); ✶✸✳ ❙❡ ξ ∈ R ❡♥tã♦ ξ = (ξ

  1

  2 3 n

  · · · , ξ

  ✶✹✳ bu r❡♣r❡s❡♥t❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ ❢✉♥çã♦ u;

  −1

  ✶✺✳ F r❡♣r❡s❡♥t❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✐♥✈❡rs❛❀

  n

  ✶✻✳ ∗ ❞❡♥♦t❛ ❛ ❝♦♥✈♦❧✉çã♦ ❡♠ t❡r♠♦s ❞❡ x ❡♠ R ❀

  k

  u ✶✼✳ ∂ r❡♣r❡s❡♥t❛ ❛ ❞❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ k ❡♠ r❡❧❛çã♦ x ❞❛ ❢✉♥çã♦ u❀

  x

  ✶✽✳ ❙❡❥❛♠ f, g : Ω → R ❞✉❛s ❢✉♥çõ❡s✳ ❯s❛✲s❡ ❛ ♥♦t❛çã♦ f . g s❡ ❡①✐t❡ > 0 g(x)

  ✉♠❛ ❝♦♥st❛♥t❡ C

  1 t❛❧ q✉❡ f(x) 6 C 1 ✱ ♣❛r❛ t♦❞♦ x ∈ Ω❀

  ✶✾✳ ❙❡❥❛♠ f, g : Ω → R ❞✉❛s ❢✉♥çõ❡s✳ ❯s❛✲s❡ ❛ ♥♦t❛çã♦ f ≈ g s❡ f . g ❡ g . f❀

  2 i,j ij

  ✷✵✳ ❙❡❥❛ R ∈ M ✉♠❛ ♠❛tr✐③ 2 × 2✳ ❉❡♥♦t❛✲s❡ ♣♦r |R| = max |R |✳

  ■❞❡♥t✐❞❛❞❡s út❡✐s

  1

  ❙❡ f, g sã♦ ❢✉♥çõ❡s ❡s❝❛❧❛r❡s ❞❡ ❝❧❛ss❡ C ✱ c é ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧

  1

  ❡ F ❡ G sã♦ ❝❛♠♣♦s ✈❡t♦r✐❛✐s t❛♠❜é♠ ❞❡ ❝❧❛ss❡ C ✱ ❡♥tã♦ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r♦✈❛❞❛s✳ ✶✳ ∇(f + g) = ∇f + ∇g; ✷✳ ∇(cf) = c∇f; ✸✳ ∇(fg) = f∇g + g∇f; ✹✳ ❞✐✈(F + G) = ❞✐✈(F ) + ❞✐✈(G); ✺✳ ❞✐✈(fF ) = f❞✐✈(F ) + ∇f · F ;

  n

  ❊♠ q✉❡ ♦ ♣♦♥t♦ · ✐♥❞✐❝❛ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧ ❡♠ R ✳

  ✶✳✷ ❉✐str✐❜✉✐çõ❡s n

  ❈♦♥s✐❞❡r❡ Ω ✉♠ ❛❜❡rt♦ ❞❡ R ✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛s ✐♥t❡❣r❛✐s r❡❛✲ ❧✐③❛❞❛s s♦❜r❡ Ω sã♦ ♥♦ s❡♥t✐❞♦ ❞❡ ▲❡❜❡s❣✉❡✱ ❛ss✐♠ ❝♦♠♦ ❛ ♠❡♥s✉r❛❜✐✲ ❧✐❞❛❞❡ ❞❛s ❢✉♥çõ❡s ❡♥✈♦❧✈✐❞❛s✳

  ❈♦♠♦ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❛s ❙❡çõ❡s ✶✳✷ ❡ ✶✳✸ ❝✐t❛♠♦s ❇r❡③✐s ❬✷❪✱ ❊✈❛♥s ❬✽❪ ❡ ▼❡❞❡✐r♦s✲❘✐✈❡r❛ ❬✶✷❪✱ ❬✶✸❪✳

  ) ❙❡❥❛ u : Ω → K ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡ s❡❥❛ (K i ❛ ❢❛♠í❧✐❛ ❞❡

  i∈I i

  t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❛❜❡rt♦s K ❞❡ Ω t❛✐s q✉❡ u = 0 q✉❛s❡ s❡♠♣r❡ ❡♠ S K

  K

  

i ✳ ❈♦♥s✐❞❡r❛✲s❡ ♦ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ K = i ✳ ❊♥tã♦ u = 0

i∈I

  q✉❛s❡ s❡♠♣r❡ ❡♠ K.

  ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ❞❡✜♥❡✲s❡ ♦ s✉♣♦rt❡ ❞❡ u✱ q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r supp (u) ✱ ❝♦♠♦ s❡♥❞♦ ♦ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ ❞❡ Ω supp (u) = Ω

  \K.

  ∞

  (Ω) ❉❡✜♥✐çã♦ ✶✳✶ ❘❡♣r❡s❡♥t❛♠♦s ♣♦r C ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s u : Ω

  → K, ❝✉❥❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡ t♦❞❛s ❛s ♦r❞❡♥s sã♦ ❝♦♥tí♥✉❛s ❡ ❝✉❥♦ s✉♣♦rt❡

  ∞

  (Ω) é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❞❡ Ω✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ C sã♦ ❝❤❛♠❛❞♦s ❞❡ ❢✉♥çõ❡s t❡st❡s✳

  ∞

  (Ω) ◆❛t✉r❛❧♠❡♥t❡✱ C é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ K ❝♦♠ ❛s ♦♣❡r❛✲

  çõ❡s ✉s✉❛✐s ❞❡ s♦♠❛ ❞❡ ❢✉♥çõ❡s ❡ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✳

  (Ω) ◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ C ∞ ∞

  (Ω) (Ω) ❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠ {ϕ k ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ C ❡ ϕ ∈ C ✳

  } k∈N ❉✐③❡♠♦s q✉❡ ϕ k

  → ϕ s❡✿ )

  ✐✮ ∃ K ⊂ Ω, K ❝♦♠♣❛❝t♦✱ t❛❧ q✉❡ supp (ϕ k ⊂ K✱ ♣❛r❛ t♦❞♦ k ∈ N❀

  n α α

  , D ϕ (x) ϕ(x) ✐✐✮ P❛r❛ ❝❛❞❛ α ∈ N k ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x ∈ Ω✳

  → D

  

  (Ω) ❉❡✜♥✐çã♦ ✶✳✸ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ C ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡✜♥✐❞❛ ❛❝✐♠❛ é ❞❡♥♦t❛❞♦ ♣♦r D(Ω) ❡ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s✳ ❉❡✜♥✐çã♦ ✶✳✹ ❯♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ Ω é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❞❡✜✲ ♥✐❞♦ ❡♠ D(Ω) ❡ ❝♦♥tí♥✉♦ ❡♠ r❡❧❛çã♦ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡✜♥✐❞❛ ❡♠ D(Ω)✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❞✐str✐❜✉✐çõ❡s s♦❜r❡ Ω é ❞❡♥♦t❛❞♦ ♣♦r D

  (Ω) ✳

  ❉❡ss❡ ♠♦❞♦✱ D

  (Ω) = {T : D(Ω) → K; T é ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦}. (Ω)

  ❖❜s❡r✈❛♠♦s q✉❡ D é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ K✳ (Ω)

  ❙❡ T ∈ D ❡ ϕ ∈ D(Ω) ❞❡♥♦t❛r❡♠♦s ♣♦r hT, ϕi ♦ ✈❛❧♦r ❞❡ T ❛♣❧✐❝❛❞♦ ♥♦ ❡❧❡♠❡♥t♦ ϕ✳

  ′ (Ω) ◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ D k (Ω)

  ❉❡✜♥✐çã♦ ✶✳✺ ❉✐③❡♠♦s q✉❡ T → T ❡♠ D s❡ , ϕ

  k hT i → hT, ϕi, ∀ ϕ ∈ D(Ω). n

  (Ω) ❉❡✜♥✐çã♦ ✶✳✻ ❙❡❥❛♠ T ∈ D ❡ α ∈ N ✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ α

  α

  T ❞❡ T ✱ ❞❡♥♦t❛❞❛ ♣♦r D ✱ é ❞❡✜♥✐❞❛ ♣♦r

  α α |α|

  T, ϕ ϕ hD i = (−1) hT, D i, ♣❛r❛ t♦❞❛ ϕ ∈ D(Ω).

  k α α

  (Ω) T = T ❈♦♠ ❡st❛ ❞❡✜♥✐çã♦ t❡♠✲s❡ q✉❡ s❡ u ∈ C ❡♥tã♦ D u D u ✱

  α

  u ♣❛r❛ t♦❞♦ |α| ≤ k, ♦♥❞❡ D ✐♥❞✐❝❛ ❛ ❞❡r✐✈❛❞❛ ❝❧áss✐❝❛ ❞❡ u✳ ❊✱ s❡ ′ ′

  

α n

T (Ω) T (Ω) .

  ❡♥tã♦ D ♣❛r❛ t♦❞♦ α ∈ N ∈ D ∈ D p

  (Ω) ✶✳✸ ❊s♣❛ç♦s L

  ❉❡✜♥✐çã♦ ✶✳✼ ❙❡❥❛♠ Ω ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ♠❡♥s✉rá✈❡❧ ❡ 1 ≤ p ≤ ∞✳

  p

  (Ω) ■♥❞✐❝❛♠♦s ♣♦r µ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡ L ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s p

  < ♠❡♥s✉rá✈❡✐s f : Ω → K t❛✐s q✉❡ kfk L (Ω)

  ∞ ♦♥❞❡✿

  1/p

  Z p p = dµ ,

  L (Ω) s❡ 1 ≤ p < ∞

  kfk |f(x)|

  Ω

  ❡

  ∞

  kfk L (Ω) |f(x)|

  x∈Ω

  • = inf / µ

  {C ∈ R {x ∈ Ω / |f(x)| > C} = 0} = inf {C > 0 : |f(x)| ≤ C q✉❛s❡ s❡♠♣r❡ ❡♠ Ω}. p + p

  : L (Ω) , 1 ❖❜s❡r✈❛çã♦ ✶✳✽ ❆s ❢✉♥çõ❡s k · k L (Ω) → R ≤ p ≤ ∞✱ sã♦ ♥♦r♠❛s✳

  p

  (Ω) ◆❛ ✈❡r❞❛❞❡ L ❞❡✈❡ s❡r ❡♥t❡♥❞✐❞♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❧❛ss❡s

  ❞❡ ❢✉♥çõ❡s ♦♥❞❡ ❞✉❛s ❢✉♥çõ❡s ❡stã♦ ♥❛ ♠❡s♠❛ ❝❧❛ss❡ s❡ ❡❧❛s sã♦ ✐❣✉❛✐s q✉❛s❡ s❡♠♣r❡ ❡♠ Ω✳

  p

  2

  (Ω) (Ω)

  ❖s ❡s♣❛ç♦s L ✱ 1 ≤ p ≤ ∞✱ sã♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ s❡♥❞♦ L ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧ ❞❛ ✐♥t❡❣r❛❧✳ ❆❧é♠

  p

  (Ω) ❞✐ss♦✱ ♣❛r❛ 1 < p < ∞✱ L é r❡✢❡①✐✈♦✳

  p ∞

  (Ω) (Ω) ❚❡♦r❡♠❛ ✶✳✾ C é ❞❡♥s♦ ❡♠ L ✱ 1 ≤ p < +∞✳

  p

  (Ω) ❚❡♦r❡♠❛ ✶✳✶✵ ✭■♥t❡r♣♦❧❛çã♦ ❞♦s ❡s♣❛ç♦s L ✮ ❙❡❥❛♠

  p q r

  1 (Ω) (Ω) (Ω) ❡♥tã♦ f ∈ L ♣❛r❛ t♦❞♦

  ≤ p < q ≤ ∞✳ ❙❡ f ∈ L ∩ L r ∈ [p, q]✳ ❆❧é♠ ❞✐ss♦✱

  

α

r p q 1−α L (Ω)

  kfk ≤ kfk L (Ω) kfk

  L (Ω)

  1

  1

  1 = α + (1

  ❝♦♠ α ∈ [0, 1] t❛❧ q✉❡ − α) ✳ r p q p (Ω) ❊s♣❛ç♦s L loc n

  ❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛♠ Ω ✉♠ ❛❜❡rt♦ ❞♦ ❡s♣❛ç♦ R ❡ 1 ≤ p < ∞✳

  p

  (Ω) ■♥❞✐❝❛♠♦s ♣♦r L ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f : Ω → R

  loc p K (Ω) K

  t❛✐s q✉❡ fχ ✱ ♣❛r❛ t♦❞♦ K ❝♦♠♣❛❝t♦ ❞❡ Ω✱ ♦♥❞❡ χ é ❛ ∈ L

  ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ K✳

  1

  (Ω) ❖❜s❡r✈❛çã♦ ✶✳✶✷ L é ❝❤❛♠❛❞♦ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡

  loc

  ✐♥t❡❣rá✈❡✐s✳

  1

  (Ω) u : D(Ω) P❛r❛ u ∈ L ❝♦♥s✐❞❡r❡♠♦s ♦ ❢✉♥❝✐♦♥❛❧ T = T → K

  loc

  ❞❡✜♥✐❞♦ ♣♦r Z , ϕ u(x) ϕ(x) dx.

  u

  hT, ϕi = hT i =

  Ω

  ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ T ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ Ω✳

  1

  (Ω) u = 0 ▲❡♠❛ ✶✳✶✸ ✭❉✉ ❇♦✐s ❘❡②♠♦♥❞✮ ❙❡❥❛ u ∈ L ✳ ❊♥tã♦ T

  loc

  s❡ ❡ s♦♠❡♥t❡ s❡ u = 0 q✉❛s❡ s❡♠♣r❡ ❡♠ Ω✳ ❆ ❛♣❧✐❝❛çã♦

  1 L (Ω) (Ω) loc −→ D

  u T u 7−→

  é ❧✐♥❡❛r✱ ❝♦♥tí♥✉❛ ❡ ✐♥❥❡t✐✈❛ ✭❞❡✈✐❞♦ ❛♦ ▲❡♠❛ ✶✳✶✸✮✳ ❊♠ ❞❡❝♦rrê♥❝✐❛

  1

  (Ω) ❞✐ss♦ é ❝♦♠✉♠ ✐❞❡♥t✐✜❝❛r ❛ ❞✐str✐❜✉✐çã♦ T u ❝♦♠ ❛ ❢✉♥çã♦ u ∈ L ✳ loc 1 p

  1

  (Ω) (Ω) (Ω) (Ω) ◆❡ss❡ s❡♥t✐❞♦ t❡♠✲s❡ q✉❡ L ✳ ❈♦♠♦ L

  loc ⊂ D ⊂ L loc p

  (Ω) t❡♠♦s q✉❡ t♦❞❛ ❢✉♥çã♦ ❞❡ L ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ Ω✱ ✐st♦

  p

  (Ω) é✱ t♦❞❛ ❢✉♥çã♦ ❞❡ L ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠❛ ❞✐str✐❜✉✐çã♦✳

  

✶✳✹ ❊s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③ ❡ ❉✐str✐❜✉✐çõ❡s ❚❡♠✲

♣❡r❛❞❛s

  P❛r❛ ❡st❛ s❡çã♦ ❝✐t❛♠♦s ▼❡❞❡✐r♦s✲❘✐✈❡r❛ ❬✶✷❪✱ ❬✶✸❪✳

  n ∞

  (R ) ❉❡✜♥✐çã♦ ✶✳✶✹ ❯♠❛ ❢✉♥çã♦ ϕ ∈ C é ❞✐t❛ r❛♣✐❞❛♠❡♥t❡ ❞❡❝r❡s✲ ❝❡♥t❡ ♥♦ ✐♥✜♥✐t♦ q✉❛♥❞♦ ♣❛r❛ ❝❛❞❛ k ∈ N t❡♠✲s❡

  2 k α

  p (ϕ) = max sup (1 + ) ϕ(x)

  k n |x| |D | < ∞, |α|≤k x∈R n

  ♦♥❞❡ α ∈ N ✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r

  

α

  lim P (x)D ϕ(x) = 0,

  |x|→∞ n

  ♣❛r❛ t♦❞♦ ♣♦❧✐♥ô♠✐♦ P ❞❡ n ✈❛r✐á✈❡✐s r❡❛✐s ❡ α ∈ N ✳

  n

  ) ❈♦♥s✐❞❡r❡♠♦s S(R ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s r❛♣✐❞❛✲

  ♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡s ♥♦ ✐♥✜♥✐t♦✳ ❙♦❜r❡ ❡ss❡ ❡s♣❛ç♦✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ s❡♠✐♥♦r♠❛s✿

  2 k α

  p (ϕ) = max sup (1 + ) ϕ(x)

  k n |x| |D |.

  α≤k x∈R n ) ◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ S(R n n

  ) ) ❯♠❛ s❡q✉ê♥❝✐❛ {ϕ n ❝♦♥✈❡r❣❡ ♣❛r❛ ϕ ❡♠ S(R s❡

  } n∈N ⊂ S(R p k (ϕ n − ϕ) ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦ ❡♠ K✱ ♣❛r❛ t♦❞♦ k ∈ N✳

  n n

  ) ) Pr♦♣♦s✐çã♦ ✶✳✶✺ ❖ ❡s♣❛ç♦ D(R é ❞❡♥s♦ ❡♠ S(R ✳

  n p n

  ) (R ) Pr♦♣♦s✐çã♦ ✶✳✶✻ ❚❡♠✲s❡ q✉❡ S(R ⊂ L ❞❡♥s❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦

  1 ≤ p ≤ ∞✳

  n

  ) ❉❡✜♥✐çã♦ ✶✳✶✼ ❈♦♥s✐❞❡r❡ S(R ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡✜✲

  n

  ) ♥✐❞❛ ❛❝✐♠❛✳ ❙❡ T : S(R

  → K é ❧✐♥❡❛r ❡ ❝♦♥tí♥✉❛✱ ❞✐③✲s❡ q✉❡ T é ✉♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳

  ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s ❝♦♠ ❛ ❝♦♥✲

  n ′

  (R ) ✈❡r❣ê♥❝✐❛ ♣♦♥t✉❛❧ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r S ✳

  ✶✳✺ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r

  ❖s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❡st❛ s❡çã♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❆❞❛♠s ❬✶❪✱ ❉❛✉tr❛②✲▲✐♦♥s ❬✼❪ ❡ ❊✈❛♥s ❬✽❪✳

  1 n

  (R ) ❉❡✜♥✐çã♦ ✶✳✶✽ ❙❡❥❛ ϕ ∈ L ✱ ❞❡✜♥✐♠♦s s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉✲

  n

  r✐❡r ❝♦♠♦ s❡♥❞♦ ❛ ❢✉♥çã♦ Fϕ ❞❡✜♥✐❞❛ ♥♦ R ♣♦r Z

  1

  −i x·ξ (Fϕ)(x) = e ϕ(ξ) dξ. n/2 n

  (2π) R

  ❖❜s❡r✈❛çã♦ ✶✳✶✾ ❚❛♠❜é♠ ❞❡♥♦t❛r❡♠♦s ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ϕ

  ❞❡ ✉♠❛ ❢✉♥çã♦ ϕ ♣♦r b ✳

  1 n

  (R ) Pr♦♣♦s✐çã♦ ✶✳✷✵ P❛r❛ u ∈ L ✱ ❡①✐st❡ C > 0 t❛❧ q✉❡

  2

  2 1 u .

  ||b || L ≤ C||u||

  L

  ❖❜s❡r✈❛çã♦ ✶✳✷✶ ❆ ❛♣❧✐❝❛çã♦ eF ❞❛❞❛ ♣♦r (eFϕ)(x) = (Fϕ)(−x), ∀ x ∈

  n

  R ✱ é ❞❡♥♦♠✐♥❛❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✐♥✈❡rs❛ ❞❡ ϕ✳ ❆❧é♠ ❞✐ss♦✱

  F F ϕ = e ϕ

  ✱ ♦♥❞❡ ϕ ❞❡♥♦t❛ ♦ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ ❞❡ ϕ✳

  n 1 n

  ) (R ) ❈♦♠♦ S(R ⊂ L ✱ ❡♥tã♦ Fϕ ❡ eFϕ ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ♣❛r❛ ϕ ∈

  n

  S (R )

  ✳ ❆❧é♠ ❞✐ss♦✱ ❛♠❜❛s sã♦ r❛♣✐❞❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡s ❞♦ ✐♥✜♥✐t♦✳ Pr♦♣♦s✐çã♦ ✶✳✷✷ ❆s ❛♣❧✐❝❛çõ❡s

  n n n n

  F : S(R ) ) ) )

  ❡ eF : S(R → S(R → S(R

  −1

  F = e sã♦ ✐s♦♠♦r✜s♠♦s ❝♦♥tí♥✉♦s ❡ F ✳

  n

  ) Pr♦♣♦s✐çã♦ ✶✳✷✸ P❛r❛ t♦❞♦ ϕ, ψ ∈ S(R ✱ t❡♠♦s

  α α n |α|

  F ϕ) = i x ϕ,

  ✐✮ F(D ❀ ∀ α ∈ N

  α α n |α|

  (Fϕ) = F( x ϕ), ✐✐✮ D ✳

  −i ∀ α ∈ N

  ❉❡✜♥✐çã♦ ✶✳✷✹ ❙❡❥❛ T ✉♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳ ❉❡✜♥✐♠♦s s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

  n

  ), hFT, ϕi = hT, Fϕi, ∀ ϕ ∈ S(R

  n

  F F T, ϕ ϕ ). he i = hT, e i, ∀ ϕ ∈ S(R

  ❖❜s❡r✈❛çã♦ ✶✳✷✺ ❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❡♠

  n

  S (R )

  ✱ t❡♠♦s q✉❡ FT ❡ eFT sã♦ ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s✳ Pr♦♣♦s✐çã♦ ✶✳✷✻ ❆s ❛♣❧✐❝❛çõ❡s

  n n n n

′ ′ ′ ′

  F : S (R ) (R ) (R ) (R )

  ❡ eF : S → S → S

  −1

  F = e sã♦ ✐s♦♠♦r✜s♠♦s ❝♦♥tí♥✉♦s ❡ F ✳

  2 n

  (R ) k = ϕχ , k P❛r❛ ϕ ∈ L ❞❡✜♥✐♠♦s ϕ B k (0) ∈ N✱ ♦♥❞❡ χ B k (0) é

  n

  ; ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ❝♦♥❥✉♥t♦ {x ∈ R k é

  |x| ≤ k}✳ ❆ss✐♠✱ Fϕ ❞❛❞❛ ♣♦r

  Z

  1

  n −i x·ξ

  (Fϕ )(x) = e ϕ(ξ) dξ, .

  k

  ∀ x ∈ R

  n/2

  (2π)

  |ξ|≤k

2 n

k (R ) k

  ➱ ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡ Fϕ ❡ q✉❡ {Fϕ k∈N é ✉♠❛ s❡q✉ê♥✲ ∈ L }

  2 n

  (R ) ❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ L ✳ ❈♦♠♦ ❡st❡ ❡s♣❛ç♦ é ❞❡ ❍✐❧❜❡rt✱ ❡st❛ s❡q✉ê♥✲ ❝✐❛ t❡♠ ✉♠ ❧✐♠✐t❡✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r Fϕ✳ ❆✐♥❞❛ ♦❜s❡r✈❛✲s❡ q✉❡ Fϕ ❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ϕ ✭✈✐st❛ ❝♦♠♦ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✮

  2 n

  (R ) ❝♦✐♥❝✐❞❡♠ ♣❛r❛ ϕ ∈ L ✳ ❆ss✐♠ ✜❝❛ ❞❡✜♥✐❞❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡

  2 n

  (R ) ❋♦✉r✐❡r ♥♦ ❡s♣❛ç♦ L ✳

  2 n

  • 1

  (R ) , C ❚❡♦r❡♠❛ ✶✳✷✼ ❙❡ u, v ∈ L ❡♥tã♦ ❡①✐st❡♠ C

  2 t❛❧ q✉❡

  ∈ R

  α α α 2 n

  D u = (iy) u u (R ) ✐✮ d ♣❛r❛ ❝❛❞❛ ♠✉❧t✐✲í♥❞✐❝❡ α t❛❧ q✉❡ D ✳ b ∈ L

  (u u ✐✐✮ \

  1

  ∗ v) = C b bv✳ u v = C

  2 ( u

  ✐✐✐✮ c b ∗ bv)✳

  −1

  ( u) ✐✈✮ u = F ✳ b ❚❡♦r❡♠❛ ✶✳✷✽ ✭❚❡♦r❡♠❛ ❞❡ P❧❛♥❝❤❡r❡❧✮ ❆s ❛♣❧✐❝❛çõ❡s

  2 n 2 n 2 n 2 n

  F : L (R ) (R ) (R ) (R )

  ❡ eF : L → L → L sã♦ ✐s♦♠♦r✜s♠♦s ❞❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt t❛✐s q✉❡ 2 2 F F 2

  = = ϕ, e ψ

  L L L

  hFϕ, Fψi hϕ, ψi he i

  2 n

  (R ) ♣❛r❛ t♦❞♦ ♣❛r ϕ, ψ ∈ L ✳

  2 n

  (R ) ❈♦r♦❧ár✐♦ ✶✳✷✾ ❙❡ ϕ ∈ L ✱ ❡♥tã♦ ||ϕ|| = ||Fϕ||✳ Pr♦♣♦s✐çã♦ ✶✳✸✵ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍❛✉s❞♦r✛✲❨♦✉♥❣✮ ❙❡ f ∈

  1

  1 p n

  L (R ) = 1

  • p q

  ❝♦♠ 1 6 p 6 2 ❡ p, q ❝♦♥❥✉❣❛❞♦s✱ ✐st♦ é✱ ✱ t❡♠♦s

  q✉❡ f 6 .

  q

  k ˆ k kfk p

  ❊①❡♠♣❧♦s

  (ϕ)(x) ✶✳ F(∆ϕ)(x) = −|x| ✿

  α α |α|

  F ϕ) = i x ϕ

  ❉❛ Pr♦♣♦s✐çã♦ ✶✳✷✸✱ F(D ✱ ❧♦❣♦ ♣❛r❛ ❝❛❞❛ j = 1, 2, . . . , n t❡♠✲s❡

  !

  2

  ∂ ϕ

  2

  

2

  2 F F F (x) = i x (ϕ)(x) = (ϕ)(x). j j

2 −x

  ∂x

  j

  ❆ss✐♠✱ ♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r t❡♠♦s q✉❡  

  !

  n n

  

2

  X ∂ ϕ ∂ ϕ

  2 X

  F F F (∆ϕ)(x) =   (x) = (x)

  2

  2

  ∂x ∂x

  j j j=1 j=1 n

  X

  

2

  2 F F = ( (ϕ)(x)) = (ϕ)(x).

  −x j −|x|

  j=1

  2

  ϕ)(x) = (ϕ)(x) ✷✳ F(∆ ✿

  |x| ❯s❛♥❞♦ ♦ ❊①❡♠♣❧♦ ✶✱ t❡♠♦s

  2

  2 F F F

  (∆ ϕ)(x) = (∆(∆ϕ))(x) = (∆ϕ)(x) −|x|

  2

  2

  4 F F = (ϕ)(x)) = (ϕ)(x).

  −|x |(−|x| |x| ✸✳ F(∇ϕ)(x) = ixF(ϕ)(x)✿

  α α |α|

  F ϕ) = i x ϕ

  ❚❛♠❜é♠ ✉s❛♥❞♦ q✉❡ F(D ✱ t❡♠♦s ∂ϕ

  F F (x) = ix j (ϕ)(x), ∀ j = 1, 2, . . . , n.

  ∂x

  j

  ❈♦♠ ✐ss♦✱ F

  (Ω) é ❝❤❛♠❛❞♦ ❞❡ ❊s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ ♦r❞❡♠ m r❡❧❛t✐✈♦ ❛♦ ❡s♣❛ç♦ L

  (Ω) t❛✐s q✉❡ ♣❛r❛ t♦❞♦ |α| 6 m, D

  α

  u ♣❡rt❡♥❝❡ ❛ L

  p

  (Ω) ✱ s❡♥❞♦ D

  α

  u ❛ ❞❡r✐✈❛❞❛ ❞✐s✲ tr✐❜✉❝✐♦♥❛❧ ❞❡ u✳ W

  m,p

  p

  (Ω) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s u ❞❡ L

  (Ω) ✳

  ❘❡s✉♠✐❞❛♠❡♥t❡✱ W

  m,p

  (Ω) = {u ∈ L

  p

  (Ω) t❛❧ q✉❡ D

  α

  u ∈ L

  p

  p

  m,p

  ( ∇ϕ)(x) = F

  (x) ✳✳✳

       

  ∂ϕ ∂x 1

  ✳✳✳

  ∂ϕ ∂x n

       

  (x) =      

  F

  ∂ϕ ∂x 1

  F

  ❛❜❡rt♦✱ m ∈ N ❡ 1 ≤ p ≤ ∞✳ ■♥❞✐❝❛r❡✲ ♠♦s ♣♦r W

  ∂ϕ ∂x n

  (x)      

  =       ix

  (ϕ)(x) ✳✳✳ ix n

  F (ϕ)(x)

        = ixF(ϕ)(x).

  ✶✳✻ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

  ❖s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡st❛ s❡çã♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❆❞❛♠s ❬✶❪✱ ❇r❡③✐s ❬✷❪✱ ❑❡s❛✈❛♥ ❬✶✵❪ ❡ ▼❡❞❡✐r♦s✲❘✐✈❡r❛ ❬✶✷❪✱ ❬✶✸❪✳ ❉❡✜♥✐çã♦ ✶✳✸✶ ❙❡❥❛♠ Ω ⊆ R

  n

  (Ω) ♣❛r❛ t♦❞♦ |α| 6 m} . m,p (Ω) ◆♦r♠❛ ❡♠ W m,p

  (Ω) P❛r❛ ❝❛❞❛ u ∈ W t❡♠✲s❡ q✉❡

  1/p

   

  X

  p α

   

  m,p = u p

  kuk kD k L (Ω)

  |α|6m

    1/p Z

  X

  α p

  =  u)(x) dx  , |(D |

  Ω |α|6m

  ❝♦♠ p ∈ [1, ∞)✱ ❡

  X

  α

  = u , kuk m,∞ kD k L (Ω)

  

|α|6m

m,p

  (Ω) ❝♦♠ p = ∞✱ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ s♦❜r❡ W ✳

  ❖❜s❡r✈❛çõ❡s✿ m,p

  (Ω), ) ✶✳ (W m,p é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ k · k

  m,2

  (Ω) ✷✳ ◗✉❛♥❞♦ p = 2✱ ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ W t♦r♥❛✲s❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❛❞♦ ♣♦r

  X

  α α m,2 2 = u, D v , u, v (Ω). m,2 L (Ω)

  hu, vi hD i ∈ W

  |α|6m m,2 m

  (Ω) (Ω) ✸✳ ❉❡♥♦t❛✲s❡ W ♣♦r H ✳

  m

  (Ω) ✹✳ H é r❡✢❡①✐✈♦ ❡ s❡♣❛rá✈❡❧✳

  2 n

  (R ) ✺✳ ❆ ♥♦r♠❛ ✉s✉❛❧ ❡♠ H é ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ❞❛❞❛ ♣♦r 2

  2

  2 = .

  • m,p

  ||u|| H ||u|| ||∆u||

  (Ω) ❖ ❊s♣❛ç♦ W m,p n

  (Ω) ❉❡✜♥✐çã♦ ✶✳✸✷ ❙❡❥❛ Ω ⊆ R ✉♠ ❛❜❡rt♦✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ W

  m,p ∞

  (Ω) (Ω) ❝♦♠♦ s❡♥❞♦ ♦ ❢❡❝❤♦ ❞❡ C ❡♠ W ✳

  ❖❜s❡r✈❛çõ❡s✿ m,p m

  (Ω) (Ω) ✶✳ ◗✉❛♥❞♦ p = 2✱ ❡s❝r❡✈❡✲s❡ H ❡♠ ❧✉❣❛r ❞❡ W ✳

  m,p m,p n

  (Ω) = W (Ω), ✷✳ ❙❡ W ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ Ω ❡♠ R ♣♦ss✉✐ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ✐❣✉❛❧ ❛ ③❡r♦✳

  m,p n m,p n

  (R ) = W (R ) ✸✳ ❱❛❧❡ q✉❡ W ✳

  m,q (Ω) ❖ ❊s♣❛ç♦ W

  1

  1 = 1

  • p q

  ❉❡✜♥✐çã♦ ✶✳✸✸ ❙✉♣♦♥❤❛ 1 6 p < ∞ ❡ q > 1 t❛❧ q✉❡ ✳

  m,p −m,q

  (Ω) (Ω) ❘❡♣r❡s❡♥t❛✲s❡ ♣♦r W ♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞❡ W ✳

  m −m

  (Ω) (Ω) ❖ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞❡ H r❡♣r❡s❡♥t❛✲s❡ ♣♦r H ✳

  ■♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈

  ❚❡♦r❡♠❛ ✶✳✸✹ ✭❚❡♦r❡♠❛ ❞❡ ❙♦❜♦❧❡✈✮ ❙❡❥❛♠ m ≥ 1 ❡ 1 ≤ p < ∞✳ 1 m

  m,p q

  > 0 (Ω) (Ω) ✐✮ ❙❡ − ❡♥tã♦ W ⊂ L ✱ ♣❛r❛ q s❛t✐s❢❛③❡♥❞♦ p n

  1 1 m =

  − ❀ q p n 1 m

  m,p q

  = 0 (Ω) (Ω), ✐✐✮ ❙❡ ❡♥tã♦ W ♣❛r❛ q ∈ [p, ∞)❀

  − ⊂ L p n 1 m

  m,p ∞

  < 0 (Ω) (Ω); ✐✐✐✮ ❙❡ ❡♥tã♦ W

  − ⊂ L p n s❡♥❞♦ ❛s ✐♠❡rsõ❡s ❛❝✐♠❛ ❝♦♥tí♥✉❛s✳

  ✶✳✼ ❉❡s✐❣✉❛❧❞❛❞❡s ■♠♣♦rt❛♥t❡s ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣

  1

  1 = 1

  • p q

  ❙❡ a ≥ 0 ❡ b ≥ 0 ❡ 1 < p, q < ∞ ❝♦♠ ❡♥tã♦

  p q

  a b . + ab

  ≤ p q

  ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r

  1

  1

  p q

  (Ω) (Ω) = 1 ❙❡❥❛♠ f ∈ L ❡ g ∈ L ❝♦♠ 1 < p < ∞ ❡ ♦✉

  • p q

  1

  q = 1 (Ω) ❡ p = ∞ ♦✉ q = ∞ ❡ p = 1✳ ❊♥tã♦ fg ∈ L ❡

  Z p q .

  L (Ω) L (Ω)

  |f(x)g(x)| dx 6 kfk kgk

  Ω

  

✶✳✽ ❚❡♦r❡♠❛ ❞❛ ❉✐✈❡r❣ê♥❝✐❛ ❡ ❋ór♠✉❧❛s ❞❡

●r❡❡♥

  ❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s ♣❛r❛ ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ Ω ❝♦♠

  2

  ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡ C ✿

  1 n

  (Ω)) ✐✳ P❛r❛ F ∈ (H ✿

  Z Z (div F )(x) dx = F (x) · η(x) dΓ, Γ = ∂Ω.

  Ω Γ

  1

  2

  (Ω), u (Ω) ✐✐✳ P❛r❛ v ∈ H ✿

  ∈ H Z Z v(x)∆u(x) dx =

  − ∇v(x) · ∇u(x) dx.

  Ω Ω

  2

  2

  (Ω) (Ω) ✐✐✐✳ P❛r❛ u ∈ H ✱ v ∈ H ✿

  Z Z v(x)∆u(x) dx = ∆v(x)u(x) dx.

  Ω Ω

  ❆ ❢✉♥çã♦ η(x) ❞❡♥♦t❛ ❛ ♥♦r♠❛❧ ❡①t❡r✐♦r ✉♥✐tár✐❛ ♥♦ ♣♦♥t♦ x ∈ ∂Ω ❡ ❛ ❢✉♥çã♦ F ✐♥t❡❣r❛❞❛ s♦❜r❡ ∂Ω é ♥♦ s❡♥t✐❞♦ ❞❛ ❢✉♥çã♦ tr❛ç♦✳

  ❆ ❢ór♠✉❧❛ ❞❡ P❡❛♥♦✲❇❛❦❡r

  ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ♦r❞✐♥ár✐❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ d

  n

  u = A(t)u, u(0) = u ∈ C dt sã♦ r❡s♦❧✈✐❞♦s ❡♠ t❡r♠♦s ❞❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ E(t, s)✱ ✐st♦ é✱ ❛ s♦✲ . ❧✉çã♦ é ❞❛❞❛ ♣♦r u(t) = E(t, 0)u ❆ ♠❛tr✐③ ❢✉♥❞❛♠❡♥t❛❧ E(t, s✮ é ❛ s♦❧✉çã♦ ♣❛r❛ d

  n×n

  E(t, s) = A(t)E(t, s), E(s, s) = I 2 .

  ∈ C dt ❙❛❜❡✲s❡ q✉❡✱ ♣❛r❛ ✉♠❛ ♠❛tr✐③ ❝♦♥st❛♥t❡ A✱ ❡ss❛ s♦❧✉çã♦ ♣♦❞❡ s❡r

  ❡①♣r❡ss❛❞❛ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ♠❛tr✐③

  ∞

  X

  1

  k E(t, s) = exp((t A .

  − s)A), exp(A) = I + k!

  k=1

  P♦ré♠✱ ♣❛r❛ ❝♦❡✜❝✐❡♥t❡s ♥ã♦ ❝♦♥st❛♥t❡s ❡ss❛ r❡♣r❡s❡♥t❛çã♦ ♥ã♦ é ✈á❧✐❞❛✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♥♦s ❢♦r♥❡❝❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ P❡❛♥♦✲❇❛❦❡r ✭✈❡r ❬✶✹❪✮✳

  1 n×n

  (R, C ). ❚❡♦r❡♠❛ ✶✳✸✺ ❙❡❥❛ A ∈ L ❊♥tã♦ ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧

  loc

  ♣❛r❛ ♦ s✐st❡♠❛ ∂ t − A(t) é ❞❛❞❛ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ P❡❛♥♦✲❇❛❦❡r

  ∞ Z t Z t Z t 1 k− 1 X

  E(t, s) = I + A(t ) A(t ) A(t )dt . . . dt dt

  1 2 k k

  2

  1

  · · ·

  s s s k=1 ✶✳✾ ▲❡♠❛ ❞❡ ▼❛rt✐♥❡③

  ❊st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✶❪✳

  • → R →

  ▲❡♠❛ ✶✳✸✻ ❙❡❥❛ E : R ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❝r❡s❝❡♥t❡ ❡ φ : R

  1 +

  R ✉♠❛ ❢✉♥çã♦ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❞❡ ❝❧❛ss❡ C t❛❧ q✉❡ φ(0) = 0 ❡

  φ(t) → +∞ q✉❛♥❞♦ t → +∞✳ ❆ss✉♠❛ q✉❡ ❡①✐st❡♠ σ ≥ 0 ❡ ω > 0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ S > 0

  Z

  1

  1+σ σ ′

  E(t) φ (t) dt < E(0) E(S).

  ω

  S

  ❊♥tã♦ E t❡♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❞❡❝❛✐♠❡♥t♦✿

  1−ωφ(t)

  , ❙❡ σ = 0✱ ❡♥tã♦ E(t) ≤ E(0)e ∀ t ≥ 0. σ 1

  1+σ

  1+σωφ(t) ✶✳✶✵ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧ ✭❱❡rsã♦ ■♥t❡❣r❛❧✮

  , ❙❡ σ > 0✱ ❡♥tã♦ E(t) ≤ E(0) ∀ t ≥ 0.

  ❙❡ ♣❛r❛ t

  1 ✱ φ(t) ≥ 0 ❡ ψ(t) ≥ 0 sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s t❛✐s

  ≤ t ≤ t q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

  t

  Z φ(t) ψ(s)φ(s)ds

  ≤ K + L

  t

  1

  s❡ ♠❛♥t❡♥❤❛ ❡♠ t ❝♦♠ K, L ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s✱ ❡♥tã♦ ≤ t ≤ t

  Z t

  

L

  ψ(s)ds

  t φ(t) .

  ≤ Ke

  ▲✐st❛ ❞❡ ❉❡✜♥✐çõ❡s ❞♦ ❚r❛❜❛❧❤♦

  ❙❡❣✉❡ ✉♠❛ ❧✐st❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s q✉❡ ❛♣❛r❡❝❡r❛♠ ♥♦ ❞❡❝♦r✲ r❡r ❞♦ tr❛❜❛❧❤♦ ❛✜♠ ❞❡ ❢❛❝✐❧✐t❛r ❛ ❧❡✐t✉r❛✿

  

n c n

  := : := : ✶✳ B M

  {ξ ∈ R |ξ| ≤ M} ❡ B M {ξ ∈ R |ξ| ≥ M}

  α

  ✷✳ b(t) = µ(1 + t) ♦♥❞❡ α ∈ (0, 1), µ > 0 Z t Z t

  1

  ♯

  b(τ ) dτ (t, s) := dτ ✸✳ Λ(t, s) := ❡ Λ

  2b(τ )

  s s 2σ

2 4δ 2δ

  (t) ✹✳ m(t, ξ) = |ξ|

  − b(t) |ξ| − b |ξ| p ✺✳ h(t, ξ) =

  −m(t, ξ)

  2 4δ 2σ

  = ✻✳ m

  • 1

  −b(t) |ξ| |ξ| p (t, ξ) = (t, ξ)

  ✼✳ h

  1

  1

  −m ✽✳ P❛r❛ N ✉♠❛ ❝♦♥st❛♥t❡ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❞❡✜♥✐✲s❡

  1+α 2δ

  Z pd := M : (1 + t) {(t, ξ) ∈ [0, ∞) × B |ξ| ≤ N},

  low 1+α 2δ

  Z := : (1 + t)

  

M

ell {(t, ξ) ∈ [0, ∞) × B |ξ| ≥ N}

  h(s, ξ)

  2δ low low

  (t, s, ξ) := exp( Λ(t, s))E (t, s, ξ) ✾✳ a

  ell −|ξ| ell

  h(t, ξ)  

  ∂ h(t, ξ) −1

  t

    (t, ξ) =

  ✶✵✳ R

  1

    2h(t, ξ)

  −1  

  −1 ∂ t h(t, ξ)

  ♯

    (t, ξ) =

  ✶✶✳ R

  1

    2h(t, ξ)

  1

  1

  ♯

  (t, ξ) := R (t, ξ) ✶✷✳ N

  1

  1

  2h(t, ξ) N −1

  (t, ξ) = (∂ (t, ξ) + N (t, ξ)R (t, ξ)) N (t, ξ) ✶✸✳ R

  2 t

  1

1

  

1

  1 Z

2

t

  h(t, ξ) (t, s, ξ) := exp h(τ, ξ) dτ

  ✶✹✳ e ell h(s, ξ)

  s

   

  1  

  Z τ 2  

  , τ , ξ) = ✶✺✳ H(τ

  1

  2

   

  

−2 h(τ

3 , ξ)dτ

  3

   

  

τ

1

  e   1 0  

  ✶✻✳ H(∞) =   0 0

  , τ , ξ) = H(τ , τ , ξ)R (τ , ξ)H(τ , τ , ξ) ✶✼✳ A(τ

  1

  2

  2

  1

  2

  1

  1

  2 A(τ 1 , τ 2 , ξ) = H(τ 1 , τ 2 , ξ)R 2 (τ 2 , ξ)

  ✶✽✳ e A( , ξ) = H( (τ , ξ)

  ✶✾✳ e

  1

  2

  

1

  ∞, τ ∞)R Z

  ∞

  ∂ h(τ, ξ)

  t

  h(τ, ξ)

  1 (τ, ξ) dτ

  ✷✵✳ β(t, ξ) := exp − h − 2h(τ, ξ)

  t

  Z

  t σ−δ) 2δ |ξ|

  h

  1 (τ, ξ) b(τ ) + dτ

  ✷✶✳ γ(t, s, ξ) := exp − |ξ| 2b(τ )

  s

    1 −i 1  

  ✷✷✳ P = √  

  2 i

  1

  t

  Z (t, s, ξ) = H(t, s, ξ) + A(t, τ, ξ)Q e (τ, s, ξ) dτ

  ✷✸✳ Q ell ell

  s

  ( Q (t, s, ξ) ✷✹✳ Q ell ell

  ∞, s, ξ) := lim

  t→∞ −1

  Q ( ✷✺✳ k(s, ξ) := β(s, ξ)P ell

  ∞, s, ξ)N(s, ξ)P

  ❈❛♣ít✉❧♦ ✷ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛ ❇❛✐①❛ ❋r❡q✉ê♥❝✐❛

  ◆❡st❡ tr❛❜❛❧❤♦ ✈❛♠♦s ❡st✉❞❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❛s s♦✲ ❧✉çõ❡s ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❝♦♠ ♦♣❡r❛❞♦r❡s ❝♦♠ ♣♦tê♥❝✐❛s ❢r❛❝✐♦♥ár✐❛s ❡ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ ❞✐ss✐♣❛t✐✈♦ ❞❡♣❡♥❞❡♥❞♦ ❞♦ t❡♠♣♦✿

  σ δ n

  u (t, x) + ( u(t, x) + 2b(t)( u (t, x) = 0, t > 0, x

  tt t

  −∆) −∆) ∈ R u(0, x) = u (x) ✭✷✳✶✮ u (0, x) = u (x)

  t

  1 α

  ❝♦♠ σ > 0, δ ∈ (0, σ) b(t) = µ(1 + t) s❡♥❞♦ µ > 0 ✱ α ∈ (0, 1) ❡ 2δ < σ(1 + α).

  ❉❡♥♦t❛♠♦s ❛ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ♣♦r

  

n

{ξ ∈ R |ξ| < M}.

  ❆♣❧✐❝❛♥❞♦ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❡♠ ✭✷✳✶✮✱ ❝♦♠ r❡s♣❡✐t♦ ❛ ✈❛✲ r✐á✈❡❧ x✱ ♦❜t❡♠♦s✿

  

σ δ

  F (u + ( u + 2b(t)( u ) = F(0).

  tt t

  −∆) −∆) ◆♦t❡ q✉❡

  σ 2σ 2σ

−1

  F (( u) = F(F ( u)) = u

  −∆) |ξ| b |ξ| b ❡✱ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦

  δ 2δ 2δ

−1

  F (( u ) = F(F ( u )) = u .

  t t t

  −∆) |ξ| b |ξ| b ❆ss✐♠✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❋♦✉r✐❡r

  2σ 2δ n

  u (t, ξ) + u(t, ξ) + 2b(t) u (t, ξ) = 0, t > 0, ξ

  tt t

  b |ξ| b |ξ| b ∈ R u(0, ξ) = u (ξ) ✭✷✳✷✮ b b u (0, ξ) = u (ξ).

  t

  1

  b b

  P❛r❛ ❡♥❝♦♥tr❛r t❛①❛s ❡①♣❧í❝✐t❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ✭✷✳✷✮ ♥♦ ❡s♣❛ç♦ ❞❡ ❋♦✉r✐❡r✱

  n

  ❞✐✈✐❞✐♠♦s ♦ R ❡♠ ❞✉❛s r❡❣✐õ❡s✱ ❛❧t❛ ❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❡ ✉t✐❧✐③❛♠♦s ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s ♣❛r❛ ❝❛❞❛ r❡❣✐ã♦✳

  ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣❧✐❝❛♠♦s ♦ ♠ét♦❞♦ ❞❡ ❞✐❛❣♦♥❛❧✐③❛çã♦ ✉s❛❞♦ ♣♦r ❉✬❆❜❜✐❝❝♦✲❊❜❡rt ❬✻❪ ♣❛r❛ ♦❜t❡r ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✷✮ ♥❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛✳ ❚❛✐s ❡st✐♠❛t✐✈❛s s❡rã♦ ✉s❛❞❛s ♣❛r❛ ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ q✉❡ s❡rá ❡♥✉♥❝✐❛❞♦ ❡ ❞❡♠♦♥str❛❞♦ ♥♦ ❈❛✲ ♣ít✉❧♦ ✹✳

  ◆♦ q✉❡ s❡❣✉❡ ❞❡♥♦t❛♠♦s✿ Z t Z t

  1 . .

  ♯ Λ(t, s) = b(τ ) dτ, Λ (t, s) = dτ.

  ✭✷✳✸✮ 2b(τ )

  

s s

  ❉❡✜♥✐♥❞♦

  2δ

  w(t, ξ) = u(t, ξ) exp Λ(t, 0) b |ξ| t❡♠♦s q✉❡

  2δ

  u(t, ξ) = w(t, ξ) exp Λ(t, 0) ; b −|ξ|

  2δ

  u t (t, ξ) = w t (t, ξ) exp Λ(t, 0) b −|ξ|

  2δ 2δ ′

  • w(t, ξ) exp Λ(t, 0) ( Λ (t, 0))

  −|ξ| −|ξ|

  2δ

  = w (t, ξ) exp Λ(t, 0)

  t

  −|ξ|

  2δ 2δ

  • w(t, ξ) exp Λ(t, 0) ( b(t));

  −|ξ| −|ξ|

  

  u (t, ξ) = w (t, ξ) exp Λ(t, 0)

  tt tt

  b −|ξ|

  2δ 2δ

  • 2w t (t, ξ) exp Λ(t, 0) ( b(t))

  −|ξ| −|ξ|

  2δ 2δ 2δ

  • w(t, ξ) exp Λ(t, 0) ( b(t))( b(t))

  −|ξ| −|ξ| −|ξ|

  2δ 2δ ′

  • w(t, ξ) exp Λ(t, 0) ( b (t))

  −|ξ| −|ξ| ❡ ❛✐♥❞❛✱ u(0, ξ) = u (ξ) = w(0, ξ) b b

  2δ

  u (0, ξ) = w (0, ξ) + u (ξ)( b(0))

  t t

  b b −|ξ|

  2δ

  w t (0, ξ) = u 1 (ξ) + µ u (ξ). ⇒ b |ξ| b

  ❯s❛♥❞♦ ❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ r❡❡s❝r❡✈❡♠♦s ✭✷✳✷✮ ❞❛ ❢♦r♠❛

  2δ 4δ 2 2δ 2σ ′

  exp Λ(t, 0) w tt (t, ξ) + b(t) b (t) + w(t, ξ) = 0.

  −|ξ| −|ξ| − |ξ| |ξ| ❆ss✐♠✱

    w (t, ξ) + m(t, ξ) w(t, ξ) = 0  tt    w(0, ξ) = u (ξ) ✭✷✳✹✮ b

     

  

   w (0, ξ) = µ u (ξ) + u (ξ)

  t

  1

  |ξ| b b ♦♥❞❡ .

  2σ 2 4δ 2δ ′

  m(t, ξ) = (t) .

  |ξ| − b(t) |ξ| − b |ξ|

  • b
  • b

  |ξ|

  2σ

  ⇔ |ξ|

  2δ

  (t) |ξ|

  ′

  4δ

  2

  6 −c b(t)

  2δ

  (t) |ξ|

  ′

  • b

  − b

  4δ

  − (1 − c) b(t)

  |ξ|

  2

  − (1 − c)b(t)

  (t) |ξ|

  ′

  − (1 − c)b

  4δ

  |ξ|

  2

  2σ

  4δ

  = |ξ|

  6 0. P❛r❛ ✉♠ ❞❛❞♦ c ∈ (0, 1)✱ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ e m(t, ξ) .

  2δ

  (t) |ξ|

  ′

  − (1 − c) b

  |ξ|

  − b(t)

  2

  n |ξ| > k}.

  = R

  c

  = B

  = ∅ ❡ B ∞

  c ∞

  ❋♦r♠❛❧♠❡♥t❡✱ ❞❡✜♥❡✲s❡✿ B = B

  {ξ ∈ R

  ▲❡♠❛ ✷✳✶ ❙❡ α ∈ (0, 1)✱ δ ∈ (0, σ) ❡ 2δ < σ(1 + α)✳ ❊♥tã♦✱ ♣❛r❛ q✉❛❧q✉❡r c ∈ (0, 1)✱ ❡①✐st❡ M > 0✱ t❛❧ q✉❡ m(t, ξ) 6

  .

  c k

  |ξ| < k} , B

  n

  {ξ ∈ R

  P❛r❛ t♦❞♦ k > 0 ❞❡♥♦t❛♠♦s✿ .

  n .

  −c b(t)

  2σ

  2

  ⇔ |ξ|

  2δ

  (t) |ξ|

  ′

  4δ

  |ξ|

  ❖❜s❡r✈❡ q✉❡ m(t, ξ) 6 −c b(t)

  

2

  ♣❛r❛ q✉❛❧q✉❡r ξ ∈ B M ❡ t > 0✳ ❉❡♠♦♥str❛çã♦✿

  2δ

  (t) |ξ|

  ′

  4δ

  |ξ|

  2δ . m(t, ξ) 6 0 P❛r❛ ♣r♦✈❛r ♦ ❧❡♠❛ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ e ✱ ♣❛r❛ t♦❞♦

  ξ M ∈ B ❡ t > 0. m(t, ξ)

  ❉❡r✐✈❛♥❞♦ e ❝♦♠ r❡❧❛çã♦ ❛ t✱ ♦❜t❡♠♦s

  4δ 2δ ′ ′′

  m t (t, ξ) = 2b(t)b (t) b (t) e −(1 − c)|ξ| − (1 − c)|ξ| s❡♥❞♦

  

2

′ 2α−1

  b(t)b (t) = µ α(1 + t)

  ′′ α−2 b (t) = µα(α .

  − 1)(1 + t) ❈♦♠ ✐ss♦ m t (t, ξ) e

  4δ

2 2δ

2α−1 α−2

  = 2µ α(1 + t) µα(α −(1 − c) |ξ| − (1 − c) |ξ| − 1)(1 + t)

  2δ 2δ α α−1 −1

  = µα(1 + t) 2µ (1 + t) + (α .

  −(1 − c) |ξ| |ξ| − 1)(1 + t) m (t, ξ) = 0 ❋❛③❡♥❞♦ e t ✱ t❡♠♦s✿

  2δ α −1

  2µ (1 + t) + (α = 0 |ξ| − 1)(1 + t)

  −1

  (α − 1)(1 + t)

  α

  = ⇒ (1 + t) −

  2δ

  2µ |ξ|

  1

  α+1 − α −2δ

  = ⇒ (1 + t) |ξ|

  2µ α +1 1

  1 − α α +1 .

  ⇒ (1 + t) = |ξ| 2µ m(t, ξ) ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ ♦ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❛ ❢✉♥çã♦ e é α +1 1

  1 − α α +1

  −1 + |ξ| 2µ m (t, ξ) > 0 ) m (t, ξ) < 0

  ❆❧é♠ ❞✐ss♦✱ e t ✱ ♣❛r❛ t♦❞♦ t ∈ [0, t ❡ e t ♣❛r❛ t♦❞♦ t > t m(t , ξ) m(t, ξ) ✳ ▲♦❣♦✱ e é ♦ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❞❛ ❢✉♥çã♦ e ✱ ♣❛r❛ t > 0

  ❡ ξ ✜①♦✳ ◆♦t❡ q✉❡✿ 1 2α α +1 !

  1

  2 2 2α 2 − α − α +1

  b(t ) = µ (1 + t ) = µ |ξ| α +1

  1 − α

  2 4δ

  2 α +1

  ) = µ ⇒ b(t |ξ| |ξ|

  2µ e 1 α +1 ! α−1

  1 − α

  −

′ α−1 α +1

  b (t ) = µα(1 + t ) = µα |ξ| α α− +1 1

  1

  2δ − α ′ α +1 (t ) = µα .

  ⇒ b |ξ| |ξ| 2µ

  ❉❡ss❛ ❢♦r♠❛✱ max m(t, ξ) = m(t , ξ) e e

  t>0 α− 1

  " 2α # α +1 α +1 4δ 4δ

  1

  1 − α − α

  2σ

  2 α +1 α +1

  = µ + µα |ξ| − (1 − c) |ξ| |ξ|

  2µ 2µ 2α α− 1 " !# α α +1 +1

  1

  1

  2σ α +1 2 − α − α

  = µ + µα |ξ| − (1 − c) |ξ| 2µ 2µ

  2σ α +1

  =

  α,µ,c

  |ξ| − C |ξ| e m(t, ξ) = |ξ|

  2σ

  • |ξ|

  e m(t, ξ) < 0 ✱

  t>0

  P♦rt❛♥t♦✱ ♣❛r❛ |ξ| 6 M✱ ❝♦♥❝❧✉í♠♦s q✉❡ e m(t, ξ) 6 max

  2σ− α +1 < 0.

  α,µ,c

  ❙❡❥❛ M > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡ −C

  2σ− α +1 .

  α,µ,c

  |ξ| α +1 −C

  |ξ| α +1 =

  α,µ,c

  − C

  t>0

  > 0. ❆ss✐♠ max

  ⇒ 2σ − 4δ 1 + α

  4δ 1 + α − 2σ < 0

  2δ < σ(1 + α) ⇒ 4δ < 2σ(1 + α) ⇒

  → ∞ q✉❛♥❞♦ µ → ∞✳ P♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡

  α,µ,c

  → 0 q✉❛♥❞♦ µ → 0 ❡ C

  ♣♦✐s α < 1 ❡ c < 1✳ ❆❧é♠ ❞✐ss♦✱ C α,µ,c é ✉♠❛ ❝♦♥st❛♥t❡ t❛❧ q✉❡ C α,µ,c

  2µ α− 1 α +1 !

  1 − α

  1 − α 2µ α +1

  2

  ❝♦♠ − c)

  • M

  ♣❛r❛ t♦❞♦ |ξ| 6 M ❡ t > 0✱ ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ❞❡♠♦♥str❛çã♦✳

  ❖❜s❡r✈❛çã♦ ✷✳✷ ❉♦ ❧❡♠❛✱ s❡❣✉❡ q✉❡✿

  2 4δ 2δ ′

  ≈ − |ξ| |ξ| ♣❛r❛ q✉❛❧q✉❡r ξ ∈ B M ✱ t > 0 ❡ M > 0 ❞❛❞♦ ♣❡❧♦ ▲❡♠❛ ✷✳✶✳

  ■♥tr♦❞✉③✲s❡ ♥❛ r❡❣✐ã♦ [0, ∞) × B M ✱ ❛ ❝✉r✈❛ ❞❛❞❛ ♣♦r✿ . 1+α 2δ

  Γ (t, ξ) = (t, ξ) / (1 + t) = N

  N M

  ∈ [0, ∞) × B |ξ| ♦♥❞❡ N > 1 é ✉♠❛ ❝♦♥st❛♥t❡ ❣r❛♥❞❡ ❛❞❡q✉❛❞❛ ❡ ❛ss✉♠✐♠♦s✱ s❡♠ ♣❡r❞❛ 1

  ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ M 6 N ✳ ◆♦t❛çã♦✿ P❛r❛ t♦❞♦ ξ ∈ B M ❞❡✜♥✐♠♦s t ξ ❝♦♠♦ ❛ ú♥✐❝❛ s♦❧✉çã♦ ♣❛r❛

  1+α 2δ

  (1 + t ) = N

  ξ ✳

  |ξ| ❖❜s❡r✈❛çã♦ ✷✳✸ ◆♦t❡ q✉❡✱ ♣❛r❛ t♦❞♦ ξ ∈ B M ✱ m(t, ξ) s❡ ❝♦♠♣♦rt❛ ❞❡

  ξ ] ξ , +

  ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ♣❛r❛ t ∈ [0, t ❡ t ∈ [t ∞)✿ ]

  ✶✳ P❛r❛ t ∈ [0, t ξ s❡❣✉❡ q✉❡✿

  2δ 2δ ′ α−1

  m(t, ξ) (t) .

  ≈ −b |ξ| ≈ −(1 + t) |ξ| ✷✳ P❛r❛ t > t ξ s❡❣✉❡ q✉❡✿

  2 4δ 2α 4δ m(t, ξ) .

  ≈ −b(t) |ξ| ≈ −(1 + t) |ξ|

  ❉❡♠♦♥str❛çã♦ ❞❡ ✶✿ ❉❡ ❢❛t♦✱ ♣❡❧♦ ▲❡♠❛ ✷✳✶✱ t❡♠♦s

  2 4δ 2δ 2δ ′ ′

  m(t, ξ) 6 b(t) + b (t) (t) .

  . −c |ξ| |ξ| −b |ξ|

  ] P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ t ∈ [0, t ξ s❡❣✉❡ q✉❡

  α+1 2δ α+1 2δ

  (1 + t) ξ ) = N |ξ| ≤ (1 + t |ξ|

  α+1 2δ ⇒ µ(1 + t) |ξ| ≤ µN. 2 2α 4δ

  µ (1 + t) |ξ|

  ⇒ ≤ µN

  2δ

α−1

  µ(1 + t) |ξ|

  2 2α 4δ 2 2δ α−1

  (1 + t) N (1 + t) ⇒ µ |ξ| ≤ µ |ξ|

  µN

  2 4δ 2δ ′

  6 b (t) . ⇒ b(t) |ξ| |ξ|

  α ❆ss✐♠✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛

  

2 4δ 2δ ′

  m(t, ξ) = (t) |ξ| − b(t) |ξ| − b |ξ|

  2 4δ 2δ ′

  > (t) −b(t) |ξ| − b |ξ|

  µN

  2δ 2δ ′ ′

  b (t) (t) >

  − |ξ| − b |ξ| α

  2δ ′

  = b (t) ,

  

1

  −c |ξ|

  µN

  = + 1 ❝♦♠ c

  1 ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✳ α 2δ ′

  (t) P♦rt❛♥t♦✱ m(t, ξ) ≈ −b ✳

  |ξ|

  ❉❡♠♦♥str❛çã♦ ❞❡ ✷✿ ❉❡ ❢❛t♦✱ ♣❡❧♦ ▲❡♠❛ ✷✳✶✱ t❡♠♦s −c

  (1 + t)

  > −µ

  2δ

  |ξ|

  α−1

  − µα(1 + t)

  4δ

  |ξ|

  2α

  2

  (1 + t)

  = −µ

  2δ

  (t) |ξ|

  ′

  − b

  4δ

  |ξ|

  2

  > −b(t)

  2

  2α

  (t) |ξ|

  |ξ|

  4δ

  |ξ|

  2

  ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✳ P♦rt❛♥t♦✱ m(t, ξ) ≈ −b(t)

  α N µ

  = 1 +

  2

  , ❝♦♠ c

  4δ

  

2

  |ξ|

  b(t)

  2

  = −c

  4δ

  |ξ|

  2α

  N (1 + t)

  − µα

  4δ

  2δ

  ′

  

2

  |ξ|

  2

  µ

  = µN ⇒

  2δ

  |ξ|

  α+1

  > µ(1 + t ξ )

  2δ

  α+1

  2α

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ t > t ξ ✱ µ(1 + t)

  4δ

  |ξ|

  2

  −b(t)

  2δ

  |ξ|

  4δ ′

  |ξ|

  (1 + t)

  |ξ|

  − b

  > µ(1 + t)

  4δ

  |ξ|

  2

  − b(t)

  2σ

  ❆ss✐♠✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ m(t, ξ) = |ξ|

  2δ .

  |ξ|

  α−1

  

  4δ

  |ξ|

  2α

  (1 + t)

  µ N

  > µN ⇒

  2δ

  |ξ|

  α−1

  µ(1 + t)

  ✳

  ❉❡✈✐❞♦ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ m(t, ξ)✱ ✉s❛♠♦s ❛ ❝✉r✈❛ ❞❛❞❛ ❛❝✐♠❛ Γ

  . = (t, ξ)

  2δ

  |ξ|

  α+1

  / (1 + t)

  

M

  ∈ [0, ∞) × B

  low ell

  N (t, ξ)

  ≤ N Z

  2δ

  |ξ|

  M / (1 + t) α+1

  = (t, ξ) ∈ [0, ∞) × B

  Z pd .

  ♣❛r❛ ❞✐✈✐❞✐r ❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❡♠ ❞✉❛s ③♦♥❛s✿ ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧ ❡ ❡❧í♣t✐❝❛✱ ❞❛❞❛s ♣♦r✿

  ≥ N .

P❡❧♦s ✐t❡♥s ✐✮ ❡ ✐✐✮✿ m(t, ξ)

  −(1 + t)

  = p −m(t, ξ).

  ✭✷✳✻✮ ❈♦♠♦ w é s♦❧✉çã♦ ❞❡ ✭✷✳✹✮ t❡♠✲s❡ q✉❡ W é s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡

  .

  low ell

  s❡ (t, ξ) ∈ Z

  

  |ξ|

  α

  (1 + t)

  pd ,

  s❡ (t, ξ) ∈ Z

  δ

  (1 + t) α− 1 2 |ξ|

          

  ❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s q✉❡ h(t, ξ) ≈

  (t, ξ)), h(t, ξ) .

  α−1

  

t

  ❞❡✜♥✐♠♦s W (t, ξ) = (ih(t, ξ)w(t, ξ) , w

  ✭✷✳✺✮ ❙❡❥❛♠ c ∈ (0, 1) ❡ M > 0 ❞❛❞♦s ♣❡❧♦ ▲❡♠❛ ✷✳✶✳ P❛r❛ t♦❞♦ ξ ∈ B M

  .

  low ell

  s❡ (t, ξ) ∈ Z

  4δ

  |ξ|

  ≈         

  −(1 + t)

  pd ,

  s❡ (t, ξ) ∈ Z

  2δ

  |ξ|

  2α s✐st❡♠❛✿   

  ∂ W (t, ξ) = A(t, ξ)W (t, ξ) t > 0

  t

  ✭✷✳✼✮  

  W (0, ξ) = W (ξ) s❡♥❞♦ W (ξ) = (ih(0, ξ)w(0, ξ), w (0, ξ))

  t

  ❡  

  ∂ h(t, ξ)

  t

  ih(t, ξ)   h(t, ξ)

  A(t, ξ) =   .

  −ih(t, ξ) P❛r❛ t♦❞♦ t > s > 0✱ ❞❡♥♦t❛✲s❡ ♣♦r E(t, s, ξ) ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧

  ❞♦ s✐st❡♠❛ ✭✷✳✼✮✱ q✉❡ é ❛ ♠❛tr✐③ s♦❧✉çã♦ ♣❛r❛   

  ∂ E(t, s, ξ) = A(t, ξ)E(t, s, ξ) t > s

  t

  ✭✷✳✽✮   E(s, s, ξ) = I.

  ❉❡ss❛ ❢♦r♠❛✱ s❡ E(t, 0, ξ) é ❛ ♠❛tr✐③ s♦❧✉çã♦ ❞❡ ✭✷✳✽✮ ♣❛r❛ s = 0 (ξ)

  ❡♥tã♦ W (t, ξ) = E(t, 0, ξ)W é ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✼✮✳ ❉❡♥♦t❛♠♦s ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ♣♦r✿

    

  E (t, s, ξ) ,  pd s❡ (s, ξ), (t, ξ) ∈ Z pd 

  E(t, s, ξ) =  

  low low

    E (t, s, ξ) . s❡ (s, ξ), (t, ξ) ∈ Z

  ell ell

  ✷✳✶ ❩♦♥❛ Ps❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧

  h(t, ξ) h(t, ξ) ih(t, ξ)

  (t) E

  01

  (t) E

  00

  E

    

  −ih(t, ξ)   

  t

  (t) E

  ∂

    

  (t)    =

  11

  ′

  (t) E

  10

  10

  11

  (t) E

  h(t, ξ) h(t, ξ) E

  01

  (t) −ih(t, ξ)E

  00

  (t) −ih(t, ξ)E

  11

  (t) + ih(t, ξ)E

  01

  t

  (t)   

  (t) ∂

  10

  (t) + ih(t, ξ)E

  00

  h(t, ξ) h(t, ξ) E

  t

  ∂

  =   

  ′

  01

  ❯s❛✲s❡ ❞✐❢❡r❡♥t❡s ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ E

  00

  (t, 0, ξ) | .

  10

  |E

  (t, 0, ξ) | . h(t, ξ)(1 + t),

  01

  , |E

  (t, 0, ξ) | . h(t, ξ) h(0, ξ)

  ✱ s❡❣✉❡ q✉❡✿ |E

  

α

  ξ ]

  ❡ t ∈ [0, t

  

M

  ✳ Pr♦♣♦s✐çã♦ ✷✳✹ P❛r❛ t♦❞♦ ξ ∈ B

  ✳ ❖ ♣ró①✐♠♦ r❡s✉❧✲ t❛❞♦ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ E pd (t, 0, ξ)

  low ell

  ❡ E

  pd

  1 h(0, ξ) (1 + t)

  |ξ|

  ′

  ∂

  (t) E

  00

  ′

  E

  ❆ss✐♠✱   

  E(t, 0, ξ) = A(t, ξ)E(t, 0, ξ) t > 0 E(0, 0, ξ) = I.

  t

  ❚❡♠♦s✱ ♣♦r ✭✷✳✽✮✱ q✉❡✿     

  2δ

  ✳ ❉❡♠♦♥str❛çã♦✿

  pd (t, 0, ξ)

  ❞❡♥♦t❛ ❛s ❡♥tr❛❞❛s ❞❛ ♠❛tr✐③ E

  ij ) i,j=0,1

  ♦♥❞❡ (E

  (t, 0, ξ) | . 1,

  11

  , |E

  (t)   

  ❡   

  E

  00

  (0) E

  01

  (0) E

  10

  (0) E

  11

  (0)    =

     1 0

  0 1    .

P❡❧❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛✱ ♦❜t❡♠♦s

  E

  E

  (t) E

  10 (0) = 0

  ✭✷✳✶✶✮     

  E

  ′

  11

  (t) = −ih(t, ξ)E

  01

  (t) E 11 (0) = 1.

  ✭✷✳✶✷✮ ■♥t❡❣r❛♥❞♦ ✭✷✳✶✶✮ ❡ ✭✷✳✶✷✮✱ t❡♠♦s✿

  10 (t)

  (t) = −ih(t, ξ)E

  − E

  10 (0) =

  −i Z t h(τ, ξ)E

  00 (τ ) dτ

  ⇒ E

  10

  (t) = −i

  Z t h(τ, ξ)E

  00

  (τ ) dτ ✭✷✳✶✸✮

  00

  10

  ′

  ′

  00

  (t) = ∂

  t

  h(t, ξ) h(t, ξ) E

  00 (t) + ih(t, ξ)E 10 (t)

  E

  00

  (0) = 1 ✭✷✳✾✮

      

  E

  01

  ′

  (t) = ∂ t h(t, ξ) h(t, ξ)

  E

  

01

  (t) + ih(t, ξ)E

  11

      

  01

  (0) = 0 ✭✷✳✶✵✮

      

  E

  (t) E h(0, ξ) h(t, ξ) = ih(0, ξ)E 11 (t).

  01 (t)

  Z t E

  = ih(0, ξ)E

  10 (t).

  ■♥t❡❣r❛♥❞♦ ❡♠ (0, t)✱ E

  00

  (t) h(0, ξ) h(t, ξ)

  − E

  00

  (0) = i Z t h(0, ξ)E

  10

  (τ ) dτ ⇒ E

  00

  (t) = h(t, ξ) h(0, ξ)

  10 (τ ) dτ.

  00

  ✭✷✳✶✺✮ Pr♦❝❡❞❡♥❞♦ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❡♠ ✭✷✳✶✵✮✱ t❡♠♦s✿

  E

  ′

  01

  (t) h(0, ξ) h(t, ξ)

  − ∂

  t

  h(t, ξ)h(0, ξ) h(t, ξ)

  01

  (t) = ih(0, ξ)E

  11

  (t) ⇒ d dt

  E

  (t) h(0, ξ) h(t, ξ)

  E

  ❡ − E −i

  = h(0, ξ) h(t, ξ)

  Z t ⇒ E

  11

  (t) =

  1 − i

  Z t h(τ, ξ)E

  01 (τ ) dτ.

  ✭✷✳✶✹✮ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✾✮ ♣♦r

  h(0,ξ) h(t,ξ)

  t❡♠✲s❡ E

  ′

  00

  (t) h(0, ξ) h(t, ξ)

  ∂

  ⇒ d dt

  

t

  h(t, ξ) h(t, ξ) E

  00

  (t) + ih(0, ξ)E

  10

  (t) ⇒ E

  ′

  00

  (t) h(0, ξ) h(t, ξ)

  − ∂

  t

  h(t, ξ)h(0, ξ) h(t, ξ)

  2 E 00 (t) = ih(0, ξ)E 10 (t)

  • ih(t, ξ)
  • ih(τ, ξ)

  • ih(τ, ξ)

  t

  10

  (s) ds dτ = i

  Z t m(τ, ξ) h(0, ξ) dτ

  − Z t m(τ, ξ)

  Z τ E

  10 (s) ds dτ.

  ❙❡❥❛ dv = m(τ, ξ) dτ ❡ u = Z τ

  E

  10

  (s) ds ✳ ■♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s✱

  E

  10

  (t) = i Z

  m(τ, ξ) h(0, ξ) dτ

  2 Z τ

  −  

  Z τ E

  10

  (s) ds Z τ m(s, ξ) ds

  t

  − Z t Z τ m(s, ξ) ds E

  10

  (τ ) dτ = i

  Z t m(τ, ξ) h(0, ξ) dτ

  − Z t

  E

  10

  (s) ds Z t m(s, ξ) ds

  10

  E

  h(0, ξ) dτ + Z t h(τ, ξ)

  (τ ) dτ

  −i Z

  ■♥t❡❣r❛♥❞♦ ❡♠ (0, t)✱ h(0, ξ) h(t, ξ)

  − E Z t

  ⇒ E

  01

  (t) = ih(t, ξ) Z

  

t

  E

  11 (τ ) dτ.

  ✭✷✳✶✻✮ ❊st✐♠❛t✐✈❛ ✶✿ E

  10

  (t) ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✶✺✮ ❡♠ ✭✷✳✶✸✮ ❡ ✉s❛♥❞♦ q✉❡ h(t, ξ) = p

  −m(t, ξ) ♦❜t❡♠♦s

  E

  10 (t) =

  t

  2

  h(τ, ξ) h(τ, ξ) h(0, ξ)

  Z τ E

  10 (s) ds dτ

  = −i

  Z

  t

  h(τ, ξ)

  2

  h(0, ξ)

  2 Z τ

  E

  10

  (s) ds dτ =

  −i Z t h(τ, ξ)

  • Z t Z τ m(s, ξ) ds E

  Z Z t Z t t m(τ, ξ)

  = i dτ m(s, ξ) ds E

  10 (τ ) dτ

  − h(0, ξ) Z

  Z t τ

  • m(s, ξ) ds E (τ ) dτ

  

10

t t Z t τ

  Z Z Z m(τ, ξ) = i dτ m(s, ξ) ds m(s, ξ) ds E (τ ) dτ

  10

  − − h(0, ξ) Z t Z t Z t m(τ, ξ)

  = i dτ m(s, ξ) ds E (τ ) dτ.

  10 ✭✷✳✶✼✮

  − h(0, ξ)

  

τ

2δ α−1 pd

  Pr♦✈❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ m(t, ξ) ≈ −(1 + t) |ξ| ❡♠ Z ✳ ❆ss✐♠✱ ✉s❛♥❞♦ ❡st❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♠ ✭✷✳✶✼✮✱ ♦❜t❡♠♦s✿

  Z Z t Z t t m(τ, ξ)

  10 (t) dτ m(s, ξ) ds E 10 (τ ) dτ

  |E | = − i h(0, ξ)

  τ

  Z t

  1

  2δ α−1

  (1 + τ ) dτ .

  |ξ| h(0, ξ) Z t Z t

  2δ α−1

  (1 + s) ds + (τ )

  10 |ξ| |E | dτ. τ

  ■♥t❡❣r❛♥❞♦✱

  t t

  Z t

  α α

  1 (1 + τ ) (1 + s)

  2δ 2δ

  • (t)

  (τ )

  10

  10

  |E | . |ξ| |ξ| |E | dτ h(0, ξ) α α

  τ α

  1 (1 + t) − 1

  2δ .

  |ξ| h(0, ξ) α Z t

  α α

  (1 + t)

  2δ − (1 + τ)

  • (τ )

  10

  |ξ| |E | dτ α

  Z t

  1

  α 2δ α 2δ

  • . (1 + t) (1 + t) (τ )

  10 |ξ| |ξ| |E | dτ.

  h(0, ξ)

  ♯ −α

  (t) = (1 + t) (t) ❈♦♥s✐❞❡r❡ E

  

10

10 |E |✳ ❆ss✐♠✱

  Z t

  1

  ♯ ♯ 2δ 2δ α

  |ξ| |ξ|

  10

  10

  h(0, ξ) P❡❧♦ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧✱ s❡❣✉❡ q✉❡✿

  Z t

  1

  ♯ 2δ 2δ α

  E (t) . exp (1 + τ ) dτ .

  10 |ξ| |ξ|

  h(0, ξ)

  ♯

  P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ E t❡♠♦s q✉❡

  10 α

  (1 + t)

  2δ 2δ α

  (t) exp (1 + t) t

  10

  |E | . |ξ| |ξ| h(0, ξ)

  α

  (1 + t)

  2δ 2δ α+1 . exp (1 + t) .

  |ξ| |ξ| h(0, ξ) ❈♦♠♦ t 6 t ξ ✱ t❡♠♦s✿

  α+1 2δ α+1 2δ

  (1 + t) 6 (1 + t ) = N.

  ξ

  |ξ| |ξ| ❈♦♠ ✐ss♦✱

  α

  (1 + t)

  2δ

  (t) exp(N )

  10

  |E | . |ξ| h(0, ξ)

  α

  (1 + t)

  2δ (t) . 10 ✭✷✳✶✽✮

  ⇒ |E | . |ξ| h(0, ξ)

  (τ ) | dτ

  11 (τ ) dτ

  11 (τ ) dτ.

  m(s, ξ) ds E

  t Z t τ

  Z

  1 −

  =

  Z τ m(s, ξ) ds E

  

−(1−α)

  m(s, ξ) ds −

  t

  Z t Z

  1 −

  =

  11 (τ ) dτ

  m(s, ξ) ds E

  ✭✷✳✶✾✮ ❯s❛♥❞♦ q✉❡ m(t, ξ) ≈ −(1 + t)

  |ξ|

  Z

  Z t Z t

  11

  ds |E

  2δ

  |ξ|

  −(1−α)

  (1 + s)

  τ

  (τ ) dτ . 1 +

  2δ

  11

  m(s, ξ) ds E

  τ

  Z t Z t

  (t) | = 1 −

  11

  ❡♠ ✭✷✳✶✾✮✱ t❡♠✲s❡✿ |E

  τ

  (s) ds Z t m(s, ξ) ds

  ❊st✐♠❛t✐✈❛ ✷✿ E

  11

  =

  11 (s) ds dτ

  E

  2 Z τ

  Z t h(τ, ξ)

  (s) ds dτ = 1 +

  E

  Z t m(τ, ξ) Z

  Z t h(τ, ξ) ih(τ, ξ) Z τ

  1 − i

  (t) =

  11

  E

  (t) ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✶✻✮ ❡♠ ✭✷✳✶✹✮ ♦❜t❡♠♦s

  11

  1 −

  

τ

  11

  11

  Z t E

  1 −

  (τ ) dτ =

  11

  − Z t Z τ m(s, ξ) ds E

  t

  (s) ds Z τ m(s, ξ) ds

  Z τ E

  E 11 (s) ds dτ. ❙❡❥❛ dv = m(τ, ξ) dτ ❡ u =

   

  1 −

  (t) =

  11

  ✳ ■♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s✱ E

  11 (s) ds

  Z τ E

  • Z t

  ✭✷✳✷✶✮

  2δ

  ξ ) α+1

  = exp(N ) ♣♦✐s (1 + t

  2δ

  |ξ|

  α+1

  Z t dτ . exp (1 + t ξ )

  |ξ|

  2δ

  α

  )

  ξ

  (1 + t

  (t) | . exp

  11

  |ξ|

  = N ✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡

  ξ

  Z τ h(s, ξ) E

  00 (s) ds dτ.

  h(s, ξ)E

  τ

  Z t Z

  = h(t, ξ) h(0, ξ)

  00 (s) ds dτ

  Z t −i

  |E

  h(t, ξ) h(0, ξ)

  00 (t) =

  ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✶✸✮ ❡♠ ✭✷✳✶✺✮ t❡♠♦s E

  00 (t)

  ✭✷✳✷✵✮ ❊st✐♠❛t✐✈❛ ✸✿ E

  (t) | . 1.

  11

  ✱ ♦❜t❡♠♦s✿ |E

  ♥❛ ú❧t✐♠❛ ❡st✐♠❛t✐✈❛ ❢♦✐ ✉s❛❞♦ q✉❡ t 6 t ξ ✳ P❡❧♦ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧ ❡ ✉s❛♥❞♦ q✉❡ t 6 t

  = 1 + Z t

  | dτ = 1 +

  α

  − (1 + τ)

  α

  (1 + t)

  2δ

  Z t |ξ|

  11 (τ )

  (τ ) | dτ .

  ! |E

  t τ

  α

  α

  (1 + s)

  2δ

  |ξ|

  α |E

  1 + |ξ|

  (τ ) | dτ

  ξ

  11

  Z t |E

  

  |ξ|

  α

  )

  . 1 + (1 + t

  2δ

  (τ ) | dτ

  11

  |E

  t

  Z

  

α

  (1 + t)

  • ih(t, ξ)
  • h(t, ξ)

  ❈♦♥s✐❞❡r❡ e E

  (t) | . exp

  dτ = exp (1 + t

  −(1−α)

  Z t (1 + τ )

  2δ

  ) |ξ|

  ξ

  (1 + t

  00

  ) |ξ|

  | e E

  ✳ P❡❧♦ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧✱ t❡♠♦s✿

  ξ

  ♣♦✐s t 6 t

  (τ ) | dτ

  00

  | e E

  2δ

  

ξ

  2δ

  −(1−α)

  α . exp

  = exp N

  2δ

  |ξ|

  α+1

  )

  

ξ

  (1 + t

  1 α

  α

  (1 + τ )

  (1 + t)

  2δ

  ) |ξ|

  ξ

  ! . exp (1 + t

  t

  α

  α

  |ξ|

  (1 + τ )

  00

  ✳ ❆ss✐♠✱ E e

  e E

  2

  Z t Z τ h(s, ξ)

  (s) ds dτ = 1 +

  00

  (t) = 1 + Z t Z τ h(0, ξ)h(s, ξ)E

  00

  2δ

  (s) ds dτ = 1 +

  |ξ|

  −(1−α)

  −m(t, ξ) ≈ (1 + t)

  (t) ✱ ❛ ❞❡✜♥✐çã♦ ❞❡ h(t, ξ) ❡ ♦ ❢❛t♦ q✉❡

  00

  E

  h(0,ξ) h(t,ξ)

  (t) =

  00

  Z t Z τ −m(s, ξ) e

  ) Z t

  . 1 + t Z t

  ξ

  | dτ . 1 + (1 + t

  00 (τ )

  | e E

  2δ

  |ξ|

  −(1−α)

  (1 + τ )

  (s) | ds dτ

  E

  00

  | e E

  2δ

  |ξ|

  −(1−α)

  Z t Z τ (1 + s)

  (s) ds dτ . 1 +

  00

  α .

  ❉❡ss❛ ❢♦r♠❛✱ h(0, ξ) |E | = | e | . 1 h(t, ξ) h(t, ξ)

  (t)

  00 ✭✷✳✷✷✮ ⇒ |E | .

  h(0, ξ) ♣♦✐s h(t, ξ) > 0✳

  (t) ❊st✐♠❛t✐✈❛ ✹✿ E

  01

  ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✶✹✮ ❡♠ ✭✷✳✶✻✮ t❡♠♦s Z t Z τ

  E (t) = ih(t, ξ) 1 h(s, ξ)E (s) ds dτ.

  01 01 ✭✷✳✷✸✮

  − i (1−α) 2 δ ❯s❛♥❞♦ q✉❡ h(t, ξ) ≈ (1 + t) ❡♠ ✭✷✳✷✸✮✱ t❡♠♦s✿

  |ξ| Z Z

  t Z t τ 01 (t) dτ h(s, ξ)E 01 (s) ds dτ

  |E | = − i ih(t, ξ) Z

  Z t Z t τ dτ + h(t, ξ) h(s, ξ)

  01 (s) dτ

  ≤ h(t, ξ) |E | ds Z t h(s, ξ) (s)

  01

  ≤ t h(t, ξ) + t h(t, ξ) |E | ds Z t (1−α) 2 δ . (1 + t)h(t, ξ) + (1 + t)h(t, ξ) (1 + s) (s) .

  01

  |ξ| |E | ds

  ♯ −1 −1

  (t) = (1 + t) h(t, ξ) (t) ❙❡❥❛ E

  

01

01 |E |✳ ❊♥tã♦✿

  Z t (1−α)

  ♯ 2 δ

  E (t) . 1 + (1 + s)

  01 (s) 01 |ξ| |E | ds

  Z t 1+α

  δ 2 −1

  = 1 + (1 + s) (1 + s) (s)

  01

  |ξ| |E | ds

  = 1 + Z t

  2δ

  Z t (1 + s)

  α

  ds . exp

  |ξ|

  2δ

  (1 + t)

  α

  t . exp

  |ξ|

  (1 + t)

  |ξ|

  α+1

  = exp(N ) ♣♦✐s✱ ❝♦♠♦ t 6 t ξ t❡♠✲s❡ q✉❡ |ξ|

  2δ

  (1 + t)

  α+1

  6 N ✳

  ▲♦❣♦✱ |E

  01

  (t) | . h(t, ξ) (1 + t).

  ✭✷✳✷✹✮ P♦rt❛♥t♦✱ ♣♦r ✭✷✳✶✽✮✱ ✭✷✳✷✵✮✱ ✭✷✳✷✷✮ ❡ ✭✷✳✷✹✮ ❛ ♣r♦♣♦s✐çã♦ ❡stá ❞❡✲

  2δ

  (t) . exp

  (1 + s) 1+α 2 |ξ|

  ♯

  δ

  h(s, ξ)E

  ♯

  01

  (s) ds . 1 +

  Z t (1 + s) 1+α 2 (1 + s) 1+α 2

  |ξ|

  2δ

  E

  01

  01

  (s) ds . 1 +

  Z t (1 + s)

  

α

  |ξ|

  2δ

  E

  ♯

  01 (s) ds.

  P❡❧♦ ▲❡♠❛ ❞❡ ●r♦♥✇❛❧❧✱ t❡♠♦s✿ E

  ♯

  ♠♦♥str❛❞❛✳

  ✷✳✷ ❩♦♥❛ ❊❧í♣t✐❝❛ low

  ❈♦♥s✐❞❡r❡ ❛❣♦r❛ (s, ξ) ∈ Z ❡ t ≥ s✳ ❉❡✜♥✐♠♦s

  ell

  h(s, ξ) .

  low 2δ low

  a (t, s, ξ) = exp Λ(t, s) E (t, s, ξ) ✭✷✳✷✺✮

  ell −|ξ| ell

  h(t, ξ)

  low low

  (t, s, ξ) ♦♥❞❡ E é ❛ s♦❧✉çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ✭✷✳✽✮ ♣❛r❛ (s, ξ) ∈ Z ❡

  ell ell

  Λ(t, s) ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✸✮✳

  pd (t, 0, ξ)

  ◆❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♦❜t✐✈❡♠♦s ❡st✐♠❛t✐✈❛s ♣❛r❛ E ✳ ◆❛

  low

  (t, s, ξ) ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ✈❛♠♦s ♦❜t❡r ❡st✐♠❛t✐✈❛s ♣❛r❛ a ❡ ❝♦♥✲

  ell low

  ell

2 low

  (t, s, ξ) s❡q✉❡♥t❡♠❡♥t❡ ♣❛r❛ E ✳ P❛r❛ ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ♣r❡❝✐s❛♠♦s

  h(t, ξ) h(t, ξ) ❡st✐♠❛r h(t, ξ)✱ ∂ t ❡ ∂ ❡♠ Z ✳ ❚❡♠♦s q✉❡

  

t ell

1

  p 2

  2σ 2 4δ 2δ ′

  h(t, ξ) = + b(t) + b (t) , −m(t, ξ) = −|ξ| |ξ| |ξ| ❝♦♥❢♦r♠❡ ❛ ❞❡✜♥✐çã♦ ❞❡ m(t, ξ).

  α 2δ α+1 2δ low

  ❈♦♠♦ h(t, ξ) ≈ (1+t) |ξ| ❡ N ≤ (1+t) |ξ| ❡♠ Z t❡♠✲s❡ 1 ell

  1

  − 2σ 2 4δ 2δ 2 4δ 2δ ′ ′ ′′

  ∂ h(t, ξ) = + b(t) + b (t) 2b(t)b (t) + b (t)

  t

  −|ξ| |ξ| |ξ| |ξ| |ξ|

  2

  1

  4δ 2δ ′ ′′

  = 2b(t)b (t) + b (t) |ξ| |ξ|

  2h(t, ξ)

  2 4δ −α −2δ 2α−1

  . (1 + t) 2µ α(1 + t) |ξ| |ξ|

  

α−2

  • µα(α

  − 1)(1 + t) |ξ|

  2δ α−1

  . (1 + t) .

  ✭✷✳✷✻✮ |ξ|

  • 2
  • 2
  • b

  • (1 + t)
  • |ξ|
  • (1 + t)
  • |ξ|
  • (1 + t)
  • (1 + t)

  2δ

  |ξ|

  α+1

  [(1 + t)

  −3

  α−2

  (1 + t)

  2δ

  |ξ|

  2δ .

  |ξ|

  α

  (1 + t)

  2δ

  |ξ|

  α−3

  

  |ξ|

  α

  (1 + t)

  2α−2

  (1 + t)

  

  2α−2

  (1 + t)

  4δ

  

  (1 + t)

  |ξ|

  −(α+1)

  α−2

  ♯

  (t, s) , ❝♦♠ σ ❡ δ ♦s ❡①♣♦❡♥t❡s ❞♦s ❧❛♣❧❛❝✐❛♥♦s ❞❛ ❡q✉❛çã♦ ✭✷✳✶✮ ❡ Λ

  ♯

  Λ

  2(σ−δ)

  (t, s, ξ) | . exp −|ξ|

  low ell

  ❡ t ≥ s ❡♥tã♦ |a

  low ell

  ✳ Pr♦♣♦s✐çã♦ ✷✳✺ ❙❡ (s, ξ) ∈ Z

  low ell

  ✭✷✳✷✼✮ ♣❛r❛ t♦❞♦ (t, ξ) ∈ Z

  (1 + t)

  −2δ

  2δ

  |ξ|

  −1 .

  N

  2δ

  |ξ|

  α+1

  (1 + t)

  −3

  α−2

  (1 + t)

  2δ

  ] . |ξ|

  |ξ|

  2α

  (1 + t)

  2∂ t h(t, ξ)b(t)b

  |ξ|

  2

  h(t, ξ)

  2δ

  (t) |ξ|

  ′′

  − ∂ t h(t, ξ)b

  2

  h(t, ξ)

  4δ

  (t) |ξ|

  ′

  2 −

  b

  1

  =

  2δ

  |ξ|

  4δ ′′

  |ξ|

  ′

  1 2h(t, ξ)

  2 t

  h(t, ξ) ✳ ❚❡♠✲s❡ ✉s❛♥❞♦ ✭✷✳✷✻✮ q✉❡

  2 t

  ❘❡st❛✲♥♦s ❡st✐♠❛r ∂

  4δ

  ′

  2δ

  

2α−1

  |ξ|

  α−2

  (1 + t)

  2δ

  |ξ|

  α−1

  4δ

  |ξ|

  2α

  (1 + t)

  4δ

  |ξ|

  (1 + t)

  (t)

  2δ

  |ξ|

  α−1

  h(t, ξ) . (1 + t)

  2δ

  (t) |ξ|

  ′′′

  (t) h(t, ξ)

  ′′

  b(t)b

  4δ

  |ξ|

  2

  (t, s) ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✸✮✳ ❉❡♠♦♥str❛çã♦✿ P❛r❛ ♣r♦✈❛r♠♦s ❡ss❡ r❡s✉❧t❛❞♦ ✈❛♠♦s ✉s❛r ✉♠ ♣r♦❝❡ss♦ ❞❡ ❞✐❛❣♦♥❛✲ ❧✐③❛çã♦✳ ❈♦♥s✐❞❡r❡ ♦ s✐st❡♠❛

     ∂ E(t, s, ξ) = A(t, ξ)E(t, s, ξ) t > s

  t

    E(s, s, ξ) = I.

  −1

  E = P EP ❙❡❥❛ e ✱ ❝♦♠

     

  1

  √ −1 −i 1 i i

  1

  2   −1  

  P = P = √ ❡     .

  2

  2 i

  1

  1

  1 ❆ss✐♠✱

  −1 −1 −1 −1 −1

  ∂ E = P ∂ e EP = P AEP = P AP EP P e = P AP

  E, e

  t t

  ♦✉ ❛✐♥❞❛✱      

  √ √ ∂ h(t,ξ) 1 t

  2

  2 −i − √ √

  ih(t, ξ)

  h(t,ξ) 2i 2i

  2

  2

        e ∂ E =

  E

  t

        e

  √ √ i

  1

  2

  2 √ √

  −ih(t, ξ)

  2

  2

  2

  2

     

  √ √ ∂ h(t,ξ) t

  1

  2

  2

−i −

√ √

  • h(t, ξ) h(t, ξ)

  h(t,ξ) 2i 2i

  2

  2

      =

  E     e

  √ √ ∂ h(t,ξ) i t

  1

  2

  2 √ √

  h(t, ξ) − h(t, ξ) −

  h(t,ξ)

  2

  2

  2

  2

   

  ∂ h(t,ξ) ∂ h(t,ξ) t t

  • h(t, ξ)

  −

  2h(t,ξ) 2h(t,ξ)

    = E

    e

  ∂ h(t,ξ) ∂ h(t,ξ) t t

  − 2h(t,ξ) 2h(t,ξ) − h(t, ξ)

  =   

  α

  t

  |∂

  (t, ξ) | =

  1

  |N

  ✭✷✳✷✾✮ ❡

  −1

  . (1 + t)

  2δ

  |ξ|

  (1 + t)

  4h(t, ξ)

  2δ

  |ξ|

  α−1

  (1 + t)

  2h(t, ξ) .

  h(t, ξ) |

  t

  |∂

  (t, ξ) | =

  1

  |R

  h(t, ξ) |

  2 .

  2δ

  low ell

  −1

  ≤ N

  −2δ

  |ξ|

  −(1+α)

  ≥ N ⇒ (1 + t)

  2δ

  |ξ|

  α+1

  ✱ (1 + t)

  ✭✷✳✸✵✮ ❈♦♠♦ (t, ξ) ∈ Z

  (1 + t)

  −2δ .

  |ξ|

  −(1+α)

  = (1 + t)

  4δ

  |ξ|

  

  (1 + t)

  2δ

  |ξ|

  

α−1

  ✳ ❆ss✐♠✱ ♣♦r ✭✷✳✷✻✮ t❡♠✲s❡

  |ξ|

  ∂ t h(t,ξ) 2h(t,ξ)

  

t

  −1 = I.

  = P IP

  −1

  E(s, s, ξ) = P E(s, s, ξ)P

  ✭✷✳✷✽✮ ❆❧é♠ ❞✐ss♦✱ e

  1    e E.

  −1

  1 −1

    

  h(t, ξ) 2h(t, ξ)

  E + ∂

  1

     e

  1 −1

    

  E = h(t, ξ)

  −h(t, ξ)    e

  E +    h(t, ξ)

     e

  ∂ t h(t,ξ) 2h(t,ξ) ∂ t h(t,ξ) 2h(t,ξ)

  −

  ∂ t h(t,ξ) 2h(t,ξ)

  −

  ❉❡✜♥✐♠♦s R

  (t, ξ) = ∂

  α

  N

  t❡♠✲s❡ q✉❡ h(t, ξ) ≈ (1+t)

  low ell

  P♦r ✭✷✳✻✮ ♣❛r❛ (t, ξ) ∈ Z

  1 (t, ξ).

  (t, ξ) ❡ N(t, ξ) = I + N

  1

  ♯

  R

  1 2h(t, ξ)

  (t, ξ) =

  1

  1    ,

  t

  −1

    

  h(t, ξ) 2h(t, ξ)

  t

  (t, ξ) = ∂

  1

  ♯

     , R

  −1 −1

    

  h(t, ξ) 2h(t, ξ)

  ✳

  ▲♦❣♦✱ ♣❛r❛ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ s❡❣✉❡ q✉❡

  1 |N

  2 ❉❡ss❛ ❢♦r♠❛ N é ❧✐♠✐t❛❞♦ ❡ ✐♥✈❡rsí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é ❧✐♠✐t❛❞❛✳

  ❙❡❥❛

  −1

  ˙ E(t, s, ξ) = N(t, ξ) e E(t, s, ξ)N (s, ξ),

  ✭✷✳✸✶✮ ❧♦❣♦

  −1

  ˙ N ∂ E(t, s, ξ) = ∂ (t, ξ) e E(t, s, ξ)N (s, ξ)

  t t −1

  • N(t, ξ)∂ E(t, s, ξ)N e (s, ξ).

  t ✭✷✳✸✷✮

  ❆ ♣r✐♠❡✐r❛ ♣❛r❝❡❧❛ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ h i

  −1 −1 −1

  N N ∂ (t, ξ) e EN (s, ξ) = ∂ (I + N (t, ξ)) (t, ξ) ˙ EN(s, ξ)N (s, ξ)

  t t

  

1

−1

  N = ∂ (t, ξ)N (t, ξ) ˙

  E,

  t 1 ✭✷✳✸✸✮ 1 (t, ξ)

  ♦♥❞❡ ❢♦✐ ✉s❛❞♦ q✉❡ N(t, ξ) = I + N ✳

  • N(t, ξ)

    h(t, ξ)

  −1

  1 −1

    

  ∂ t h(t, ξ) 2h(t, ξ)

     +

  1 −1

    

  (t, ξ) ˙ E =

     N

  −1

     N

  1   

  −1

  1 −1

    

  ∂ t h(t, ξ) 2h(t, ξ)

     +

  1   

  −1

    

  1   

  (t, ξ) = h(t, ξ)DN(t, ξ) − h(t, ξ)N(t, ξ)D

  1

  R

  ✭✷✳✸✹✮ ❆❣♦r❛✱ ♥♦t❛♠♦s q✉❡

  (t, ξ) ˙ E.

  −1

     N

  −1

  (t, ξ) ˙ E

  1 −1

    

  ∂ t h(t, ξ) 2h(t, ξ)

     +

  1 −1

    

  (t, ξ)   h(t, ξ)

  1

  1 −1

  (t, ξ))   h(t, ξ)

  ✭✷✳✸✺✮

     e

  1    e

  −1

  1 −1

    

  h(t, ξ) 2h(t, ξ)

  

t

  E + ∂

  1 −1

  −1

    

    h(t, ξ)

  (s, ξ) = N(t, ξ)

  −1

  E(t, s, ξ)N

  N (t, ξ)∂ t e

  ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ s❡❣✉♥❞❛ ♣❛r❝❡❧❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

  ❯t✐❧✐③❛♥❞♦ ✭✷✳✷✽✮ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ e E

  E    N

  (s, ξ) = N(t, ξ)h(t, ξ)

  1

    

  (s, ξ) = (I + N

  −1

  (t, ξ) ˙ EN(s, ξ)N

  −1

  1    N

  −1

  1 −1

  h(t, ξ) 2h(t, ξ)

    

  t

  ∂

  (s, ξ)

  −1

  (t, ξ) ˙ EN(s, ξ)N

  −1

     N

  1 −1

  • N

  ♦♥❞❡ D =

  h(t, ξ) 2h(t, ξ)

  1

  (t, ξ) t❡♠✲s❡ q✉❡ ∂

  t

  h(t, ξ) 2h(t, ξ)

    

  1 −1

  −1 1   = h(t, ξ)DN

  1

  (t, ξ) −h(t, ξ)N

  1

  (t, ξ)D+ ∂

  t

     1 0

  0 1    .

  0 1    .

  ❙✉❜st✐t✉✐♥❞♦ ❛ ✐❞❡♥t✐❞❛❞❡ ❛♥t❡r✐♦r ❡♠ ✭✷✳✸✹✮ ♦❜t❡♠♦s N

  (t, ξ)∂ t e E(t, s, ξ)N

  −1

  (s, ξ) =

    h(t, ξ)

    

  1 −1

     + h(t, ξ)

    

  1 −1

     N

  1

  ❈♦♠♦ N(t, ξ) = I + N

     1 0

    

     1 0

  1    .

  ❆❧é♠ ❞✐ss♦ R

  1

  (t, ξ) = ∂

  t

  h(t, ξ) 2h(t, ξ)

    

  1 −1

  −1

  1    −

  ∂

  t

  h(t, ξ) 2h(t, ξ)

  0 1    .

  h(t, ξ) 2h(t, ξ)

  ✭✷✳✸✻✮ ❉❡ ✭✷✳✸✺✮ ❡ ✭✷✳✸✻✮ ♦❜t❡♠♦s

  R

  1 (t, ξ) = h(t, ξ)DN(t, ξ)

  − h(t, ξ)N(t, ξ)D =

  ∂

  t

  h(t, ξ) 2h(t, ξ)

    

  1 −1

  −1

  1    −

  ∂

  t

  (t, ξ)

  • h(t, ξ)N
  • N
  • N

    

  E + ∂ t h(t, ξ)

     ˙

  1 −1

    

  (t, ξ) ˙ E = h(t, ξ)

  −1

     N

    

  −1 −1

  1

    

  0 1    +

     1 0

  2h(t, ξ)   

  (t, ξ) ∂ t h(t, ξ)

  1

  0 1   

     1 0

  2h(t, ξ) (I + N

  (t, ξ))

  E +   

     ˙

  −1

  (t, ξ)N

  1

  (t, ξ)R

  1

  E + N

  2h(t, ξ) ˙

  E + ∂ t h(t, ξ)

  1 −1

  1

  = h(t, ξ)   

  (t, ξ) ˙ E ✭✷✳✸✼✮

  −1

     N

    

  −1 −1

  2h(t, ξ)   

  (t, ξ) ∂ t h(t, ξ)

  ∂ t h(t, ξ) 2h(t, ξ)

     ˙

  (t, ξ) ˙ E.

  0 1   

  1 −1

  2h(t, ξ)   

  (t, ξ) ∂ t h(t, ξ)

  1

   

  1 −1

  (t, ξ)  

  1

     1 0

  1   

  h(t, ξ) 2h(t, ξ)

  t

  ∂

     +

  1 −1

  (t, ξ)   

  1

  − h(t, ξ)N

  −1

     N

  1 −1

  (t, ξ) ∂ t h(t, ξ)

    

  (t, ξ) ˙ E = h(t, ξ)

  −1

     N

  1   

  −1

  1 −1

  2h(t, ξ)   

  1

  −1

  0 1   

  2h(t, ξ)    1 0

  (t, ξ)) + ∂ t h(t, ξ)

  1

     (I + N

  1 −1

    

    h(t, ξ)

  (t, ξ) ˙ E =

  • N
  • N

  ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✸✸✮ ❡ ✭✷✳✸✼✮ ❡♠ ✭✷✳✸✷✮ ❡ ❝♦♥s✐❞❡r❛♥❞♦

  −1

  N ♦❜t❡♠♦s

   

  1 ∂ h(t, ξ)

  t

    ˙

  ˙ ∂ E(t, s, ξ) = R (t, ξ) ˙ E + h(t, ξ) E + E.

  t 2 ✭✷✳✸✽✮

    ˙ 2h(t, ξ)

  −1 ❉❡✜♥❛

  ˙ E(t, s, ξ) = e (t, s, ξ)Q (t, s, ξ)

  ell ell ✭✷✳✸✾✮

  ♦♥❞❡ 1 2 Z t h(t, ξ) e (t, s, ξ) = exp h(τ, ξ) dτ

  ell

  ✭✷✳✹✵✮ h(s, ξ)

  s

  (t, s, ξ) ❡ Q ell é ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

        0 0  

      

  ∂ Q(t, s, ξ) = (t, ξ)  t

  2

  −2h(t, ξ)   + R  Q(t, s, ξ), t > s 0 1      Q(s, s, ξ) = I.

  ✭✷✳✹✶✮ P❛r❛ ♣r♦✈❛r♠♦s q✉❡ Q(t, s, ξ) é s♦❧✉çã♦ ❞❡ ✭✷✳✹✶✮ ✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡

  ❡s❝r❡✈❡r

  −1

  ˙ Q(t, s, ξ) = (e (t, s, ξ)) E(t, s, ξ),

  ell

  • h(t, ξ)
  • 3 2 h(s, ξ) 1 2
  • (e

  (∂

  −1

  h(t, ξ)   

  1 −1

     +

  ∂

  t

  h(t, ξ) 2h(t, ξ)

  2 (t, ξ) e ell (t, s, ξ)Q(t, s, ξ)

  = − (e

  ell

  (t, s, ξ))

  −1

  

t

  ell

  e

  ell

  (t, s, ξ)) + h(t, ξ)   

  1 −1

    

  2h(t, ξ)

  2

  (t, ξ) Q(t, s, ξ) =

  − h(s, ξ) h(t, ξ) 1 2 exp

  − Z t

  s

  h(τ, ξ) dτ exp Z t

  s

  (t, s, ξ))

  (∂ t e ell (t, s, ξ)) e ell (t, s, ξ) Q(t, s, ξ)

  h(τ, ξ) dτ

  h(τ, ξ) dτ . ❈♦♠ ✐ss♦

  ♦♥❞❡ (e ell (t, s, ξ))

  −1

  = h(s, ξ) h(t, ξ) 1 2 exp

  − Z t

  s

  h(τ, ξ) dτ ❡ ∂

  t

  e

  ell

  (t, s, ξ) = ∂ t h(t, ξ)

  2h(s, ξ) 1 2 h(t, ξ) 1 2

  Z

  t s

  ∂ t Q(t, s, ξ) = −(e

  ell (t, s, ξ)) −2

  ell (t, s, ξ)) −2

  (∂ t e ell (t, s, ξ)) ˙ E(t, s, ξ)

  ell

  (t, s, ξ))

  −1

  ∂

  t

  ˙ E(t, s, ξ). ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✸✽✮✱ ✭✷✳✸✾✮✱ ∂

  t e ell (t, s, ξ)

  ❡ (e

  ell (t, s, ξ)) −1

  ❡s❝r✐t♦s ❛❝✐♠❛ t❡♠♦s✱ ∂ t Q(t, s, ξ) =

  −(e

  • (e
  • R
  • ∂ t h(t, ξ)
  • R

  • h(t, ξ)
  • 3 2 h(s, ξ) 1 2h(t, ξ)
  • R
  • R

  (t, s, ξ) é s♦❧✉çã♦ ❞♦ s✐st❡♠❛

  ell

  (s, s, ξ))

  −1

  ˙ E(s, s, ξ) = I. ▲♦❣♦✱ t❡♠♦s q✉❡

  ˙ E(t, s, ξ)

  ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ˙

  E(t, s, ξ) = e

  ell

  (t, s, ξ)Q

  ell

  (t, s, ξ) ✱ ♦♥❞❡ Q ell

  ❞❛❞♦ ❡♠ ✭✷✳✹✶✮✳ ◆❛ s❡q✉ê♥❝✐❛ ✈❛♠♦s ❡st✐♠❛r |R

  2 (t, ξ) Q(t, s, ξ).

  

2

  (t, ξ) |✳ P❡❧❛s ❡st✐♠❛t✐✈❛s ❛♥t❡r✐♦r❡s r❡st❛ ❡st✐♠❛r |∂ t

  N

  1

  (t, ξ) |✳ ❚❡♠♦s q✉❡

  N

  1

  (t, ξ) = ∂ t h(t, ξ) 4h(t, ξ)

  2

    

  −1

  ❈♦♠♦ ˙E(s, s, ξ) = I t❡♠✲s❡ q✉❡ Q(s, s, ξ) = (e

  0 1    Q(t, s, ξ) + R

  ∂

  h(t, ξ) 2h(t, ξ)

  t

  h(t, ξ) 2h(s, ξ) 1 2 h(t, ξ) 1 2

    

  1 −1

    

  t

  h(t, ξ) 2h(t, ξ)

  2 (t, ξ) Q(t, s, ξ)

  = −

  ∂

  t

  − h(t, ξ)    1 0

  −2 h(t, ξ)    0 0

  0 1    + h(t, ξ)

    

  1 −1

    

  t

  h(t, ξ) 2h(t, ξ)

  2 (t, ξ) Q(t, s, ξ)

  = h(t, ξ)   

  −2    + R

  2

  (t, ξ) Q(t, s, ξ) =

  1   

  • (1 + t)

  (t, ξ) + N

  . (1 + t)

  (t, ξ) |

  −1

  (t, ξ)) N

  1

  (t, ξ)R

  1

  1

  |ξ|

  N

  t

  (t, ξ) | = | (∂

  2

  (t, ξ) s❡r ❧✐♠✐t❛❞♦✱ ♦❜t❡♠♦s |R

  ❊ ✜♥❛❧♠❡♥t❡✱ ♣♦r ✭✷✳✷✾✮✱ ✭✷✳✸✵✮✱ ❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛ ❡ ♦ ❢❛t♦ ❞❡ N −1

  −2δ .

  −(α+2)

  −2δ

  −(α+2)

  1+α

  6

  −2δ

  |ξ|

  −(1+α)

  > N ⇒ (1 + s)

  2δ

  |ξ|

  ✱ ❡♥tã♦ (1 + s)

  −(α+2)

  low ell

  ✭✷✳✹✷✮ ❚❡♠♦s q✉❡ (s, ξ) ∈ Z

  −2δ .

  |ξ|

  −(α+2)

  (1 + t)

  −2δ .

  |ξ|

  |ξ|

  . (1 + t)

  1 N .

  t

  1    .

  −1

    

  4

  h(t, ξ) 4h(t, ξ)

  2

  h(t, ξ)

  h(t, ξ)

  α

  2

  ∂

  2

  (t, ξ) = h(t, ξ)

  1

  N

  t

  ❧♦❣♦ ∂

  ❯s❛♥❞♦ ✭✷✳✷✻✮✱ ✭✷✳✷✼✮ ❡ ♦ ❢❛t♦ q✉❡ h(t, ξ) ≈ (1 + t)

  |ξ|

  8δ

  |ξ|

  |ξ|

  4α

  (1 + t)

  6δ

  |ξ|

  2α−2+α

  6δ

  

2α+α−2

  2δ

  (1 + t)

  (t, ξ) | .

  1

  N

  t

  s❡❣✉❡ q✉❡ |∂

  low ell

  ❡♠ Z

  • (1 + t)
  • (1 + s)

  s h(τ, ξ) .

  h(τ, ξ)    0 0

  ell (t, s, ξ)

  | ≤ exp   

  Z t

  s

    −2h(τ, ξ)

     0 0

  0 1    + R

  2 (τ, ξ)

     dτ

    

  ≤ exp   

  −2 Z t

  s

  0 1    dτ

    

  s

  R

  2 (τ, ξ)dτ

    

  ≤ exp −2 Z t

  s

  h(τ, ξ) +

  1 N = exp

  −2 Z t

  s

  h(τ, ξ) exp

  1 N . exp

  −2 Z t

  ❆ss✐♠✱ ♣♦r ✭✷✳✹✸✮ ❡ ❞♦ ❢❛t♦ q✉❡ h(t, ξ) é ♥ã♦ ♥❡❣❛t✐✈♦ ♦❜t❡♠♦s |Q

  (τ, ξ)    dτ

  ❈♦♠ ✐ss♦ ❡ ✭✷✳✹✷✮✱ s❡❣✉❡ q✉❡ Z t

  −2δ

  s

  |R | dτ .

  Z t

  s −(2+α)

  |ξ|

  −2δ

  = |ξ|

  −2δ

  − (1 + t)

  −(1+α)

  1 + α

  −(1+α)

  1 + α . |ξ|

  (1 + s)

  2

  −(1+α)

  6

  1 N .

  ✭✷✳✹✸✮ ❆ s♦❧✉çã♦ Q ell

  (t, s, ξ) ❞♦ s✐st❡♠❛ ✭✷✳✹✶✮ é ❞♦ t✐♣♦

  Q

  ell

  (t, s, ξ) = exp   

  Z t

  s

    −2h(τ, ξ)

     0 0

  0 1    + R

  • Z t

  (t, s, ξ) ▲♦❣♦✱ Q ell é ❧✐♠✐t❛❞♦✳ ❆❧é♠ ❞✐ss♦✱

  Z t |Q | . exp −2 ✭✷✳✹✹✮

  s

  P♦r ✭✷✳✸✶✮ ❡ ✭✷✳✸✾✮ s❡❣✉❡ q✉❡

  −1

  N (t, ξ) e E(t, s, ξ)N (s, ξ) = e ell (t, s, ξ)Q ell (t, s, ξ).

  −1

  E = P EP ❈♦♠♦ e ✱

  low −1 −1

  N N (t, ξ)P E (t, s, ξ)P (s, ξ) = e (t, s, ξ)Q (t, s, ξ)

  ell ell ell

  ♦✉ ❛✐♥❞❛

  low −1 −1

  N E (t, s, ξ) = P (t, ξ)e (t, s, ξ)Q (t, s, ξ)N(s, ξ)P.

  ell ell ell −1

  ❏á ✈✐♠♦s q✉❡ N, N sã♦ ❧✐♠✐t❛❞♦s✳ ❆ss✐♠✱ t❡♠♦s

  low

  (t, s, ξ) (t, s, ξ) (t, s, ξ)

  ell ell |E ell | . |e | |Q |.

  P♦r ✭✷✳✹✵✮ ❡ ✭✷✳✹✹✮✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛ 1 2 Z t h(t, ξ)

  low (t, s, ξ) exp h(τ, ξ) dτ .

  |E ell | . − h(s, ξ)

  s

  ❯s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛ ❡♠ ✭✷✳✷✺✮ ♦❜t❡♠♦s 1 2 Z t h(s, ξ)

  low 2δ

  |a ell | . − |ξ| h(t, ξ)

  s

  ✭✷✳✹✺✮

  low

  ❈♦♠♦ (s, ξ), (t, ξ) ∈ Z ❡ t > s t❡♠♦s q✉❡

  ell α 2δ

  h(s, ξ) (1 + s) |ξ| .

  1.

  ✭✷✳✹✻✮ ≈

  α 2δ

  h(t, ξ) (1 + t) |ξ|

  ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡

  ′ α−1

  b (t) µα(1 + t) α

  −1

  = = (1 + t)

  α

  2b(t) 2µ(1 + t)

  2 α

  α 2δ −α−1 −2δ

  = (1 + t) (1 + t) |ξ| |ξ|

  2 C

  

α

−1

  N h(t, ξ) ≤

  2 ✭✷✳✹✼✮

  ≤ 2h(t, ξ),

  low α+1 2δ

  ♣♦✐s✱ ❡♠ Z t❡♠♦s (1 + t) ell |ξ| ≥ N.

  P❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦ ❡♥❝♦♥tr❛r❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ❛♣r♦✲ ♣r✐❛❞❛ ♣❛r❛ h(t, ξ)✳ P❛r❛ t❛❧✱ ✉s❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ h(t, ξ) ❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ y

  √ √ x + y 6 x + , √ ∀ x > 0, y > −x. 2 x

  ❉❡ss❛ ❢♦r♠❛✱ q

  2 4δ 2δ 2σ ′

  |ξ| |ξ| − |ξ|

  2δ 2σ

  q ′ b (t) |ξ| − |ξ|

  6 b(t) |ξ| p

  • 2 4δ

  2 4δ

  2 b(t) |ξ|

  2δ 2σ ′

  b (t)

  2δ |ξ| − |ξ|

  = b(t) |ξ|

  2b(t) |ξ|

  ′ 2(σ−δ)

  b (t)

  2δ |ξ| + = b(t) .

  |ξ| − 2b(t) 2b(t)

  ❊♥tã♦ ♣♦r ✭✷✳✹✼✮ t❡♠✲s❡

  2(σ−δ)

  |ξ|

  

.

  ✭✷✳✹✽✮ −h(t, ξ) ≤ b(t)|ξ| −

  2b(t) ❆ss✐♠✱ ✉s❛♥❞♦ ✭✷✳✹✽✮ ❡ ✭✷✳✹✻✮ ❡♠ ✭✷✳✹✺✮ ♦❜t❡♠♦s

  t

  Z

  2(σ−δ) low |ξ|

  (t, s, ξ) dτ |a ell | . exp −

  2b(τ )

  s ♯ 2(σ−δ)

  = exp Λ (t, s) −|ξ|

  ♦ q✉❡ ♣r♦✈❛ ❛ ♣r♦♣♦s✐çã♦✳

  ❈❛♣ít✉❧♦ ✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛ ❆❧t❛ ❋r❡q✉ê♥❝✐❛

  ◆❡st❡ ❝❛♣ít✉❧♦ ❡♥❝♦♥tr❛♠♦s t❛①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✷✮ ♥❛ r❡❣✐ã♦ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✳ P❛r❛ t❛❧✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ✉t✐❧✐③❛♠♦s ❛ ❡q✉❛çã♦ ✭✷✳✷✮ ❡ ❞❡t❡r♠✐♥❛♠♦s ❛ ❡♥❡r❣✐❛ ❞❡ss❡ s✐st❡♠❛✳

  c n

  = / ❊♠ t♦❞♦ ❝❛♣ít✉❧♦ ❝♦♥s✐❞❡r❛♠♦s ξ ∈ B M

  {ξ ∈ R |ξ| ≥ M}✱ ❝♦♠ M > 0

  ❞❛❞♦ ♣❡❧♦ ▲❡♠❛ ✷✳✶✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✷✮ ♣♦r bu t ♦❜t❡♠♦s

  

2σ 2δ

  2 u (t) u (t) + u(t) u (t) + 2b(t) u (t) = 0. tt t t t

  b b |ξ| b b |ξ| |b | P❛r❛ bv = bv(t, ξ) t❡♠♦s d d

  2

  |bv| bv bv) = bv bv + bv bv bvbv bvbv bv bv dt dt 1 d

  2 = Re( ). t

  ⇒ |bv| bv bv 2 dt ❚♦♠❛♥❞♦ ❛ ♣❛rt❡ r❡❛❧ ❞❛ ❡q✉❛çã♦ ❡ ✉s❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♦❜✲ t❡♠♦s

  1 d

  2 2σ 2 2δ

  2 + u t (t) u(t) + 2b(t) u t (t) = 0.

  |b | |ξ| |b | |ξ| |b | 2 dt ❙❡❥❛ 0 ≤ S ≤ T ≤ +∞✳ ■♥t❡❣r❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♥♦ ✐♥t❡r✈❛❧♦

  [S, T ] ✱ t❡♠♦s

  Z T

  1

  2 2σ 2 2δ

  2

  • u (T ) u(T ) + 2 b(t) u (t) dt

  t t

  |b | |ξ| |b | |ξ| |b |

  2 S

  1

  2 2σ

  2 + = u (S) u(S) . t

  |b | |ξ| |b |

  2 ❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❛❝✐♠❛ q✉❡

  Z T

  2δ

  2 2 2σ

  2 b(t) u + (t) dt . u (S) u(S) . t t ✭✸✳✶✮

  |ξ| |b | |b | |ξ| |b |

  S

  ❆ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ é ❞❛❞❛ ♣♦r✿

  1

  1

  2 2σ

  2 E(t, ξ) = u + (t) u(t) . t

  |b | |ξ| |b |

  2

  2 u b

  ❖ ♣ró①✐♠♦ ♣❛ss♦ s❡rá ♠✉❧t✐♣❧✐❝❛r ❛ ❡q✉❛çã♦ ✭✷✳✷✮ ♣♦r ✱ ▲♦❣♦✱

  2b(t)

  1

  1

  2σ 2 2δ

tt t

  b |ξ| |b | |ξ| b b b 2b(t) 2b(t) n o

  d

  2 (t) u(t) = u (t) u(t) u (t) .

  ◆♦t❡ q✉❡ bu tt t t ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ b dt b b −|b | ❛ ♣❛rt❡ r❡❛❧✱ ❛ ✐❞❡♥t✐❞❛❞❡ ❛❝✐♠❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

  1

  2σ 2 2δ

  • u(t) Re u(t) u t (t) |ξ| |b | |ξ| b b

  2b(t)

  1 1 d

  2 = u t (t) Re u t (t) u(t) .

  |b | − b b 2b(t) 2b(t) dt

  ■♥t❡❣r❛♥❞♦ ❛ ✐❞❡♥t✐❞❛❞❡ ❛❝✐♠❛ ❞❡ S ❛ T t❡♠✲s❡ Z T Z T

  1

  2σ 2 2δ

  u(t) dt + Re u(t) u (t) dt

  t

  |ξ| |b | |ξ| b b 2b(t)

  S S

  Z T Z T

  1 1 d

  2 = u t (t) dt Re u t (t) u(t) dt.

  |b | − b b 2b(t) 2b(t) dt

  S S

  ■♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s ♦ ú❧t✐♠♦ t❡r♠♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❛❝✐♠❛ ♦❜t❡♠♦s Z T Z T

  1

  2σ

2 2δ

  u(t) dt = Re u(t) u (t) dt

  t

  |ξ| |b | − |ξ| b b 2b(t)

  S S

  Z T

  1

  1

  2

  • 2b(t) 2b(T )

  u t (t) dt Re u t (T ) u(T ) |b | − b b

  S

  Z T

  ′

  1 b (t) + Re u (S) u(S) Re u (t) u(t) dt.

  t t

  b b − b b

  2

  2b(S) 2b(t)

  S

  • b
  • |ξ|
  • |ξ|

  1 2b(t)

  2δ

  1 M

  dt ≤

  2

  |b u t (t) |

  2δ

  M

  −2δ

  M

  S

  Z T

  dt ≤

  2

  |b u t (t) |

  S

  S

  Z T

  dt; ❼

  2

  (t) |

  t

  |b u

  2δ

  b(t) |ξ|

  S

  dt + Z T

  2

  |b u(t) |

  2σ

  |ξ|

  Z T

  b(t) |ξ|

  S

  2

  |ξ|

  2δ .

  ❡ |ξ|

  2σ

  ≤ |ξ|

  2σ

  ≤ 1✱ M

  1 2b(t)

  dt; ♥❛s ❡st✐♠❛t✐✈❛s ❛❝✐♠❛ ✉s❛♠♦s t❛♠❜é♠ q✉❡

  2

  |b u(t) |

  2σ

  2

  |b u t (t) |

  (t) b(t)

  2δ

  S

  |b u

  t

  (t) |

  2

  dt; ❼ −

  Z T

  b

  ′

  ′

  (t) 2b(t)

  b u t (t) b u(t) dt .

  Z T

  S

  b

  1 4b(t)

  Z T

  2σ .

  |b u

  2

  |b u(S) |

  2σ

  

2

  |b u t (S) |

  Re b u t (S) b u(S) .

  1 2b(S)

  ; ❼

  2

  |b u(T ) |

  2σ

  2

  (T ) |

  t

  (T ) b u(T ) .

  Z T

  1

  ❯s❛♥❞♦ ❛s três ❞❡s✐❣✉❛❧❞❛❞❡s ❛❜❛✐①♦ |ξ| ≥ M ⇒ M

  σ

  ≤ |ξ|

  σ

  Re(z) ≤ |Re(z)| ≤ |z|, ab

  ≤

  2 a

  t

  2

  2

  , ✈❛♠♦s ❡st✐♠❛r s❡♣❛r❛❞❛♠❡♥t❡ ♦s t❡r♠♦s ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✿

  ❼ −

  1 2b(T )

  Re b u

  ; ❼ −

  S

  dt ≤

  Z T

  2

  (t) |

  t

  |b u

  2δ

  b(t) |ξ|

  S

  dt + Z T

  2

  |b u(t) |

  2δ

  |ξ|

  1 4b(t)

  S

  | dt ≤

  |ξ|

  |b u(t) | |b u t (t)

  2δ

  Re b u t (t) b u(t) dt

  ≤ Z T

  

S

  |ξ|

  2δ

  | dt =

  b(t) 1 2 |b u t (t)

  Z T

  S

  1 b(t) 1 2 |ξ|

  δ

  |b u(t) | |ξ|

  δ

  • |ξ|

  • |b u
  • |ξ|
  • Z T
  • Z T
  • |ξ|
  • |ξ|

  (1 + t)

  T S

  −α

  −α

  (1 + t)

  α µ

  dt =

  −α−1

  (1 + t)

  S

  Z T

  dt = α µ

  2α

  2

  α µ

  µ

  α−1

  µα(1 + t)

  S

  dt = Z T

  2

  (t) b(t)

  ′

  b

  S

  ❆✐♥❞❛✱ t❡♠♦s q✉❡ Z T

  2 dt.

  ! =

  (1 + T )

  ′

  S

  2 .

  |b u(S) |

  2σ

  2

  (S) |

  t

  |b u

  2 dt .

  |b u(t) |

  2σ

  |ξ|

  1 4b(t)

  ❉❡ss❛ ❢♦r♠❛✱ Z T

  −α

  1 µ .

  ≤

  −α

  (1 + S)

  1 µ

  α =

  −α

  (1 + S)

  α µ

  −α ≤

  −α

  (1 + S)

  −α −

  (t) b(t)

  b

  ✭✸✳✷✮

  t

  2

  (t) |

  t

  |b u

  2δ

  b(t) |ξ|

  S

  2

  |b u(S) |

  2σ

  2

  (S) |

  2

  S

  |b |

  2σ

  |ξ|

  2

  |b |

  2

  |b |

  2σ

  |ξ|

  1 4b(t)

  S

  P♦rt❛♥t♦✱ s❡❣✉❡ ❞❛s ❡st✐♠❛t✐✈❛s ❛❝✐♠❛ Z T

  dt

  b

  2 Z T S

  2 dt .

  |b u(S) |

  2σ

  2

  (S) |

  t

  2

  |b u(S) |

  2σ

  2

  (S) |

  t

  |b u

  |b u(t) |

  ′

  2σ

  |ξ|

  1 4b(t)

  S

  Z T

  ❯s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ✭✸✳✶✮ ❡ ♦ ❢❛t♦ ❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ s❡r ❞❡✲ ❝r❡s❝❡♥t❡ ♦❜t❡♠♦s

  2 dt.

  |b u(t) |

  2σ

  

2

  |b u t (t) |

  2

  (t) b(t)

  • |b u
  • |ξ|

  • |ξ|
  • |ξ|

  ✭✸✳✹✮

  "Z

  2

  (t) |

  t

  |b u

  2δ+γ

  b(t) |ξ|

  |ξ|≥M

  S

  |ξ|≥M

  F (t) dt = Z T

  S

  ❯s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ✭✸✳✸✮ s❡❣✉❡ q✉❡ Z T

  dξ, s❡♥❞♦ q✉❡ γ s❡rá ❡s❝♦❧❤✐❞♦ ❛❞❡q✉❛❞❛♠❡♥t❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ r❡s✉❧✲ t❛❞♦ ♣r✐♥❝✐♣❛❧✳

  2

  |b u(t) |

  2σ+γ

  2

  dξ

  1 b(t) |ξ|

  

t

  (S) |

  h (S).

  2CE

  dξ =

  2

  |b u(S) |

  2σ+γ

  2

  t

  2σ+γ

  |b u

  

γ

  |ξ|

  |ξ|≥M

  ≤ C Z

  dξ # dt

  2

  |b u(t) |

  (t) |

  |b u

  |ξ|≥M

  t

  ✭✸✳✸✮ ❉❡✜♥✐♠♦s ❛❣♦r❛ ♦ ❢✉♥❝✐♦♥❛❧

  2 .

  |b u(S) |

  2σ

  2

  (S) |

  |b u

  γ

  2 .

  |b |

  2σ

  1 b(t) |ξ|

  2 Z T S

  |b |

  2δ

  F (t) := Z

  b(t) |ξ|

  S

  |b u(t) |

  |ξ|

  ❯s❛♥❞♦ ✭✸✳✶✮ ❡ ✭✸✳✷✮ t❡♠♦s Z T

  1

  (t) =

  h

  ❡ ❛ ❡♥❡r❣✐❛ ♥❛ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ E

  2 dξ.

  2σ+γ

  2δ+γ

  1 b(t) |ξ|

  |ξ|≥M

  dξ + Z

  

2

  (t) |

  t

  |b u

  |ξ|

  • |ξ|

  |ξ|≥M

  • Z
  • |ξ|

  2δ 2δ

  ❚❛♠❜é♠✱ ✉t✐❧✐③❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ b(t) ≥ 1 ❡ M ♦❜té♠✲s❡ ≤ |ξ|

Z Z

  1

  1

  1

  1

  1

  γ 2 2σ+γ

  2

  |ξ| |b | |ξ| |b | b(t) 2 b(t) 2 b(t)

  

|ξ|≥M |ξ|≥M

  Z Z

  1

  1

  1

  γ 2 2σ+γ

  2

  u (t) dξ + u(t) dξ

  t

  ≤ |ξ| |b | |ξ| |b |

  2 2 b(t)

  |ξ|≥M |ξ|≥M

  Z Z

  1

  1

  1

  2δ+γ 2 2σ+γ

  2

  u (t) dξ + u(t) dξ

  t

  ≤ |ξ| |b | |ξ| |b |

  2δ

  2M 2 b(t)

  

|ξ|≥M |ξ|≥M

  ! Z Z

  1

  2δ+γ 2 2σ+γ

  2

  b(t) u t (t) dξ + u(t) dξ ≤ C |ξ| |b | |ξ| |b | b(t)

  |ξ|≥M |ξ|≥M = CF (t).

  ■♥t❡❣r❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❞❡ ❙ ❛ ❚✱ ❡ ✉s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ✭✸✳✹✮ t❡♠♦s

  Z T Z T

  1 E h (t) dt F (t) dt 1 E h (S).

  ≤ C ≤ C b(t)

  S S

  R t

  1

  dτ ❆ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ▼❛rt✐♥❡③ ✭✈❡r ❬✶✶❪✮ ♣❛r❛ σ = 0✱ φ(t) =

  b(τ ) −1 1−ωφ(t)

h (t) h (0)e ,

  ❡ ω = C s❡❣✉❡ q✉❡ E ≤ E ∀ t ≥ 0 ✐st♦ é✱ !

  Z Z

  1

  γ 2 2σ+γ

  2 E (t) u (0) dξ + u(0) dξ h t

  ≤ |ξ| |b | |ξ| |b |

  2

  |ξ|≥M |ξ|≥M

  Z t

  1

  1−c dτ

  b(τ ) e Z t

  1 Z Z

  1−c dτ

  1

  γ

2 2σ+γ

  2

  b(τ ) u dξ + u dξ e

  1

  ≤ |ξ| |b | |ξ| |b | n n

  2 R R Z Z 1−α

  γ 2 2σ+γ

  2 −C (1+t)

  . u dξ + u dξ e .

  1 ✭✸✳✺✮ R R n n |ξ| |b | |ξ| |b |

  ❈❛♣ít✉❧♦ ✹ ❘❡s✉❧t❛❞♦ Pr✐♥❝✐♣❛❧

  ◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞♦ tr❛❜❛❧❤♦✳ ❆♥✲ t❡s ♣♦ré♠✱ ✉s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s ✈❛♠♦s ♦❜t❡r ❛❧❣✉♠❛s ❡st✐♠❛t✐✈❛s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

  P❛r❛ t ≤ t ξ ❥á ♦❜s❡r✈❛♠♦s q✉❡ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ❡q✉✐✈❛❧ê♥❝✐❛

  α 2δ

  h(t , ξ) )

  ξ ξ

  ≈ (1 + t |ξ|

  α+1 2δ ) = N.

  ❡ q✉❡ t ξ é ❛ ú♥✐❝❛ s♦❧✉çã♦ ♣❛r❛ (1 + t ξ ❉❡ss❛ ✐❣✉❛❧❞❛❞❡ |ξ|

  ❞❡❝♦rr❡ q✉❡ α +1 1 N N

  α+1 (1 + t ) = (1 + t ) = . ξ ξ 2δ

  ⇒

  2δ 1+α

  |ξ| |ξ|

  • 1 h(0, ξ)
  • h(t

  , 0, ξ) | + |E

  ξ

  (1 + t

  , 0, ξ) | .

  ξ

  (t

  11

  ξ

  1 h(t

  (t

  01

  , ξ) |E

  ξ

  1 h(t

  ♦✉ ❛✐♥❞❛✱

  ) +

  ξ

  | . h(t

  |ξ| 1+α

  |ξ|

  −2δ .

  |ξ|

  − 2δα 1+α

  |ξ|

  − α α +1

  = N 1 α +1

  , ξ) . (1 + t

  −2δ

  |ξ|

  −α

  )

  ξ

  ) + (1 + t

  ξ

  ξ , ξ)(1 + t ξ ) + 1,

  11 (t ξ , 0, ξ)

  − 1+α .

  10 (t ξ , 0, ξ) | .

  |ξ|

  α

  (1 + t ξ )

  , ξ) h(0, ξ)

  ξ

  h(t

  | + |E

  h(t

  00 (t ξ , 0, ξ)

  |E

  / |ξ| ≤ M} t❡♠✲s❡

  n

  = {ξ ∈ R

  ❯s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ❡ ❛ Pr♦♣♦s✐çã♦ ✷✳✹ ♣♦❞❡♠♦s ♦❜t❡r ❛❧✲ ❣✉♠❛s ❡st✐♠❛t✐✈❛s✿ ❊st✐♠❛t✐✈❛ ✶✿ P❛r❛ ξ ∈ B M

  2δ .

  ξ

  | + |E

  00 (t ξ , 0, ξ)

  01 (t ξ , 0, ξ)

  |E

  1. ❊st✐♠❛t✐✈❛ ✷✿ P❛r❛ ξ ∈ B M t❡♠✲s❡

  | .

  10 (t ξ , 0, ξ)

  | + |E

  , ξ) |E

  , ξ) h(0, ξ)

  ξ

  ♦✉ ❛✐♥❞❛✱ h(0, ξ) h(t

  , ξ) h(0, ξ) ,

  ξ

  2 h(t

  , ξ) h(0, ξ) =

  ξ

  • N

  ❊st✐♠❛t✐✈❛ ✸✿ P❛r❛ ξ ∈ B M t❡♠✲s❡ |ξ|

  2(σ−δ)

  , 0) = |ξ|

  2(σ−δ)

  Z t ξ

  1 2b(τ ) dτ

  = |ξ|

  2(σ−δ)

  2µ Z t ξ

  (1 + τ )

  −α

  dτ =

  |ξ|

  2µ (1 + τ )

  (t

  1−α

  1 − α

  t ξ

  = |ξ|

  2(σ−δ)

  2µ(1 − α)

  (1 + t ξ )

  1−α

  − 1 . |ξ|

  2(σ−δ)−2δ 1−α 1+α

  = |ξ|

  2 σ− 1+α

  ξ

  ♯

  2δ

  α + 1

  |ξ|

  2δ

  Z t ξ =

  |ξ|

  2δ

  Z t ξ µ(1 + τ )

  α

  dτ =

  |ξ|

  2δ

  µ (1 + τ )

  α+1

  t ξ

  Λ

  ≤ µ

  α + 1 (1 + t

  ξ

  )

  α+1

  |ξ|

  2δ

  = µN

  α + 1 .

  1. ❊st✐♠❛t✐✈❛ ✹✿ P❛r❛ ξ ∈ B M t❡♠✲s❡

  |ξ|

  2(σ−δ)

  . 1, ♣♦✐s σ(1 + α) > 2δ✳

  

low

  ❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ (t, ξ) ∈ Z ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✺ t❡♠♦s ✉♠❛

  

ell

low

  (t, s, ξ) ❡st✐♠❛t✐✈❛ ♣❛r❛ |E |✱ ♦✉ s❡❥❛✱ t❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ❝❛❞❛

  ell low

  (t, s, ξ) ❡♥tr❛❞❛ ❞❛ ♠❛tr✐③ |E

  ell |✳ ❱❛♠♦s ❡s❝r❡✈❡r ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ❛

  ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❝♦❧✉♥❛ ❞❡ E(t, 0, ξ) ✉s❛♥❞♦ ❛ ♥♦t❛çã♦ ❥á ❛♣r❡s❡♥t❛❞❛

  T T

  = (1, 0) = (0, 1) ❛♥t❡r✐♦r♠❡♥t❡ e ❡ e

  

1 ✿

2(σ−δ) ♯

low Λ (t,0) −|ξ|

  (t, 0, ξ) |a ell | . e 2δ 2(σ−δ) ♯ h(0, ξ)

  Λ(t,0) low Λ (t,0)

−|ξ| −|ξ|

  e E (t, 0, ξ) ⇒ ell

  . e h(t, ξ) 2δ 2(σ−δ) ♯ h(0, ξ)

  low Λ(t,0) Λ (t,0) −|ξ| −|ξ|

  E (t, 0, ξ)e e . e ✭✹✳✶✮

  ⇒ ell h(t, ξ) ❡ 2δ 2(σ−δ) h(0, ξ)

  low Λ(t,0) Λ (t,0) −|ξ| −|ξ|

  E (t, 0, ξ)e e . e

  ell 1 ✭✹✳✷✮

  h(t, ξ) 2δ 2δ 2(σ−δ) ♯

  1

  low Λ(t,0) Λ (t,0) −|ξ| − −|ξ| 1+α

  (t, 0, ξ)e . e , E

  1 e

  ⇒ ell |ξ| h(t, ξ)

  2δ .

  ♣♦✐s h(0, ξ) ≈ |ξ| P❛r❛ ❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

  α

  ❚❡♦r❡♠❛ ✹✳✶ ❙❡❥❛ σ > 0✱ δ ∈ (0, σ)✱ b(t) = µ(1 + t) ❝♦♠ µ > 0 ❡ α

  ∈ (0, 1)✱ ❡ ❛ss✉♠❛ q✉❡ 2δ < σ(1 + α)✳ ❙❡❥❛ 1 ≤ p ≤ 2 ≤ q ≤ ∞✱

  q ′

  q = ❡ k > 0✳ ❉❡✜♥✐♠♦s

  q−1

  1

  1 θ(k, n, p, q) := k + n ,

  ✭✹✳✸✮ − p q

  • Ct

  ku k

  > 0 ✱ ✐st♦ é✱

  θ > 2δ 1 + α

  .

  ✭✹✳✺✮ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡

  θ < 2δ 1 + α

  ✭✹✳✻✮ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ k |ξ|

  k

  b u(t) k L q

  (B M ) ≤ Ct −k 01

  

L

p (R n )

  ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ t♦❞♦ t > 0✱ ❞❛❞♦ q✉❡ k

  −k 12

  ku

  1

  k

  L p (R n ) ✭✹✳✼✮

  ♣❛r❛ t♦❞♦ t > 0✱ ♦♥❞❡ −k

  12 > 0

  ✱ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ ✭✹✳✻✮✳ ◆♦ ❝❛s♦ ❡s♣❡❝✐❛❧

  θ = 2δ 1 + α

  11

  (R n ) ✭✹✳✹✮

  ✭✹✳✽✮

  − j 2δ 1 + α

  ❡ k j1 =

  1 − α

  2(σ − δ)

  θ − j

  2δ 1 + α , k

  j2

  = 1 + α 2δ

  θ − j = 1 + α

  2δ θ

  , ♣❛r❛ j = 0, 1. ❊♥tã♦ ❛ s♦❧✉çã♦ ♣❛r❛ ✭✷✳✷✮ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ót✐♠❛ k |ξ|

  k L p

  k

  b u(t) k

  L q (B M )

  ≤ Ct

  −k 01

  ku k L p

  (R n )

  −k 11

  ku

  1

  • Ct

  ♦❜té♠✲s❡ 1 1

  k 01 − q −k p q p n p n

  k |ξ| b k ≤ Ct ku k L (R ) ku k L (R )

  L (B M )

  ✭✹✳✾✮ ♣❛r❛ t♦❞♦ t > 0✳ ❉❡♠♦♥str❛çã♦✿

  > 0 ❙❡❥❛ α ∈ (0, 1) ❡ t ≥ t ✳ ❈♦♥s✐❞❡r❡ p q

  ′ ′

  p = , q = p q − 1 − 1

  ❡

  1

  1

  1

  1

  1 = = . − −

  ′ ′

  r q p p q ◆♦ ❈❛♣ít✉❧♦ ✷ ❞❡✜♥✐♠♦s

  2δ

  w(t, ξ) = u(t, ξ) exp Λ(t, 0) b |ξ| ❡

  W (t, ξ) = (ih(t, ξ)w(t, ξ) , w (t, ξ))

  t

  ❡ ♦❜s❡r✈❛♠♦s q✉❡ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✼✮ é ❞❛❞❛ ♣♦r W (t, ξ) = E(t, 0, ξ)W (ξ)

  (ξ) = (ih(0, ξ)w(0, ξ), w (0, ξ)) ❝♦♠ W t ✳

  (R n )

  k

  −|ξ|

Λ(t,0)

  e

  −1

  ih(t, ξ)

  k

  E(t, 0, ξ)e b u (ξ) k L q

  T

  e

  −|ξ| Λ(t,0)

  e

  −1

  h(0, ξ)h(t, ξ)

  ≤ k |ξ|

  T

  b u (ξ) k L q

  2δ

  µ |ξ|

  1

  E(t, 0, ξ)e

  T

  e

  −|ξ|

Λ(t,0)

  e

  −1

  ih(t, ξ)

  k

  e

  E(t, 0, ξ)e

  1 (ξ)

  2δ

  kb u k L p

  L r

  E(t, 0, ξ)e k

  T

  e

  −|ξ| Λ(t,0)

  e

  −1

  h(0, ξ)h(t, ξ)

  k

  ≤ k |ξ|

  b u (ξ) k L q

  µ |ξ|

  1

  1

  E(t, 0, ξ)e

  T

  e

  −|ξ|

Λ(t,0)

  e

  −1

  ih(t, ξ)

  k

  (ξ) k L q

  1

  b u

  − |ξ|

  b u

  ❯t✐❧✐③❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❛❝✐♠❛ ❡ ❧❡♠❜r❛♥❞♦ q✉❡ e = (1, 0)

  = e

  b u (ξ) + b u

  2δ

  |ξ|

  E(t, 0, ξ) ih(0, ξ) b u (ξ), µ

  T

  e

  −|ξ| Λ(t,0)

  − i h(t, ξ) e

  P♦rt❛♥t♦✱ b u(t, ξ) =

  t (0, ξ)].

  E(t, 0, ξ)[ih(0, ξ)w(0, ξ), w

  T

  |ξ| Λ(t,0)

  (ξ) , ✭✹✳✶✵✮

  E(t, 0, ξ)W (ξ) ⇒ ih(t, ξ) b u(t, ξ)e

  T

  E(t, 0, ξ)W (ξ) ⇒ ih(t, ξ)w(t, ξ) = e

  T

  W (t, ξ) = e

  T

  s❡q✉❡ ❞✐r❡t❛♠❡♥t❡ q✉❡ e

  T

  = (0, 1)

  1

  ❡ e

  T

  1

  ♦♥❞❡ ♥❡ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ❛s ❡①♣r❡ssõ❡s ♣❛r❛ w(0, ξ) ❡ w t (0, ξ)

  1

  −|ξ| Λ(t,0)

  E(t, 0, ξ)e

  T

  e

  −|ξ|

Λ(t,0)

  e

  −1

  ih(t, ξ)

  k

  − |ξ|

  E(t, 0, ξ)e b u (ξ)

  T

  e

  e

  ❞❛❞❛s ❡♠ ✭✷✳✹✮✳ ◆❛ ❡st✐♠❛t✐✈❛ ❛❜❛✐①♦ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦ ✉s❛♠♦s L

  −1

  h(0, ξ)h(t, ξ)

  k

  = k |ξ|

  b u(t, ξ) k L q

  k

  ❆ss✐♠✱ s❡❣✉❡ ❞❡ ✭✹✳✶✵✮ ❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r k |ξ|

  ) ✳

  M

  (B

  s

  = L

  s

  • k |ξ|
  • k |ξ|
  • k |ξ|

  k

  E(t, 0, ξ)e

  P❛r❛ ❡st✐♠❛r ❛s ♥♦r♠❛s ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✹✳✶✶✮✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♥❛ r❡❣✐ã♦ ❞❡ ❜❛✐①❛ ❢r❡q✉ê♥❝✐❛ ❞♦✐s ❝❛s♦s✿ ③♦♥❛ ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧ ❡ ③♦♥❛ ❡❧í♣t✐❝❛✳

  s❡♥❞♦ q✉❡ ❛ ú❧t✐♠❛ ❡st✐♠❛t✐✈❛ s❡❣✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍❛✉s❞♦r✛✲ ❨♦✉♥❣✳

  (R n )

  ku k L p

  L r

  k

  1

  T

  {(1 + t)

  e

  −|ξ|

Λ(t,0)

  e

  −1

  h(t, ξ)

  k+2δ

  L p (R n ) ✭✹✳✶✶✮

  k

  ❉❡✜♥✐♠♦s N (t) := min

  −

α +1

  ku

  00

  (t, 0, ξ) | . |ξ|

  00

  |E

  −1

  h(0, ξ)h(t, ξ)

  k

  ⇒ |ξ|

  (t, 0, ξ) | . h(t, ξ) h(0, ξ)

  |E

  N 1 , M } ≤ N(0) = M

  ) ✳ ◆♦t❡ q✉❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✹✱ t❡♠♦s q✉❡

  N (t)

  (B

  r

  ❈♦♥s✐❞❡r❡ L

  pd

  , (t, ξ) ∈ Z

  ✐st♦ é✱ |ξ| < N(t)✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ t < t ξ . ❈❛s♦ ✶✿ ❩♦♥❛ ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧✿ ξ ∈ B M

  1

  L r

  k

  T

  k |ξ|

  (R n ) .

  kb u k L p

  L r

  k

  1

  E(t, 0, ξ)e

  e

  h(0, ξ)h(t, ξ)

  −|ξ|

Λ(t,0)

  e

  −1

  h(t, ξ)

  k+2δ

  L p (R n )

  k r kb k

  k |ξ|

  −1

  k −1 −|ξ| Λ(t,0)

  h(t, ξ)

  1

  E(t, 0, ξ)e

  T

  e

  −|ξ| Λ(t,0)

  e

  −1

  k

  e

  (R n )

  ku k L p

  L r

  E(t, 0, ξ)e k

  T

  e

  −|ξ| Λ(t,0)

  T

  • k |ξ|

  k

  • k |ξ|

  ❡ |E | . h(t, ξ)(1 + t)

  k k

−1

  h(t, ξ) (t, 0, ξ) (1 + t)

  01

  ⇒ |ξ| |E | . |ξ| ❡♠ ♣❛rt✐❝✉❧❛r

  

k+2δ k 2δ

−1

  h(t, ξ)

  01 (t, 0, ξ) (1 + t)

  |ξ| |E | . |ξ| |ξ|

  k 2δ α+1

  (1 + t) ≤ |ξ| |ξ|

  k k N . .

  ≤ |ξ| |ξ| P♦❞❡♠♦s ❛✐♥❞❛ ❛✜r♠❛r q✉❡ ♥❛ ③♦♥❛ ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧

  Λ(t,0)

−|ξ|

  e ≈ 1. ❉❡ ❢❛t♦✱

  Z t

  2δ 2δ α

  Λ(t, 0) = µ(1 + τ ) dτ |ξ| |ξ|

  µ

  α+1 2δ

  (1 + t) ≤ |ξ|

  α + 1 µ

  N ≤

  α + 1 ≤ µN.

  Λ(t,0) −µN −|ξ|

  ❆ss✐♠✱ e ≤ e ≤ 1✳

  • k |ξ|
  • k |ξ|

  k

  − α +1

  ♣♦✐s N(t) . (1 + t)

  

−k

02

  ≤ t

  − α +1 k+ n r

  N (t) n r . (1 + t)

  k

  ) 1 r . N (t)

  N (t)

  volume (B

  dξ 1 r ≤ N(t)

  k

  kr

  |ξ|

  B N (t)

  = Z

  L r (B N (t) )

  k

  k

  ✭✹✳✶✷✮ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ N(t) ❡ q✉❡ |ξ| < N(t)✱ t❡♠♦s k |ξ|

  (R n ) .

  k L p

  ✳ ❆ss✐♠✱ ✉s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛ ❡♠ ✭✹✳✶✷✮ s❡❣✉❡ q✉❡ k |ξ|

  h(0, ξ)h(t, ξ)

  (B N (t) ) ku

  e

  k

  1

  ku

  L r (B N (t) )

  k

  1

  E(t, 0, ξ)e

  T

  e

  −|ξ| Λ(t,0)

  −1

  −1

  h(t, ξ)

  k

  L p (R n )

  ku k

  L r (B N (t) )

  E(t, 0, ξ)e k

  T

  e

  −|ξ| Λ(t,0)

  e

  1

  k L r

  L p (R n )

  

T

  −|ξ| Λ(t,0)

  e

  −1

  h(t, ξ)

  k

  L p (R n )

  ku k

  L r (B N (t) )

  E(t, 0, ξ)e k

  e

  

T

  −|ξ| Λ(t,0)

  e

  −1

  h(0, ξ)h(t, ξ)

  k

  ) ✿ k |ξ|

  N (t)

  (B

  r

  ❯t✐❧✐③❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ✈❛♠♦s ❡st✐♠❛r ♦s t❡r♠♦s q✉❡ ❛♣❛r❡✲ ❝❡♠ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✹✳✶✶✮ ❝♦♥s✐❞❡r❛♥❞♦ L

  e

  E(t, 0, ξ)e

  k

  1

  (B N (t) ) ku k L p (R n )

  k L r

  k

  k |ξ|

  L p (R n ) .

  ku k

  L r (B N (t) )

  k

  1

  E(t, 0, ξ)e

  T

  e

  −|ξ|

Λ(t,0)

  e

  −1

  h(t, ξ)

  k+2δ

  (R n )

  k L p

  1

  (B N (t) ) ku

  k L r

  • (1 + t) k |ξ|

  • k |ξ|
  • k |ξ|

  • (1 + t) t
  • t

  k

  e

  T

  E(t, 0, ξ)e k

  L r (B M \B N (t) )

  ku k

  L p (R n )

  h(t, ξ)

  e

  −1

  e

  −|ξ| Λ(t,0)

  e

  T

  E(t, 0, ξ)e

  −|ξ| Λ(t,0)

  −1

  k

  − α +1

  −|ξ| Λ(t,0)

  |E

  01 (t, 0, ξ)

  | . |ξ|

  k

  |ξ|

  e

  h(0, ξ)h(t, ξ)

  −|ξ| 2(σ−δ) Λ (t,0) .

  ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ ❢❡✐t♦ ♥♦ ❈❛s♦ ✶✱ ✈❛♠♦s ❡st✐♠❛r ♦s t❡r♠♦s ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✹✳✶✶✮ ❡♠ B M

  \ B

  N (t) ✉s❛♥❞♦ ❛s ❡st✐♠❛t✐✈❛s ❛❝✐♠❛✿

  k |ξ|

  k

  1

  L r (B M \B N (t) )

  −1

  e

  −|ξ| 2(σ−δ) Λ (t,0)

  k

  L r (B M \B N (t)

)

  ku k

  L p (R n )

  k− α +1

  −|ξ| 2(σ−δ) Λ (t,0)

  k

  k

  L r

(B

M \B N (t) )

  ku

  1

  k

  L p (R n ) .

  e

  k |ξ|

  ku

  e

  1

  k

  L p (R n )

  k+2δ

  h(t, ξ)

  −1

  −|ξ| Λ(t,0)

  L p (R n ) .

  e

  

T

  E(t, 0, ξ)e

  1

  k

  L r (B M \B N (t) )

  ku k

  e

  h(t, ξ)

  ✭✹✳✶✹✮

  ku k

  ku

  1

  k

  L p (R n )

  . t

  −k 02

  L p (R n )

  L p (R n )

  −k 12

  ku

  

1

  k

  L p (R n ) .

  ✭✹✳✶✸✮ ❈❛s♦ ✷✿ ❩♦♥❛ ❡❧í♣t✐❝❛✿ ξ ∈ B

  −k 02

  ku k

  , (t, ξ) ∈ Z

  T

  k+2δ

  h(t, ξ)

  −1

  e

  −|ξ|

Λ(t,0)

  e

  E(t, 0, ξ)e

  −k 02

  1

  k

  L r (B N (t) )

  ku k

  L p (R n )

  . t

  M

  low ell

  k

  e

  −|ξ| 2(σ−δ) Λ (t,0)

  ❡

  1 h(t, ξ) E

  low ell

  (t, 0, ξ)e

  1

  −|ξ|

Λ(t,0)

  k

  .

  |ξ|

  − α +1

  e

  −|ξ| 2(σ−δ) Λ (t,0)

  ⇒ |ξ|

  e

  (t, 0, ξ) | . |ξ|

  ❈♦♥s✐❞❡r❡ L

  . e

  r B M \B N (t) .

  P❡❧❛s ❡st✐♠❛t✐✈❛s ✭✹✳✶✮ ❡ ✭✹✳✷✮ s❡❣✉❡ q✉❡ h(0, ξ) h(t, ξ)

  E

  low ell

  (t, 0, ξ)e e

  −|ξ|

Λ(t,0)

  −|ξ| 2(σ−δ) Λ (t,0)

  00

  ⇒ |ξ|

  k

  h(0, ξ) h(t, ξ)

  −1

  e

  −|ξ| Λ(t,0)

  |E

  • k |ξ|
  • k |ξ|

  • k |ξ|

  P❛r❛ ❡st✐♠❛r 2δ 2(σ−δ) ♯

  j Λ (t,0) k− +1 −|ξ| α r

  k |ξ| k L (B M )

  \B N (t)

  ✈❛♠♦s ❝♦♥s✐❞❡r❛r q✉❛tr♦ ❝❛s♦s✿ j = 0 ♦✉ ✭✹✳✺✮ é ✈❡r❞❛❞❡✐r❛❀ j = 1✱ ✭✹✳✺✮ ♥ã♦ é ✈❡r❞❛❞❡✐r❛ ❡ r = +∞❀ j = 1✱ ✭✹✳✻✮ é ✈❡r❞❛❞❡✐r❛ ❡ r ∈ [1, +∞)❀ j = 1

  ✱ ✭✹✳✽✮ é ✈❡r❞❛❞❡✐r❛ ❡ r ∈ [1, +∞)✳ ❈❛s♦ ✷✳✶✿ j = 0 ♦✉ ✭✹✳✺✮ é ✈❡r❞❛❞❡✐r❛

  ❱❛♠♦s ❡st✐♠❛r 2δ 2(σ−δ) ♯

  j Λ (t,0) k− +1 −|ξ| α r

  e k |ξ| k L (B M (t) )

  \B N 1 Z 2δ ♯ 2(σ−δ) r r j Λ (t,0) k− −|ξ| α +1 = e dξ .

  |ξ|

  B M \B N (t)

  ❆♣❧✐❝❛♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ 1 η = ξ (Λ(t, 0)) t❡♠♦s k

  k k −

  = Λ(t, 0) |ξ| |η|

  ❡ ❝♦♠♦ (σ−δ)

  2(σ−δ) 2(σ−δ) δ

  = (Λ(t, 0)) −|η| −|ξ|

  ) α +1 .

  )

  −|ξ| 2(σ−δ) Λ (t,0)

  k

  L r (B M \B N (t) )

  = Z

  B M \B N (t)

  |ξ|

  k− α +1 j

  e

  −|ξ|

2(σ−δ)

Λ (t,0) r

  dξ 1 r . Λ(t, 0)

  − 1 2δ ( k− α +1 j

  − n 2δr

  k− α +1 j

  Z

  B M \B N (t)

  |η|

  k− α +1 j

  e

  −c|η| 2(σ−δ) r

  dη 1 r . Λ(t, 0)

  − ( n r +k− α +1 j

  ) 1 . (1 + t)

  −

  ( n r

  e

  ❉❡ss❛ ❢♦r♠❛✱ s❡♥❞♦ η = ξ (Λ(t, 0)) 1 t❡♠✲s❡ ♣❛r❛ t♦❞♦ t ≥ t q✉❡ k |ξ|

  s❡❣✉❡ q✉❡

  (1 + t)

  −|ξ| 2(σ−δ) Λ (t,0) −|η| 2(σ−δ) (Λ(t,0)) (σ−δ) δ Λ (t,0)

  ❯s❛♥❞♦ q✉❡ σ(1 + α) > 2δ s❡❣✉❡ ♣❛r❛ t♦❞♦ t ≥ t q✉❡ −|η|

  2(σ−δ)

  (Λ(t, 0))

  − (σ−δ) δ

  Λ

  ♯

  (t, 0) . −|η|

  2(σ−δ)

  (1 + t)

  − (σ−δ) δ (α+1)

  1−α

  −c|η| 2(σ−δ) .

  = −|η|

  2(σ−δ)

  (1 + t) σ (α+1)+δ(α+1)+δ (1−α) δ . −|η|

  2(σ−δ)

  (1 + t) 2δ−σ(α+1) δ ≤ −|η|

  2(σ−δ)

  , ❧♦❣♦✱ e

  −|ξ| 2(σ−δ) Λ (t,0)

  = e

  −|η| 2(σ−δ)

(Λ(t,0))

(σ−δ) δ Λ (t,0)

  ≤ e

  • k− α +1 j

  ◆♦t❡ q✉❡ ⇔

  . (1 + t)

  ❈♦♠♦ (t, ξ) ∈ Z

  low ell

  ✈❛❧❡ q✉❡ (1 + t)

  1+α

  |ξ|

  2δ

  ≥ N✳ ▲♦❣♦ |ξ|

  −2δ

  1+α

  k

  ⇒ |ξ|

  −1 .

  (1 + t) 1+α ⇒ |ξ|

  − 1+α −k

  . (1 + t) 1+α 1+α −k , ♣♦✐s θ = k +

  n r

  ≤

  2δ 1+α

  L (B M \B N (t) ) .

  −|ξ| 2(σ−δ)

Λ

(t,0)

  σ(α + 1) − δ(α + 1) > 2δ − δ(α + 1)

  e

  ⇔ (σ

  − δ)(α + 1) > 2δ − δ − δα ⇔

  α + 1 δ

  >

  1 − α

  σ − δ .

  ✭✹✳✶✺✮ P♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦ t ≥ t t❡♠✲s❡ k |ξ|

  k− α +1 j

  −|ξ| 2(σ−δ) Λ (t,0)

  e

  k L r

  (B M

\B N

(t) )

  . (1 + t)

  − ( n r +k− α +1 j

  ) 1−α 2(σ−δ) ≤ t

  −k j 1 .

  ✭✹✳✶✻✮ ❈❛s♦ ✷✳✷✿ j = 1✱ ✭✹✳✺✮ ♥ã♦ é ✈❡r❞❛❞❡✐r❛ ❡ r = +∞

  ◆❡ss❡ ❝❛s♦ q✉❡r❡♠♦s ❡st✐♠❛r k |ξ|

  k− α +1

  ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ✭✹✳✺✮ ♥ã♦ é s❛t✐s❢❡✐t❛✳

  ❆ss✐♠ 2δ (1+α)k 12 12

  

k− 1− −k −k

1+α 2δ . (1 + t) = (1 + t) . t .

  |ξ| P♦rt❛♥t♦✱ 2δ 2(σ−δ) ♯

  k− Λ (t,0)

1+α −|ξ| −k

12 e t .

  L (B ) . ✭✹✳✶✼✮

  k |ξ| k M \B N (t) ❈❛s♦ ✷✳✸✿ j = 1✱ ✭✹✳✻✮ é ✈❡r❞❛❞❡✐r❛ ❡ r ∈ [1, +∞)

  ❙❡ ✭✹✳✻✮ é ✈❡r❞❛❞❡✐r❛ ❡♥tã♦ ❡①✐st❡ ǫ > 0 t❛❧ q✉❡ n ǫ 2δ

  • k +

  ≤ r r 1 + α ❛ss✐♠✱

  2δ n ǫ

  • 1 + α r r
  • k + − ≤ 0,

  ❡ ♣♦r ✉♠ ❝á❧❝✉❧♦ ❢❡✐t♦ ❛❝✐♠❛ α +1

  −1

  . (1 + t) |ξ| 2δ α +1 2δ n ǫ n ǫ

  • k+ − ( −k− − 1+α r r 2δ 1+α r r
  • . (1 + t) ).

  ⇒ |ξ|

  • n r
  • ǫ r
  • n

    r

  • ǫ r

  . (1 + t)

  − α +1

r k−

1+α

  |ξ|

  −n−ǫ

  dξ 1 r = (1 + t)

  − α +1 2δ k− 1+α

  Z

  B M \B N (t)

  |ξ|

  −n−ǫ

  dξ 1 r . (1 + t)

  − α +1 k− 1+α

  − α +1 k+ n r +1

  B M \B N (t)

  . t

  −k 12

  , ✭✹✳✶✽✮

  ♦♥❞❡ ✉s❛♠♦s q✉❡ Z

  B M \B N (t)

  |ξ|

  −n−ǫ

  dξ 1 r ≤ C.

  ❈❛s♦ ✷✳✹✿ j = 1✱ ✭✹✳✽✮ é ✈❡r❞❛❞❡✐r❛ ❡ r ∈ [1, +∞) ◆❡ss❡ ❝❛s♦ t❡♠✲s❡ q✉❡ k + n r

  = 2δ 1 + α

  ⇒ |ξ|

  (1 + t)

  Z

  r −

1+α

  dξ 1 r .

  ❯s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛ ♦❜t❡♠♦s k |ξ|

  k− α +1 −|ξ| 2(σ−δ) Λ (t,0)

  k L r

  

(B M

\B N (t) )

  = Z

  B M \B N (t)

  |ξ|

  k− α +1 r

  e

  −|ξ| 2(σ−δ) Λ (t,0) r

  dξ 1 r =

  Z

  B M \B N (t)

  |ξ|

  r k− 1+α

n

r

ǫ r

  e

  −|ξ| 2(σ−δ) Λ (t,0) r

  dξ 1 r

  Z

  B M \B N (t)

  |ξ|

  r k− 1+α

  |ξ|

  −n−ǫ

  • n r
  • ǫ r
  • n r
  • ǫ r
  • n r
  • ǫ r

  • k+ n r = 1.
  • k+

    n

    r

    n r

  • k+

    n

    r

  ✱ ❛ss✉♠✐♥❞♦ ✭✹✳✺✮✱ ♣♦r ✭✹✳✶✹✮ ❡ ✭✹✳✶✻✮ s❡❣✉❡ q✉❡ k |ξ|

  e

  −|ξ| Λ(t,0)

  e

  −1

  h(0, ξ)h(t, ξ)

  k

  low ell

  E(t, 0, ξ)e k

  ❉❡♠♦♥str❛çã♦ ❞❛ ❡st✐♠❛t✐✈❛ ✭✹✳✹✮ P❛r❛ ❛ ③♦♥❛ ❡❧í♣t✐❝❛✱ ✐st♦ é✱ (t, ξ) ∈ Z

  , ∀ 0 < t < t .

  L p (R n )

  k

  1

  L p (R n )

  

T

  L r (B M \B N (t) )

  k

  E(t, 0, ξ)e

  k

  1

  ku

  L r (B M \B N (t) )

  k

  1

  

T

  ku k

  e

  −|ξ| Λ(t,0)

  e

  −1

  h(t, ξ)

  k

  L p (R n )

  b u(t, ξ) k L q ≤ K ku k

  ♦ r❡s✉❧t❛❞♦ é ✐♠❡❞✐❛t♦ ♣♦✐s ♣♦r ✭✹✳✶✶✮ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ K > 0 t❛❧ q✉❡ k |ξ|

  L p (R n )

  k− α +1 r

  |ξ|

  B M \B N (t)

  Z

  dξ 1 r =

  −|ξ| 2(σ−δ) Λ (t,0) r

  e

  |ξ|

  e

  B M \B N (t)

  = Z

  

(B M

\B N (t) )

  k L r

  k− α +1 −|ξ| 2(σ−δ) Λ (t,0)

  ❆ss✐♠✱ k |ξ|

  r − 1+α

  −|ξ| 2(σ−δ) Λ (t,0) r

  > 0 ✳ P❛r❛ 0 < t < t

  dξ 1 r

  ✭✹✳✶✾✮ ❯s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ♣♦❞❡♠♦s ✜♥❛❧♠❡♥t❡ ♣r♦✈❛r ❛s ❡st✐♠❛✲ t✐✈❛s ❞♦ t❡♦r❡♠❛✳ Pr♦✈❛♠♦s ❝❛❞❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ t ≥ t

  dξ 1 r . (log(1 + t)) 1 r .

  −n

  |ξ|

  B M \B N (t)

  Z

  −|ξ| 2(σ−δ) Λ (t,0) r

  dξ 1 r =

  (e

  −n

  |ξ|

  r − 1+α

  |ξ|

  B M \B N (t)

  Z

  • K ku
  • k |ξ|

  2δ k+2δ Λ(t,0) T −1 −|ξ| r p n

  • h(t, ξ) e e E(t, 0, ξ)e

  1 L (B ) L (R )

  k |ξ| k M \B N (t) ku k 2(σ−δ) ♯

  k Λ (t,0) −|ξ| r p n

  . e

  L (B ) L (R )

  k |ξ| k M \B N (t) ku k 2δ ♯ 2(σ−δ)

  Λ (t,0) k− −|ξ| α +1 e r p n

  • L (B )

  1 L (R )

  k |ξ| k M \B N (t) k ku

  −k 01 −k p n p n 11 t + t , .

  . L (R )

  1 L (R ) ✭✹✳✷✵✮

  ku k ku k ∀ t ≥ t ❯s❛♥❞♦ ✭✹✳✶✸✮ ❡ ✭✹✳✷✵✮ ❡♠ ✭✹✳✶✶✮ ❝♦♥❝❧✉í♠♦s q✉❡

  k q

  u(t, ξ) k |ξ| b k L (B M )

  k Λ(t,0) T −1 −|ξ| r p n

  h(0, ξ)h(t, ξ) e e E(t, 0, ξ)e

  L (B ) L (R )

  ≤ k |ξ| k N (t) ku k

  k Λ(t,0) T −1 −|ξ|

  • 1 L (B )

  h(t, ξ) e e E(t, 0, ξ)e r p n

  1 L (R )

  k |ξ| k N (t) ku k

  k+2δ Λ(t,0) T −1 −|ξ|

  • r p n

  h(t, ξ) e e E(t, 0, ξ)e

  1 L (B ) L (R )

  k |ξ| k N (t) ku k

  k Λ(t,0) T −1 −|ξ|

  • L (B ) L (R )

  h(0, ξ)h(t, ξ) e e E(t, 0, ξ)e r p n

  k |ξ| k M \B N (t) ku k

  k Λ(t,0) T −1 −|ξ| r p n

  • h(t, ξ) e e E(t, 0, ξ)e

  1 L (B )

  1 L (R )

  k |ξ| k M \B N (t) ku k

  k+2δ Λ(t,0) T −1 −|ξ|

  • 1 L (B ) L (R ) −k 02 −k p n p n
  • 12

      h(t, ξ) e e E(t, 0, ξ)e r p n k |ξ| k M \B N (t) ku k

      t + t . L (R )

      01 ku k ku k 11

      −k −k p n p n +t + t , .

      L (R )

      ku k ku k ∀ t ≥ t ◆♦t❡ q✉❡ s❡❣✉❡ ❞❡ ✭✹✳✶✺✮ q✉❡ 1 1 + α

      − α <

      2(σ 2δ − δ) 1 1 + α

      − α θ < θ

      ⇒ 2(σ 2δ

      − δ) k < k .

      01 02 ✭✹✳✷✷✮

      ⇒

    • Ct

      e

      e

      k

      k |ξ|

      (B M \B N (t) ) ku k L p (R n ) .

      k L r

      1

      E(t, 0, ξ)e

      

    T

      −|ξ| Λ(t,0)

      k

      e

      −1

      h(t, ξ)

      k+2δ

      L p (R n )

      k

      1

      ku

      −|ξ| 2(σ−δ) Λ (t,0)

      L r (B M \B N (t) )

      k

      . t

      L

    p

    (R

    n ) ✭✹✳✷✸✮

      k

      1

      ku

      −k 12

      L p (R n )

      ku k

      −k 01

      (R n )

      ku k

      k L p

      1

      (B M \B N (t) ) ku

      k L r

      −|ξ| 2(σ−δ) Λ (t,0)

      e

      k− α +1

      L p (R n )

      L r (B M \B N (t) )

      1

      ♣❛r❛ t♦❞♦ t ≥ t ✳

      11

      L p (R n )

      ku k

      (B M ) ≤ Ct −k 01

      b u(t, ξ) k L q

      k

      ❯s❛♥❞♦ ❛s ♦❜s❡r✈❛çõ❡s ❛❝✐♠❛ ❡♠ ✭✹✳✷✶✮ ♦❜t❡♠♦s ♣❛r❛ t♦❞♦ t ≥ t ✿ k |ξ|

      12 .

      < k

      α + 1 ⇒ k

      ku

      − 2δ

      α + 1 1 + α 2δ

      − 2δ

      2(σ − δ)

      1 − α

      > 0, ❛ss✐♠

      2δ α+1

      ❆✐♥❞❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ✭✹✳✺✮ t❡♠♦s θ −

      −k 11

      1

      E(t, 0, ξ)e

      E(t, 0, ξ)e k L r

      T

      e

      −|ξ| Λ(t,0)

      e

      −1

      h(t, ξ)

      k

      (B M \B N (t) ) ku k L p (R n )

      T

      k

      e

      −|ξ| Λ(t,0)

      e

      −1

      h(0, ξ)h(t, ξ)

      k

      P❛r❛ ❛ ③♦♥❛ ❡❧í♣t✐❝❛ ❛ss✉♠✐♥❞♦ ✭✹✳✻✮✱ ♣♦r ✭✹✳✶✹✮✱ ✭✹✳✶✻✮ ♣❛r❛ j = 0✱ ✭✹✳✶✼✮ ♣❛r❛ j = 1 ❡ r = +∞✱ ✭✹✳✶✽✮ ♣❛r❛ j = 1 ❡ r ∈ [1, +∞) s❡❣✉❡ q✉❡ k |ξ|

      , ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❡ ✭✹✳✹✮✳ ❉❡♠♦♥str❛çã♦ ❞❛ ❡st✐♠❛t✐✈❛ ✭✹✳✼✮

      L p (R n )

    • k |ξ|
    • k |ξ|
    • k |ξ|
    • t
    • k |ξ|

    • k |ξ|
    • k |ξ|

      −1

      ku k

      −k 02

      . t

      (B M \B N (t) ) ku k L p (R n )

      k L r

      1

      E(t, 0, ξ)e

      T

      e

      −|ξ| Λ(t,0)

      e

      h(t, ξ)

      −k 12

      k+2δ

      L p (R n )

      k

      1

      ku

      L r (B M \B N (t) )

      k

      1

      E(t, 0, ξ)e

      T

      e

      L p (R n )

      ku

      e

      b u(t, ξ) k L q

      , ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❡ ✭✹✳✼✮✳ ❉❡♠♦♥str❛çã♦ ❞❛ ❡st✐♠❛t✐✈❛ ✭✹✳✾✮

      L p (R n )

      k

      1

      ku

      −k 12

      L p (R n )

      ku k

      −k 01

      . t

      (B M )

      k

      1

      ♣❛r❛ t♦❞♦ t ≥ t ✳ ❯s❛♥❞♦ ✭✹✳✷✷✮ ❡♠ ✭✹✳✷✹✮ t❡♠♦s ♣❛r❛ t ≥ t q✉❡ k |ξ|

      

    (R

    n

    ) ✭✹✳✷✹✮

      k L p

      1

      ku

      −k 12

      (R n )

      ku k L p

      −k 01

      L p

    (R

    n

    )

      k

      −|ξ| Λ(t,0)

      −1

      P❛r❛ ❛ ③♦♥❛ ❡❧í♣t✐❝❛ ❛ss✉♠✐♥❞♦ ✭✹✳✽✮✱ ♣♦r ✭✹✳✶✹✮✱ ✭✹✳✶✻✮ ♣❛r❛ j = 0✱ ✭✹✳✶✼✮ ♣❛r❛ j = 1 ❡ r = +∞✱ ✭✹✳✶✾✮ ♣❛r❛ j = 1 ❡ r ∈ [1, +∞) s❡❣✉❡ q✉❡

      E(t, 0, ξ)e k

      E(t, 0, ξ)e

      T

      e

      −|ξ| Λ(t,0)

      e

      −1

      h(t, ξ)

      k

      L p (R n )

      ku k

      L r (B N (t) )

      

    T

      k L r

      e

      −|ξ| Λ(t,0)

      e

      −1

      h(0, ξ)h(t, ξ)

      k

      ≤ k |ξ|

      L q (B M )

      b k

      k

      ❯s❛♥❞♦ ✭✹✳✶✸✮ ❡ ✭✹✳✷✸✮ ❡♠ ✭✹✳✶✶✮ ❝♦♥❝❧✉í♠♦s q✉❡ k |ξ|

      1

      (B N (t) ) ku

      h(t, ξ)

      ku k

      k

      (B M \B N (t) ) ku k L p (R n )

      E(t, 0, ξ)e k L r

      

    T

      e

      −|ξ| Λ(t,0)

      e

      −1

      h(0, ξ)h(t, ξ)

      k

      L p (R n )

      L r (B N (t) )

      1

      k

      1

      E(t, 0, ξ)e

      T

      e

      −|ξ| Λ(t,0)

      e

      −1

      h(t, ξ)

      k+2δ

      (R n )

      k L p

    • k |ξ|
    • k |ξ|
    • t
    • t
    • t
    • t

    • k |ξ|
    • k |ξ|
    • k |ξ|
    • (log(1 + t))
    • 1 r ku

        e

        ku k

        L p (R n )

        k

        h(0, ξ)h(t, ξ)

        −1

        e

        −|ξ| Λ(t,0)

        T

        k

        E(t, 0, ξ)e k

        L r (B M \B N (t) )

        ku k

        L p (R n )

        k

        h(t, ξ)

        −1

        L r (B N (t) )

        1

        −|ξ| Λ(t,0)

        k

        T

        E(t, 0, ξ)e

        1

        k

        L r (B N (t) )

        ku

        1

        L p (R n )

        E(t, 0, ξ)e

        k+2δ

        h(t, ξ)

        −1

        e

        −|ξ| Λ(t,0)

        e

        T

        e

        e

        −|ξ| Λ(t,0)

        1

        L p (R n )

        . t

        −k 02

        ku k

        L p (R n )

        −k 12

        ku

        k

        L r (B M \B N (t) )

        L p

      (R

      n

      )

        −k 01

        ku k

        L p (R n )

        1

        k

        L p (R n ) .

        ku k

        k

        T

        L p (R n )

        E(t, 0, ξ)e

        1

        k

        L r (B M \B N (t) )

        ku

        1

        k

        k+2δ

        1

        h(t, ξ)

        −1

        e

        −|ξ| Λ(t,0)

        e

        T

        E(t, 0, ξ)e

        e

        e

        ✭✹✳✷✻✮

        E(t, 0, ξ)e

        k+2δ

        h(t, ξ)

        −1

        e

        −|ξ| Λ(t,0)

        e

        

      T

        1

        k

        k L r

        (B M \B N (t) ) ku k L p (R n ) .

        k |ξ|

        k

        e

        −|ξ| 2(σ−δ) Λ (t,0)

        k

        L p (R n )

        1

        ku k

        −1

        k |ξ|

        k −1 −|ξ| Λ(t,0)

        T

        k L r

        (B M \B N (t) ) ku k L p (R n )

        k

        h(t, ξ)

        e

        ku

        −|ξ| Λ(t,0)

        e

        T

        E(t, 0, ξ)e

        1

        k

        L r (B M \B N (t) )

        L r (B M \B N (t) )

        L p (R n )

        −1

        −|ξ| Λ(t,0)

        b u(t, ξ) k L q

        (B M )

        ≤ k |ξ|

        k

        h(0, ξ)h(t, ξ)

        −1

        e

        e

        ✳ ❯s❛♥❞♦ ✭✹✳✶✸✮ ❡ ✭✹✳✷✺✮ ❡♠ ✭✹✳✶✶✮ ❝♦♥❝❧✉í♠♦s q✉❡ ♣❛r❛ t♦❞♦ t ≥ t ✿ k |ξ|

        

      T

        E(t, 0, ξ)e k

        L r (B N (t) )

        ku k

        L p (R n )

        k

        h(t, ξ)

        k

        = 1 ≤ (log(1 + t)) 1 r

        k− α +1

        . t

        e

        −|ξ| 2(σ−δ) Λ (t,0)

        k L r

        (B M \B N (t) ) ku

        1

        k L p

        (R n )

        −k 01

        −k 12

        ku k

        L p (R n )

        1

        k

        L p (R n ) ✭✹✳✷✺✮

        ♣❛r❛ t♦❞♦ t ≥ t > 0

        ✱ ♣♦✐s ♥❡ss❡ ❝❛s♦ ❛ss✉♠✐♥❞♦ ✭✹✳✽✮ t❡♠✲s❡ t

      • k |ξ|
      • k |ξ|
      • k |ξ|
      • k |ξ|
      • k |ξ|
      • t
      • t
      • (log(1 + t))
      • 1 r ku

          ❯s❛♥❞♦ ✭✹✳✷✷✮ ❡♠ ✭✹✳✷✻✮ t❡♠♦s ♣❛r❛ t ≥ t q✉❡ 1

          k ′ r 01 q −k p n p n

          k |ξ| b k ku k L (R ) ku k L (R )

          L (B M )

          > 0 ♣❛r❛ t♦❞♦ t ≥ t ✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❡ ✭✹✳✾✮✳

          ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✹✳✶ ❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✸ ♣r♦✈❛✲ ♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞♦ tr❛❜❛❧❤♦✳

          α

          ❚❡♦r❡♠❛ ✹✳✷ ❙❡❥❛ σ > 0✱ δ ∈ (0, σ)✱ b(t) = µ(1 + t) ❝♦♠ µ > 0 ❡ α

          ∈ (0, 1)✱ ❡ ❛ss✉♠❛ q✉❡ 2δ < σ(1 + α)✳ ❙❡❥❛ 1 ≤ p ≤ 2 ❡ k > 0✳ ❉❡✜♥✐♠♦s

          1

          1 ¯

          θ(k, n, p) := k + n , − p

          2 ❡ 1 2δ

          − α ¯ k = θ ,

          j1

          − j 2(σ 1 + α

          − δ) 1 + α 1 + α 2δ ¯ ¯ k = θ θ ,

          j2

          − j = − j 2δ 2δ 1 + α ♣❛r❛ j = 0, 1.

          ❊♥tã♦ ❛ s♦❧✉çã♦ ♣❛r❛ ✭✷✳✶✮ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ót✐♠❛ k 2 −k −k 2 n p n p n 01 11 u(t) + Ct

          L (R ) L (R )

          1 L (R )

          k(−∆) k ≤ Ct ku k ku k 1−α k n k−σ n −C (1+t)

        • e

          H (R )

          1 H (R )

          ku k ku k

        • Ct
        • ku k
        • ku

        • C(log(1 + t))
        • 1 p<
        • ku k H k
        • ku
        • k |ξ|

          b f t❡♠♦s k(−∆) k 2 u(t) k

          2δ

          |ξ|

          −1

          = F

          f .

          ❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P❧❛♥❝❤❡r❡❧ ❡ ❛ ✐❞❡♥t✐❞❛❞❡ (−∆)

          δ

          −C (1+t) 1−α .

          e

          (R n )

          k H k−σ

          1

          

        (R

        n )

          L 2 (R n )

          = kF

          −1

          ( |ξ|

          

        k

          b u(t)) k

          L 2 (R n ) .

          k |ξ|

          k

          b u(t) k

          L 2 (R n )

          ≤ k |ξ|

          k

          b u(t) k L 2

          (B M )

          

        k

          b u(t) k L 2

          L p (R n )

          1

          k

          k

          ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ t♦❞♦ t &gt; 0✱ ❞❛❞♦ q✉❡ k

          11

          &gt; 0 ✱ ✐st♦ é✱

          ¯ 2δ 1 + α

          P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ¯

          θ &lt; 2δ 1 + α t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ♣❛r❛ t♦❞♦ t &gt; 0✿ k(−∆) k 2 u(t) k

          L 2 (R n )

          ≤ Ct

          −k 01

          ku k

          

        L

        p (R n )

          −k 12

          ku

          1

          L p (R n )

          ku

          ♦❜té♠✲s❡ ♣❛r❛ t♦❞♦ t &gt; 0✿ k(−∆) k 2 u(t) k

          − 1 q

          

        L

        p (R n )

          ku k

          −k 01

          ≤ Ct

          L 2 (R n )

          2δ 1+α

          H k

        (R

        n )

          ◆♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ¯θ =

          −C (1+t) 1−α .

          e

          H k−σ (R n )

          k

          1

          (B C M ) .

          ❊s❝♦❧❤❡♥❞♦ γ = 2k − 2σ ❡♠ ✭✸✳✺✮ ♦❜t❡♠♦s Z

          2k

          2

          |ξ| |b |

          |ξ|≥M

          Z Z 1−α

          2 2k

          2 2k−2σ −C (1+t)

          . u dξ + u dξ e

          n n |ξ| |b | |ξ| |b | ❧♦❣♦ 1−α

          k 2 C k n k−σ n −C (1+t) u(t) . e + .

          L (B ) H (R )

          1 H (R )

          k |ξ| b k ku k ku k M ❖ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛ ❡ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✳

          ❆♣ê♥❞✐❝❡ ❆ ❘❡s✉❧t❛❞♦s ❆❞✐❝✐♦♥❛✐s

          Z

              

          3

          , ξ) dτ

          3

          h(τ

          τ 1 τ 2

          

        −2

          ◆❡st❡ ❛♣ê♥❞✐❝❡ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ♦✉tr♦s r❡s✉❧t❛❞♦s q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♣r♦✈❛r r❡s✉❧t❛❞♦s ❞♦ ♣❡r✜❧ ❛ss✐♥tót✐❝♦ ❞❛ ❡q✉❛çã♦ ✭✶✮✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♣❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ✷ ❞❛❞♦ ❡♠ ❬✻❪✱ q✉❡ ♣r♦✈❛ q✉❡ ❛ s♦❧✉çã♦ ❞❡ ✭✶✮ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❞✐❢✉sã♦ ❛♥ô♠❛❧❛✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ✷ ♥ã♦ ❝♦♥t❛ ♥❡st❡ ❛♣ê♥❞✐❝❡✳

          1 0 e

          , ξ) =     

          2

          , τ

          1

          ♦♥❞❡ H(τ

          ❙❡❥❛ Q(t, s, ξ) = H(t, s, ξ) e Q(t, s, ξ)

          ❡ Q(t, s, ξ) s♦❧✉çã♦ ❞❡ ✭✷✳✹✶✮✳ ◆♦t❡ q✉❡ H(t, s, ξ)H(s, t, ξ) = I. ❉❡ ❢❛t♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ H(t, s, ξ)✱ t❡♠♦s  

           

          1

          1  

            Z t Z s

             

          H(t, s, ξ)H(s, t, ξ) =  

           

          −2 h(τ, ξ) dτ −2 h(τ, ξ) dτ

           

          s t

          0 e e  

          1  

          Z t Z s  

          =  

          −2 h(τ, ξ) dτ −2 h(τ, ξ) dτ

           

          s t

          0 e e  

          1    Z t Z t 

          =  

          −2 h(τ, ξ) dτ + 2 h(τ, ξ) dτ

           

          s s

          0 e   1 0  

          =   = I.

          0 1 ❆ss✐♠✱ ♣❡❧❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛

          Q(t, s, ξ) = H(t, s, ξ) e Q(t, s, ξ)

          −1

          Q(t, s, ξ) = H(t, s, ξ) Q(t, s, ξ) ⇒ e Q(t, s, ξ) = H(s, t, ξ)Q(t, s, ξ).

          ⇒ e ❊ ❡♥tã♦ Q e (t, s, ξ) = H (s, t, ξ)Q(t, s, ξ) + H(s, t, ξ)Q (t, s, ξ).

          t t t ✭❆✳✶✮

          ❙✉❜st✐t✉✐♥❞♦ ✭✷✳✹✶✮ ❡♠ ✭❆✳✶✮ ♦❜t❡♠♦s    

          Z s e   e Q (t, s, ξ) = Q(t, s, ξ)

          t

            H(t, s, ξ)

          −2 h(τ, ξ) dτ t

          0 2h(t, ξ) e     0 0    

        • H(s, t, ξ) (t, ξ) Q(t, s, ξ)

          2

          −2h(t, ξ)   + R  H(t, s, ξ) e 0 1      

          Z s =   

          2h(t, ξ)   H(t, s, ξ) h(τ, ξ) dτ

          −2 t

          0 e     0 0    

        • H(s, t, ξ)

          −2h(t, ξ)   H(t, s, ξ)  0 1 # e

        • H(s, t, ξ)R

          

        2 (t, ξ)H(t, s, ξ) Q(t, s, ξ)

              

          1   

           

          s

          Z Z t   

          = 2h(t, ξ)     

            h(τ, ξ) dτ

          −2 −2 h(τ, ξ) dτ

            

          t s

          0 e e  

             

          1

          1   0 0

            Z s

             Z t   

          − 2h(t, ξ)    

           

          −2 h(τ, ξ) dτ −2 h(τ, ξ) dτ

            0 1

          t s

          e 0 e

        • H(s, t, ξ)R (t, ξ)H(t, s, ξ)] e Q(t, s, ξ)

          2

               0 0 0 0     

          = 2h(t, ξ)   − 2h(t, ξ)   0 1 0 1

        • H(s, t, ξ)R (t, ξ)H(t, s, ξ)] e Q(t, s, ξ)

          2 = [H(s, t, ξ)R (t, ξ)H(t, s, ξ)] e Q(t, s, ξ).

          2

          . , τ , ξ) = H(τ , τ , ξ)R (τ , ξ)H(τ , τ , ξ). ❉❡✜♥❛ A(τ

          1

          2

          2

          1

          2

          1

          1 2 ❆ss✐♠✱ ❛ ✐❣✉❛❧✲

          ❞❛❞❡ ❛❝✐♠❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❞❛ ❢♦r♠❛ e Q (t, s, ξ) = A(t, s, ξ) e Q(t, s, ξ).

          t

          Q(s, s, ξ) = ◆♦t❡ q✉❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ H(t, s, ξ) ❡ ✭✷✳✹✶✮ s❡❣✉❡ q✉❡ e H(s, s, ξ)Q(s, s, ξ) = I.

          Q(t, s, ξ) Q(t, s, ξ) P♦rt❛♥t♦✱ t❡♠♦s q✉❡ Q(t, s, ξ) = H(t, s, ξ) e ❝♦♠ e s♦❧✉çã♦ ❞♦ s✐st❡♠❛

             Q e (t, s, ξ) = A(t, s, ξ) e Q(t, s, ξ), t &gt; s

          t

            e Q(s, s, ξ) = I.

          ❙❡♥❞♦ q✉❡ A(t, s, ξ) ∈ L ✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✺✱ e é ❞❛❞♦ ♣♦r

          loc ∞ Z t Z τ 1 Z τ k− 1 X

          e Q(t, s, ξ) = I + A(τ , s, ξ) A(τ , s, ξ) A(τ , s, ξ) dτ . . . dτ

          1 2 k k

          1

          · · ·

          

        s s s

        k=1

          = t ❝♦♠ τ ✳ ▲♦❣♦

          " Q ell (t, s, ξ) = H(t, s, ξ) I +

          #

          ∞

          Z t Z τ 1 Z τ k− 1 X

        • A(τ

          1 , s, ξ) A(τ 2 , s, ξ) A(τ k , s, ξ) dτ k . . . dτ

          1

          · · ·

          s s s k=1

          " Z t

          = H(t, s, ξ) I + A(τ , s, ξ) dτ

          1

          1 s

          # Z t Z τ 1 A(τ , s, ξ) A(τ + , s, ξ) dτ dτ + . . .

          1

          

        2

          2

          1 s s Z t = H(t, s, ξ) + H(t, s, ξ)A(τ

          1 , s, ξ) dτ

          1 s

          Z t Z τ 1 + H(t, s, ξ)A(τ , s, ξ) A(τ , s, ξ) dτ dτ + . . .

          1

          2

          2 1 ✭❆✳✷✮ s s

          ❱❛♠♦s ❡s❝r❡✈❡r ❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ s❡❣✉♥❞♦ ❡ ♦ t❡r❝❡✐r♦ t❡r♠♦ ❞❡ss❛ s♦♠❛✱ ♦s ♦✉tr♦s s❡❣✉❡♠ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦t❡ q✉❡

             

          1

          1  

            Z s

           Z t   

          H(t, s, ξ)H(s, τ, ξ) =  

            h(τ , ξ) dτ −2 h(τ , ξ) dτ  −2

          

        1

        1 

          1

          1 τ s

          e 0 e  

          1  

          Z t Z s  

          =  

          −2 h(τ , ξ) dτ h(τ , ξ) dτ

          

          

        1

          1

          1 1 

          − 2

          s τ

          e  

          1  

          Z t   = = H(t, τ, ξ).

           

          −2 h(τ , ξ) dτ

          

          

        1

        1  τ

          0 e , s, ξ)

          ❆ss✐♠✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ A(τ

          1 ❡ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♣♦❞❡♠♦s

          ❡s❝r❡✈❡r ♦ s❡❣✉♥❞♦ t❡r♠♦ ❞❛ ❢♦r♠❛ Z t Z t

          H(t, s, ξ)A(τ , s, ξ) dτ = H(t, s, ξ)H(s, τ , ξ)R (τ , ξ)H(τ , s, ξ) dτ

          1

          1

          1

          2

          1

          1

          1 s s

          Z t = H(t, τ

          1 , ξ)R 2 (τ 1 , ξ)H(τ 1 , s, ξ) dτ 1 . s

          1 .

          e A(t, τ

          , s, ξ) dτ

          2

          dτ

          1

          = Z t

          s

          1

          , ξ)H(τ

          , ξ) Z τ 1

          s

          e A(τ

          1

          , τ

          2

          , ξ)H(τ

          2

          2

          , s, ξ) dτ

          1

          2

          dτ

          1

          = Z

          t s

          A(t, τ e

          , ξ) Z

          (τ

          τ 1 s

          H(τ

          1

          , τ

          2

          , ξ)R

          2

          2

          2

          2

          s

          dτ

          1 + . . .

          = H(t, s, ξ) + Z t

          s

          e A(t, τ

          1 , ξ) H(τ 1 , s, ξ)

          A(τ e

          , s, ξ) dτ

          1

          , τ

          2

          , ξ)H(τ

          2

          , s, ξ) dτ

          2

          2

          2

          dτ

          s

          1 .

          ✭❆✳✹✮ ❙✉❜st✐t✉✐♥❞♦ ✭❆✳✸✮ ❡ ✭❆✳✹✮ ❡♠ ✭❆✳✷✮ t❡♠♦s

          Q ell (t, s, ξ) = H(t, s, ξ) + Z t

          s

          e A(t, τ

          1 , ξ)H(τ 1 , s, ξ) dτ

          1

          e A(t, τ

          , ξ)H(τ

          1

          , ξ) Z τ 1

          s

          e A(τ

          1

          , τ

          2

          , s, ξ) dτ

          , ξ)H(τ

          ❙❡❥❛ e A(τ

          s

          Z τ 1

          s

          A(τ

          2 , s, ξ) dτ 2 dτ

          1

          = Z t

          H(t, s, ξ)H(s, τ

          H(t, s, ξ)A(τ

          1

          , ξ)R

          2

          (τ

          

        1

          , ξ)H(τ

          1

          1 , s, ξ)

          s

          s

          , ξ)R

          1

          , τ

          2

          , ξ) = H(τ

          1

          , τ

          2

          

        2

          ❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ♦ t❡r❝❡✐r♦ t❡r♠♦ s❡❣✉❡ q✉❡ Z t

          (τ

          2 , ξ).

          ▲♦❣♦✱ Z t

          s

          Z t

          

        s

          e ✭❆✳✸✮

          , s, ξ) Z τ 1

          A(τ

          2

          , ξ) Z τ 1

          dτ

          1

          = Z t

          s

          e A(t, τ

          1

          s

          , s, ξ) dτ

          H(τ

          1

          , s, ξ)H(s, τ

          2

          , ξ)R

          2

          (τ

          2

          2

          2

          1

          , s, ξ) dτ

          2

          dτ

          1

          = Z t

          s

          H(t, τ

          , ξ)R

          A(τ

          2

          (τ

          1

          , ξ)H(τ

          1

          , s, ξ) Z τ 1

          s

        • Z t
        • . . . dτ
        • Z τ
        • 1 h(τ, ξ) dτ      .

            s

            ∞)R

            2

            , s, ξ) dτ

            2

            dτ

            1 + . . .

            s❡♥❞♦ H(

            ∞) =    1 0

            0 0    , e

            A( ∞, τ

            1 , ξ) = H(

            2 (τ 1 , ξ).

            2

            ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ❞❡ Q ell (

            ∞, s, ξ) é ♥✉❧❛✳ ❉❡♠♦♥str❛çã♦✿

            P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ H(t, s, ξ) t❡♠✲s❡ lim

            t→∞

            H(t, s, ξ) =     

            1 lim

            

          t→∞

            e

            −2

            Z t

            , ξ)H(τ

            , τ

            ❊♥tã♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❞❛ ❢♦r♠❛ Z t

            ell

            s

            e ✭❆✳✺✮

            Pr♦♣♦s✐çã♦ ❆✳✶ P❛r❛ t♦❞♦ (s, ξ) ∈ Z

            low ell

            ❡①✐st❡ Q ell (

            ∞, s, ξ) .

            = lim

            t→∞

            Q ell (t, s, ξ) ❞❡✜♥✐❞♦ ♣♦r

            Q

            ( ∞, s, ξ) = H(∞) +

            1

            Z

            ∞ s

            e A(

            ∞, τ, ξ)H(τ, s, ξ) dτ

            ∞ s

            e A(

            ∞, τ

            1

            , ξ) Z τ 1

            

          s

            e A(τ

          • Z

            1 + . . .

            2

            P♦r ✭❆✳✻✮✱ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ❞❡ Q

            2 , ξ).

            (τ

            2

            , ξ) = H( ∞)R

            2

            (τ

            2

            , ξ)R

            H(t, τ

            ∞, s, ξ) é ♥✉❧❛✳ ❉❡ ❢❛t♦✱ Q

            t→∞

            , ξ) = lim

            2

            e A(t, τ

            t→∞

            = lim

            , ξ) .

            2

            A( ∞, τ

            , ξ) ❡ ❞♦s ❝á❧❝✉❧♦s ❛❝✐♠❛ q✉❡ e

            ell (

            ell

            , τ

            

          s

            dτ

            2

            , s, ξ) dτ

            2

            , ξ)H(τ

            2

            , τ

            1

            e A(τ

            , ξ) Z τ 1

            ( ∞, s, ξ) = H(∞) +

            1

            (τ

            2

            H( ∞)R

            ∞ s

            (τ, ξ)H(τ, s, ξ) dτ

            

          2

            H( ∞)R

            ∞ s

            Z

            2

            1

            ❉❡s❡♥✈♦❧✈❡♥❞♦ ♦ ❡❧❡♠❡♥t♦ a

            Z t

            α

            (1 + τ )

            s

            Z t

            t→∞

            lim

            −2C |ξ|

            h(τ, ξ) dτ ≤ e

            s

            

          t→∞

            −2C |ξ|

            lim

            −2

            h(τ, ξ) dτ = e

            s

            Z t

            −2

            e

            t→∞

            lim

            22 ❞❛ ♠❛tr✐③ ❛❝✐♠❛✱ t❡♠♦s

            dτ = e

            lim

            A(τ

            2C |ξ|

            ✭❆✳✻✮ ❊✱ ♣♦rt❛♥t♦✱ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ e

            0 0   

            H(t, s, ξ) =    1 0

            t→∞

            ∞) = lim

            ▲♦❣♦ H(

            α + 1 = 0.

            α+1

            (1 + s)

            α + 1 e

            t→∞

            α+1

            (1 + t)

            t→∞

            lim

            −2C |ξ|

            ! = e

            t s

            α + 1

            α+1

            (1 + τ )

          • Z

                Z

            ∞

            1 0 1 0    

            = (τ, ξ)H(τ, s, ξ) dτ +

            2

              +   R

            s

            0 0 0 0 Z Z τ

            ∞ 1

            1 0   (τ + , ξ) A(τ e , τ , ξ)H(τ , s, ξ) dτ dτ + . . .

            2

            1

            1

            2

            2

            2

            1

             R

            s s

            0 0 ◆❛ s❡❣✉♥❞❛ ✐❣✉❛❧❞❛❞❡✱ ❛ ✐♥t❡❣r❛❧ ❞❛ s❡❣✉♥❞❛ ♣❛r❝❡❧❛ é ✉♠❛ ♠❛tr✐③ 2×2✱

              1 0  

            (τ, ξ)H(τ, s, ξ) ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞♦ ♣r♦❞✉t♦ ❞❡

            2 ✱

              ♣❡❧❛ ♠❛tr✐③ R 0 0 r❡s✉❧t❛♥❞♦ ♥❛ ✐♥t❡❣r❛❧ ❞❡ ✉♠❛ ♠❛tr✐③ 2 × 2 ❝♦♠ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♥✉❧❛✳

              1 0  

            ◆❛ t❡r❝❡✐r❛ ♣❛r❝❡❧❛✱ t❡♠♦s ♦ ♣r♦❞✉t♦ ❞❛ ♠❛tr✐③   ♣❡❧❛ ♠❛tr✐③ 0 0

            R τ 1 e

            2

            2 (τ 1 , ξ) A(τ 1 , τ 2 , ξ)H(τ 2 , s, ξ) dτ 2 dτ

            1

            × 2✱ R ✳ P♦rt❛♥t♦ ❛ t❡r❝❡✐r❛

            s

            ♣❛r❝❡❧❛ s❡rá ❛ ✐♥t❡❣r❛❧ ❞❡ ✉♠❛ ♠❛tr✐③ 2 × 2 ❝♦♠ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♥✉❧❛✳ ❙❡❣✉✐♥❞♦ ❡ss❡ r❛❝✐♦❝í♥✐♦✱ t❡♠♦s ✉♠❛ s♦♠❛ ✐♥✜♥✐t❛ ❞❡ ♠❛tr✐③❡s ♦♥❞❡ t♦✲

            ( ❞❛s ♣♦ss✉❡♠ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♥✉❧❛✱ ♣♦rt❛♥t♦✱ Q ell

            ∞, s, ξ) é ✉♠❛ ♠❛tr✐③

            2 × 2 ❝♦♠ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳

            ( (t, s, ξ) ❱❛♠♦s ♠♦str❛r q✉❡ |Q ell ell

            ∞, s, ξ)−Q | ✈❛✐ ♣❛r❛ ③❡r♦✱ q✉❛♥❞♦ t → ∞✳ ❈♦♠♦ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ♠❛tr✐③❡s 2 × 2✱ ✈❛♠♦s ❝♦♥s✐✲

            T T

            = (1, 0) = (0, 1) ❞❡r❛r e ❡ e

            1 ❡ ♠♦str❛r q✉❡ ❛ ♣r✐♠❡✐r❛ ❡ ❛ s❡❣✉♥❞❛ ell ( ell (t, s, ξ)

            ❧✐♥❤❛ ❞❡ |Q ∞, s, ξ) − Q | ✈ã♦ ❛ ③❡r♦✱ q✉❛♥❞♦ t → ∞✳ ❆ss✐♠✱

            ❜❛st❛ ♠♦str❛r q✉❡

            T

            e (Q ell ( ell (t, s, ξ)) ∞, s, ξ) − Q −→ 0, q✉❛♥❞♦ t → ∞

            1 + . . .

            , ξ) Z τ 1

            2

            , ξ)H(τ

            2

            , τ

            1

            e A(τ

            s

            1

            2

            ∞, τ

            e A(

            ∞ s

            ∞, τ, ξ)H(τ, s, ξ) dτ

            e A(

            ∞ s

            Z

            H( ∞) +

            , s, ξ) dτ

            dτ

            = e

            e A(τ

            dτ

            2

            , s, ξ) dτ

            2

            , ξ)H(τ

            2

            , τ

            

          1

            s

            1 + . . .

            , ξ) Z τ 1

            1

            e A(t, τ

            s

            e A(t, τ, ξ)H(τ, s, ξ) dτ

            s

            Z t

            − H(t, s, ξ) +

            T

            ell (t, s, ξ))

            ❡ e

            2

            1 0 e

                

            0 0    −

               1 0

                

            T

            , ξ)) = e

            , τ

            Z τ 1

            1

            (H( ∞) − H(τ

            T

            ❖❜s❡r✈❡ q✉❡ e

            ell (t, s, ξ)) −→ 0, q✉❛♥❞♦ t → ∞.

            (Q ell ( ∞, s, ξ) − Q

            1

            T

            −2

            τ 2

            (Q ell ( ∞, s, ξ) − Q

            ✭❆✳✼✮ ❱❛♠♦s ❛❣♦r❛ ✉t✐❧✐③❛r ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣❛r❛ ❡st✐♠❛r

            T

            (t, s, ξ)) |. ❚❡♠♦s q✉❡ e

            ell

            ( ∞, s, ξ) − Q

            ell

            (Q

            T

            |e

            T .

            h(τ, ξ) dτ     

            = (0, 0)

            h(τ, ξ) dτ     

            τ 2

            Z τ 1

            −2

            −e

            = 1 0     

                

          • Z
          • Z t
          e A( ∞, τ, ξ)H(τ, s, ξ) dτ.

            t

            Z

            T

            = e

            e A(t, τ, ξ)H(τ, s, ξ) dτ

            s

            ∞, τ, ξ)H(τ, s, ξ) dτ − Z t

            e A(

            ∞ s

            T

            T →∞

            T &gt; t ✱ ❝♦♠ t ✜①♦✱ t❡♠♦s e

            ✳ ❈❛❧❝✉❧❛r❡♠♦s ❛❣♦r❛ ❛ ❞✐❢❡r❡♥ç❛ ❞❛❞❛ ♥❛ s❡❣✉♥❞❛ ♣❛r❝❡❧❛✳ P❛r❛

            T

            P♦r ✭❆✳✼✮✱ ❛ ♣r✐♠❡✐r❛ ♣❛r❝❡❧❛ ❞❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❡♠ ✭❆✳✽✮ é (0, 0)

            ❋❛r❡♠♦s ♦s ❝á❧❝✉❧♦s ♣❛r❛ ❛s três ♣r✐♠❡✐r❛s ♣❛r❝❡❧❛s✱ ❛s ♦✉tr❛s s❡✲ ❣✉❡♠ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳

            1 .

            dτ

            2

            lim

            Z T

            2

            s

            Z T

            T →∞

            lim

            T

            H(τ, s, ξ) dτ

            ∞, τ, ξ) − e A(t, τ, ξ) i

            h e A(

            Z t

            s

            T →∞

            lim

            T

            ! = e

            e A(t, τ, ξ)H(τ, s, ξ) dτ

            s

            ∞, τ, ξ)H(τ, s, ξ) dτ − Z t

            e A(

            , s, ξ) dτ

            , ξ)H(τ

            = e

            − (II)) + . . . ✭❆✳✽✮ ♦♥❞❡

            , ξ) Z τ 1

            1

            A( e ∞, τ

            ∞ s

            Z

            T

            (I) = e

            e A(t, τ, ξ)H(τ, s, ξ) dτ + ((I)

            A(τ e

            s

            Z t

            ∞, τ, ξ)H(τ, s, ξ) dτ −

            e A(

            ∞ s

            Z

            (H( ∞) − H(t, s, ξ)) +

            T

            s

            1

            2

            Z t

            , τ

            1

            A(τ e

            s

            , ξ) Z τ 1

            1

            A(t, τ e

            s

            T

            , τ

            ❡ (II) = e

            1

            dτ

            2

            , s, ξ) dτ

            2

            , ξ)H(τ

            2

          • e

            ◆♦t❡ q✉❡ e ∞, τ, ξ) − e ∞)R − H(t, τ, ξ)R = (H( (τ, ξ).

            2

            ∞) − H(t, τ, ξ)) R ❆ss✐♠✱ ♣♦r ✭❆✳✼✮ t❡♠✲s❡

            Z Z t

            ∞ T

            e e e A( A(t, τ, ξ)H(τ, s, ξ) dτ ∞, τ, ξ)H(τ, s, ξ) dτ −

            s s

            Z T

            T

            = e lim A( e ∞, τ, ξ)H(τ, s, ξ) dτ

            T →∞ t

            Z

            ∞ T

            e = e A(

            ∞, τ, ξ)H(τ, s, ξ) dτ. ✭❆✳✾✮

            t

            ❘❡st❛✲♥♦s ❝❛❧❝✉❧❛r ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✭■✮ ❡ ✭■■✮✱ q✉❡ é ✉♠ ❝❛s♦ ♣❛rt✐✲ ❝✉❧❛r ❞♦ ❝á❧❝✉❧♦ ❢❡✐t♦ ❛❝✐♠❛✱ ❝♦♥❢♦r♠❡ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ❛❜❛✐①♦✿

            Z Z τ

            ∞ 1 T

            e e e A( , ξ) A(τ , τ , ξ)H(τ , s, ξ) dτ dτ

            1

            1

            2

            2

            2

            1

            ∞, τ

            s s

            Z t Z τ 1 e e A(t, τ , ξ) A(τ , τ , ξ)H(τ , s, ξ) dτ dτ

            1

            1

            2

            2

            2

            1

            −

            s s

            Z Z Z t τ 1

            ∞ T

            e e e = e A(

            1 , ξ) A(t, τ 1 , ξ) A(τ 1 , τ 2 , ξ)H(τ 2 , s, ξ) dτ 2 dτ

            1

            ∞, τ −

            s s s

            Z Z τ

            ∞ 1 T

            e e = e A( , ξ)H(τ, s, ξ) A(τ , τ , ξ)H(τ , s, ξ) dτ dτ ,

            1

            1

            2

            2

            2

            1

            ∞, τ

            t s

            ♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ ✭❆✳✾✮✳

          • e
          • e

            ell

            1

            ∞, τ

            e A(

            ∞ t

            Z

            T

            ∞, τ, ξ)H(τ, s, ξ) dτ

            e A(

            ∞ t

            Z

            T

            (t, s, ξ)) ≤ e

            ( ∞, s, ξ) − Q

            

          s

            ell

            (Q

            T

            ✳ ❆ss✐♠✱ t❡♠♦s q✉❡ e

            1

            | ∈ L

            2 , τ 1 , ξ)

            A(τ

            ✭❆✳✶✵✮ ❧♦❣♦ | e

            , ξ) |,

            1

            (τ

            , ξ) Z τ 1

            e A(τ

            , ξ) | = |R

            2

            3 , τ 4 , ξ)H(τ

          4 , s, ξ) dτ

          4 dτ 3 dτ

            e A(τ

            s

            , s, ξ) Z τ 3

            3

            , ξ)H(τ

            3

            , τ

            2

            e A(τ

            

          s

            , ξ) Z τ 2

            ∞, τ

            1

            e A(

            ∞ t

            Z

            T

            1

            dτ

            2

            , s, ξ) dτ

            2

            , ξ)H(τ

            2

            , τ

            2

            1

            2 + . . .

            , ξ) Z

            T

            1

            dτ

            2

            , s, ξ) dτ

            2

            , ξ)H(τ

            2

            , τ

            1

            A(τ e

            

          τ

          1

          s

            1

            ∞ t

            A( e ∞, τ

            t

            Z ∞

            T

            ∞, τ, ξ)H(τ, s, ξ) dτ

            e A(

            ∞ t

            Z

            T

            ∞, s, ξ) − Q = e

            T

            ❊♥tã♦ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭❆✳✽✮ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

            Z

            A( e ∞, τ

            (τ

            | ∈ L

            2

            , ξ) | |R

            1

            , τ

            2

            , ξ) | ≤ |H(τ

            1

            , τ

            2

            A(τ

            ✱ ❛ss✐♠ | e

            1

            2 (τ, ξ)

            2

            ❆♥t❡s ❞❡ ❡st✐♠❛r♠♦s ♦ t❡r♠♦ ❛❝✐♠❛ ✈❛❧❡ ❧❡♠❜r❛r q✉❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ H(t, s, ξ)✱ |H(t, s, ξ)| = 1 ❡ ♣♦r ✭✷✳✹✸✮ |R

            3 , τ 4 , ξ)H(τ

          4 , s, ξ)dτ

          4 dτ 3 dτ 2 + . . .

            e A(τ

            s

            , s, ξ) Z τ 3

            3

            , ξ)H(τ

            3

            , τ

            2

            A(τ e

            s

            , ξ) Z τ 2

          • e
          • e

            ❱❛♠♦s ❡st✐♠❛r s❡♣❛r❛❞❛♠❡♥t❡ ❝❛❞❛ t❡r♠♦ ❞❡ss❛ s♦♠❛✳ P❛r❛ ♦ ♣r✐✲ ♠❡✐r♦ t❡♠♦s

            Z Z

            ∞ ∞ T T

            e A( A(

            ∞, τ, ξ)H(τ, s, ξ) dτ | | e ∞, τ, ξ)| |H(τ, s, ξ)| dτ e 6 |e

            t t

            Z

            ∞

            A( ≤ | e ∞, τ, ξ)| dτ.

            t

            P❛r❛ ♦ s❡❣✉♥❞♦ t❡r♠♦✱ ✉s❛♥❞♦ ✭❆✳✶✵✮ ❡ ✭✷✳✹✸✮ s❡❣✉❡ q✉❡ Z Z τ 1

            ∞ T

            A( e , ξ) A(τ e , τ , ξ)H(τ , s, ξ) dτ dτ

            1

            1

            

          2

            2

            2

            1

            ∞, τ e

            t s

            Z τ Z ∞ 1 T

            6 A( , ξ) A(τ , τ , ξ) , s, ξ) dτ

            1

            1

            2

            2

            2

            1

            |e | | e ∞, τ | | e | |H(τ | dτ

            t s

            Z Z τ

            ∞ 1

            = A( , ξ) A(τ , τ , ξ) dτ

            1

            1

            2

            2

            1

            | e ∞, τ | | e | dτ

            t s

            Z Z τ

            ∞ 1 A( , ξ) (τ , ξ) dτ

            1

            

          2

            2

            2

            1

            ≤ | e ∞, τ | |R | dτ

            t s

            Z

            ∞

            C A(

            1 , ξ) 1 .

            ≤ | e ∞, τ | dτ N

            t

            ❆ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦ t❡r❝❡✐r♦ t❡r♠♦✱ s❡❣✉❡ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♣❛r❛ ♦ s❡❣✉♥❞♦✱ ❝♦♠♦ s❡❣✉❡

            Z Z τ 2

            ∞ T

            e e A( , ξ) A(τ , τ , ξ)H(τ , s, ξ)

            2

            2

            

          3

            3

            ∞, τ | e

            t s

            Z τ 3 e A(τ

            3 , τ 4 , ξ)H(τ 4 , s, ξ)dτ 4 dτ 3 dτ

            2 s τ

            Z ∞ Z 2 T

            6 A( , ξ) A(τ , τ , ξ) , s, ξ)

            2

            2

            3

            3

            |e | | e ∞, τ | | e | |H(τ |

            t s

            Z τ 3 A(τ , τ , ξ) , s, ξ) dτ dτ

            3

            4

            4

            4

            3

            2

            | e | |H(τ | dτ

            s

            τ τ

            Z ∞ Z 2 Z 3 A( , ξ) A(τ , τ , ξ) A(τ , τ , ξ) dτ dτ

            6

            2

            

          2

            3

            3

            4

            4

            3

            2

            | e ∞, τ | | e | | e | dτ

            t s s

            Z

            2 ∞

            C A( , ξ) .

            2

            2

            ≤ | e ∞, τ | dτ

            2 N t

            ❖s ♦✉tr♦s t❡r♠♦s sã♦ t♦❞♦s ❡st✐♠❛❞♦s ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳ ❈♦♠ ✐ss♦✱

            T

            (Q ( (t, s, ξ))

            ell ell

            |e ∞, s, ξ) − Q | Z Z

            ∞ ∞

            C A( A( , ξ)

            1

            1

            ≤ | e ∞, τ, ξ)| dτ + | e ∞, τ | dτ N

            t t

            Z

            2 ∞

            C A( , ξ) + . . .

          • 2

            2

            | e ∞, τ | dτ

            2 N t

            Z

            ∞

            A( ◆♦t❡ q✉❡ ♦ t❡r♠♦

            | e ∞, τ, ξ)| dτ ❛♣❛r❡❝❡ ❡♠ t♦❞♦s ♦s t❡r♠♦s

            t

            ❞❛ s♦♠❛✱ ❛ss✐♠✱ ❝♦❧♦❝❛♥❞♦ ❡st❡ ❡♠ ❡✈✐❞ê♥❝✐❛✱ ♦❜t❡♠♦s

            2 C C T

            (Q ( (t, s, ξ)) 1 + +

          • ell ell

            |e ∞, s, ξ) − Q | ≤ · · · N N Z

            ∞

            A( | e ∞, τ, ξ)| dτ

            t ∞ k

            Z

            ∞

            X C = A( | e ∞, τ, ξ)| dτ.

            N

            t k=0

            ❚♦♠❛♥❞♦ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ t❡♠♦s q✉❡ ❛ sér✐❡ ❛❝✐♠❛ é ❝♦♥✲ ✈❡r❣❡♥t❡✱ ✐st♦ é✱

            ∞ k

            X C = C,

            N

            k=0

            ❡✱ ♣♦rt❛♥t♦✱ Z

            ∞ T

            (Q ( (t, s, ξ)) A(

            ell ell

            |e ∞, s, ξ) − Q | ≤ C | e ∞, τ, ξ)| dτ

            t

            .

            | + e

            1 H(t, s, ξ) + e

          T

            

          1

          Z t s

            A(t, τ, ξ)Q e

            ell (τ, s, ξ) dτ.

            ❊♥tã♦✱ ❧❡♠❜r❛♥❞♦ q✉❡ Q

            ell (t, s, ξ)

            é ❧✐♠✐t❛❞♦ |e

            T

            1 Q ell

            (t, s, ξ) | 6 |e

            T

            1 H(t, s, ξ)

            T

            (t, s, ξ) = e

            1 Z t s

            e A(t, τ, ξ)Q

            ell

            (τ, s, ξ) dτ

            6 |e

            T

            1 H(t, s, ξ)

            | + C

            1 Z t s

            |e

            T

            1

            T

            1 Q ell

            Z ∞

            1+α .

            t

            |R

            2

            (τ, ξ) | dτ

            = lim Z T

            t

            |R

            2

            (τ, ξ) | dτ .

            |ξ|

            −2δ

            (1 + t)

            ✭❆✳✶✶✮ ❆ ú❧t✐♠❛ ❡st✐♠❛t✐✈❛ s❡❣✉❡ ❞❡ ✭✷✳✹✸✮✳

            T

            P❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦✱ r❡st❛ ♣r♦✈❛r q✉❡ e

            T

            1

            (Q

            ell

            ( ∞, s, ξ) − Q

            ell

            (t, s, ξ)) −→ 0, q✉❛♥❞♦ t → ∞. ❈♦♠♦ ♣r♦✈❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ❞❡ Q ell

            ( ∞, s, ξ) é ♥✉❧❛✳

            ❆ss✐♠✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ |Q

            ell

            (t, s, ξ) | −→ 0, q✉❛♥❞♦ t → ∞. P♦r ✭❆✳✺✮ t❡♠♦s q✉❡ e

            e A(t, τ, ξ) | dτ.

            ❈♦♠♦ ❢❡✐t♦ ♣❛r❛ ♦ ♦✉tr♦ ❝❛s♦✱ ✈❛♠♦s ❡st✐♠❛r ❝❛❞❛ t❡r♠♦ ❞❡ss❛ s♦♠❛ s❡♣❛r❛❞❛♠❡♥t❡✳ ❆ss✐♠✱ ♣❛r❛ ♦ ♣r✐♠❡✐r♦✱ t❡♠♦s e

            α

            α

            dτ = e

            −C|ξ|

            Z t (1 + τ )

            α

            dτ e

            C|ξ|

            Z s (1 + τ )

            α

            dτ = C(s) e C µ

            |ξ|

            Z t µ(1 + τ )

            dτ = C(s) e

            s

            −C 2 |ξ|

          Λ(t,0)

            ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ ✭✷✳✸✮✱ ❡ C(s) = e

            C|ξ|

            Z s (1 + τ )

            α

            dτ . P❛r❛ ♦ s❡❣✉♥❞♦ t❡r♠♦ ✈❛♠♦s s❡♣❛r❛r ❡♠ ❞♦✐s ❝❛s♦s✿ s ∈

            t

            2

            , t ❡ s

            ∈ 0,

            t

            2

            (1 + τ )

            Z

            T

            T

            0 1     

            1 e

            −2

            Z t

            

          s

            h(τ, ξ) dτ     

            =      e

            −2

            Z t

            s

            h(τ, ξ) dτ     

            ❧♦❣♦✱ |e

            | = e

            dτ −

            −2

            Z t

            s

            h(τ, ξ) dτ ≤ e

            −C|ξ|

            Z t

            s

            (1 + τ )

            α

            dτ = e

            −C|ξ|

            Z t (1 + τ )

            α

            ✳ t

            , t ❈❛s♦ ✶✿ s ∈

            Z t

            −2 h(τ , ξ) dτ

            1

            1 T τ

            H(t, τ, ξ) |e

          1 | = e ≤ 1.

            ❆ss✐♠✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛ ❡ ♣♦r ✭✷✳✹✸✮ s❡❣✉❡ q✉❡

            t t

            Z Z

            T T

            A(t, τ, ξ) e H(t, τ, ξ)R (τ, ξ)

            2

            |e

            1 | dτ = |e 1 | dτ s s

            Z t Z t

            T

            H(t, τ, ξ)

            2 (τ, ξ) 2 (τ, ξ)

            1

            ≤ |e | |R | dτ ≤ |R | dτ 2 t t 2

            −2δ −2δ −2δ

            e C C C

            |ξ| |ξ| |ξ| = . ≤ ≤ ✭❆✳✶✷✮

            1+α 1+α 1+α t 1+t

            (1 + t) 1 +

            2

            

          2

          t

            ❈❛s♦ ✷✿ s ≤

            2

            ◆❡ss❡ ❝❛s♦✱ ❝♦♥s✐❞❡r❡ t Z t Z Z t 2 T T T e e e

            A(t, τ, ξ) A(t, τ, ξ) A(t, τ, ξ) |e

            1 | dτ = |e 1 | dτ + |e t 1 | dτ. s s 2

            ◆♦t❡ q✉❡ ❛ s❡❣✉♥❞❛ ✐♥t❡❣r❛❧ é ❡st✐♠❛❞❛ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ ❢❡✐t❛ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ♦✉ s❡❥❛✱

            Z t Z t

            −2δ

            e C

            T T |ξ|

            e A(t, τ, ξ) H(t, τ, ξ)R 2 (τ, ξ) . |e

            1 | dτ = |e 1 | dτ ≤ t t α+1 2 2 (1 + t) P❛r❛ ❛ ♣r✐♠❡✐r❛ ✐♥t❡❣r❛❧✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ Z t h(τ , ξ) dτ

            −2

            1

            1 τ

            f (τ ) = e é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✳ t t ❆ss✐♠ Z Z 2 2 T T e

            A(t, τ, ξ) H(t, τ, ξ)R (τ, ξ)

            2

            |e

            1 | dτ = |e 1 | dτ s s t Z t

            Z 2 −2 h(τ , ξ) dτ

            1

            1 τ

            = e

            2 (τ, ξ)

            |R | dτ

            s

            Z t t

            −2 h(τ , ξ) dτ t 2

            

          1

            1 Z 2 2 (τ, ξ)

            ≤ e |R | dτ

            s

            Z t

            

          −2 h(τ , ξ) dτ

          t

            1

            1 C 2 e

            ≤ N

            Z t

            2δ α

          −2c (1 + τ ) dτ

            1

            1 2 t |ξ| e

            .

            Z t Z t

            2δ α 2δ α −c (1 + τ ) dτ (1 + τ ) dτ

            1

            1

            1

            1 2 t t |ξ| − c |ξ| 2

            = e t Z Z t 2

            2δ α 2δ α

            (1 + τ ) dτ (1 + τ ) dτ

            −c

            1

            1

            1

            1

            |ξ| − c |ξ| 2 t ≤ e c 2δ α Z t µ

          |ξ| µ(1 + τ

          1 ) dτ

            1

            = e

            Λ(t,0) −c |ξ|

            = e .

            ▲♦❣♦✱ ♥❡ss❡ ❝❛s♦✱ t Z t Z Z t 2 T T T e e e

            |e

            1 | dτ = |e 1 | dτ + |e t 1 | dτ s s 2 −2δ

            |ξ| Λ(t,0)

            −c |ξ| . + e .

            ✭❆✳✶✸✮

            

          α+1

            (1 + t) P❡❧♦ ❝❛s♦ ✶✱ ✭❆✳✶✷✮ ❡ ♣❡❧♦ ❝❛s♦ ✷✱✭❆✳✶✸✮✱ ♦❜t✐✈❡♠♦s q✉❡

          −2δ T Λ(t,0) |ξ|

            

          −c |ξ|

          Q (t, s, ξ) + . ell ✭❆✳✶✹✮

            |e

            1 | . e α+1

            (1 + t) ◆♦t❡ q✉❡ ✭❆✳✶✶✮ ❡ ✭❆✳✶✹✮ ✈ã♦ ❛ ③❡r♦ q✉❛♥❞♦ t → ∞✱ ❡✱ ♣♦rt❛♥t♦✱ ❛

            ♣r♦♣♦s✐çã♦ ❡stá ♣r♦✈❛❞❛✳ ❉❡✜♥✐♠♦s

            2 4δ 2σ

          1 (t, ξ) = .

          • m −b(t) |ξ| |ξ|

            (t, ξ) ❆✜r♠❛çã♦ ❆✳✷ ❉❛❞♦ m

            1 ❝♦♠♦ ❛❝✐♠❛✱ ✈❛❧❡ q✉❡ 2 4δ

            m (τ, ξ)

            1

            ≈ −b(τ) |ξ| ❉❡♠♦♥str❛çã♦✿ P❛r❛ ξ ∈ B M ❡ t ≤ t ξ ♣♦r ✭✷✳✺✮ t❡♠✲s❡

            2 4δ m(t, ξ) .

            ≈ −b(t) |ξ|

          • |ξ|

            −2δ

            2σ ≤ 0.

            ❈♦♥s✐❞❡r❡ c &lt; 1✳ P❛r❛ (t, ξ) ∈ Z

            low ell

            ✈❛❧❡ q✉❡ (1 + t)

            α+1

            |ξ|

            2δ ≥ N.

            ❆ss✐♠✱ (1 + t)

            α+1

            ≥ N|ξ|

            ⇔ (1 + t)

            |ξ|

            ≥ N 1 α +1 |ξ|

            − α +1

            ⇔ (1 + t)

            2α

            ≥ N α +1 |ξ|

            − 4δα α +1

            ⇔ (c

            − 1)(1 + t)

            2α

            ≤ (c − 1)N α +1 |ξ|

            4δ

            

            − 4δα α +1 .

            (t, ξ) = −b(t)

            ❚❛♠❜é♠ ♣❡❧❛s ❞❡✜♥✐çõ❡s ❞❡ m(t, ξ) ❡ m

            1

            (t, ξ) t❡♠♦s ≥ m(t, ξ). ❈♦♠ ❡ss❡s ❞♦✐s r❡s✉❧t❛❞♦s✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ m

            1 (t, ξ) &amp;

            −b(t)

            2

            |ξ|

            4δ .

            P♦r ♦✉tr♦ ❧❛❞♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ ∃ c &gt; 0 t❛❧ q✉❡ m

            1

            2

            (1 + t)

            |ξ|

            4δ

            2σ

            ≤ −c b(t)

            2

            |ξ|

            4δ

            q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ♠♦str❛r q✉❡ (c

            − 1) µ

            2

          • |ξ|
          • |ξ|

          • C M |ξ| α +1
          • (c

            ∂

            ≈ −b(τ)

            2

            |ξ|

            4δ

            ✭❆✳✶✺✮ ♣❛r❛ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

            ❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦✿ β(t, ξ) .

            = exp Z

            ∞ t

            h(τ, ξ) − h

            1

            (τ, ξ) −

            t

            . P♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦ ξ ∈ B M ❡ τ &gt; t ξ ❝♦♥❝❧✉í♠♦s q✉❡ m

            h(τ, ξ) 2h(τ, ξ) dτ ,

            ✭❆✳✶✻✮ ♦♥❞❡ h

            1

            (τ, ξ) .

            = p −m

            1 (τ, ξ).

            P❡❧❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ m

            1

            (τ, ξ) s❡❣✉❡ q✉❡ h

            1

            (τ, ξ) ≈ b(τ)|ξ|

            1 (τ, ξ)

            4δ 1+α

            ❈♦♠ ✐ss♦ − 1) µ

            |ξ|

            2 2α

            |ξ|

            4δ

            |ξ|

            2σ

            ≤ (c − 1) µ

            2 α +1

            |ξ|

            − 4δα α +1

            |ξ|

            4δ

            2σ

            ≤ 0 ♣❛r❛ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ◆♦t❡ q✉❡ ♥❛ ♣❡♥ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ❛ ❝♦♥❞✐çã♦ 2δ &lt; σ(1 + α), ♦✉ s❡❥❛✱ 2σ &gt;

            = (c − 1) µ

            2 N α +1

            |ξ| α +1

            2σ

            ≤ (c − 1) µ

            2 N α +1

            |ξ| α +1

            = |ξ| α +1

            C

            M

            − 1) µ

            2 N α +1

            2δ .

            ◆❛ s❡q✉ê♥❝✐❛ ✈❛♠♦s ♣r♦✈❛r três ❧❡♠❛s q✉❡ s❡rã♦ ✉s❛❞♦s ♥❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✻✳

            1 (τ, ξ)

            ▲❡♠❛ ❆✳✸ P❛r❛ h(τ, ξ)✱ h ❡ b(τ) ❝♦♠♦ ❞❡✜♥✐❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ✈❛❧❡ q✉❡

            2 2δ 2δ ′ ′

            b (τ ) b (τ ) |ξ| |ξ|

            1 (τ, ξ) 1 (τ, ξ)

            −

            2

            h(τ, ξ) − h

            2h

            1 (τ, ξ) h 1 (τ, ξ) low

            ♣❛r❛ t♦❞♦ (τ, ξ) ∈ Z ✳

            ell

            ❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❢ór♠✉❧❛ ❞❛ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f(x) s❡❣✉❡ q✉❡

            1

            2

          ′ ′′

            f (x) = f (a) + f (a)(x f (a)(x + . . .

            − a) + − a) 2!

            ❊ ❛✐♥❞❛✱ ✉t✐❧✐③❛♥❞♦ ♦ r❡st♦ ❞❡ ▲❛❣r❛♥❣❡✱ t❡♠♦s

            1

            2 ′ ′′

            f (x) = f (a) + f (a)(x f (a )(x , − a) + − a)

            2! ♦♥❞❡ a ❡stá ❡♥tr❡ x ❡ a✳

            √ x (τ, ξ) ❈♦♥s✐❞❡r❡ f(x) = ✱ x = −m(τ, ξ) ❡ a = −m

            1 ✳ ◆♦t❡ q✉❡

            x (τ, ξ)

            1

            − a = −m(τ, ξ) + m

            2σ 2 4δ 2δ 2 4δ 2σ ′

            = + b(τ ) + b (τ ) −|ξ| |ξ| |ξ| − b(τ) |ξ| |ξ|

          • 2δ ′ = b (τ ) .

            |ξ|

            ❆ss✐♠

            )

            1

            8 h

            2

            )

            2δ

            (τ ) |ξ|

            ′

            = (b

            3

            8 √ a

            2

            2δ

            3

            (τ ) |ξ|

            ′

            (b

            (τ, ξ) ≤

            1

            2h

            2δ

            (τ ) |ξ|

            

            (τ, ξ) − b

            1

            ❡ ❡♥tã♦ h(τ, ξ) − h

            (τ, ξ)

            = Ch

            3 .

            ′

            low ell

            ♣❛r❛ t♦❞♦ (τ, ξ) ∈ Z

            −2δ

            |ξ|

            −(α+2)

            (τ, ξ) . (1 + τ )

            1

            2h(τ, ξ)h

            

          1 (τ, ξ)]

            [h(τ, ξ) − h

            2δ

            (τ ) |ξ|

            ✈❛❧❡ q✉❡ b

            1

            (τ, ξ) ❡ b(τ) ❝♦♠♦ ❞❡✜♥✐❞♦s ❛♥t❡r✐♦r♠❡♥t❡

            1

            ❖ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳ ▲❡♠❛ ❆✳✹ P❛r❛ h(τ, ξ)✱ h

            2 .

            2

            (τ, ξ)

            1

            h

            2δ

            (τ ) |ξ|

            ′

            (τ, ξ) b

            ❈♦♠♦ a é ✉♠ ✈❛❧♦r ❡♥tr❡ x ❡ a✱ t❡♠♦s q✉❡ a &gt; a

            8 p a

            1

            (τ ) |ξ|

            ⇒ h(τ, ξ) − h

            3

            8 p a

            2

            )

            2δ

            (τ ) |ξ|

            ′

            − (b

            1 (τ, ξ)

            2 p −m

            2δ

            ′

            (τ, ξ) − b

            b

            

          1 (τ, ξ) +

            −m

            ❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s✱ t❡♠♦s p −m(τ, ξ) = p

            2

            − a)

            3

            4 √ a

            1

            1 2!

            − a) −

            2 √ a

            1

            ′

            2

            1

            )

            2δ

            (τ ) |ξ|

            ′

            (b

            (τ, ξ) =

            1

            2h

            2δ

            (τ ) |ξ|

            ′

            (τ, ξ) − b

            ⇒ h(τ, ξ) − h

            (τ ) |ξ|

            3

            8 p a

            2

            )

            2δ

            (τ ) |ξ|

            ′

            − (b

            (τ, ξ) =

            1

            2h

            2δ

            ✳

            ❉❡♠♦♥str❛çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡ ♥♦t❡ q✉❡✱ ♣♦r ❞❡✜♥✐çã♦

            2σ 2 4δ 2δ ′

            m(τ, ξ) = (τ ) |ξ| − b(t) |ξ| − b |ξ|

            ❡

            

          2 4δ m (τ, ξ) = .

            1

            |ξ| − b(τ) |ξ| ❆ss✐♠ m(τ, ξ) 6 (τ, ξ) (τ, ξ) 6 h(τ, ξ).

            1

            1

            −m ⇒ h (τ, ξ)

            P❡❧❛s ❡q✉✐✈❛❧ê♥❝✐❛s ♣❛r❛ h(τ, ξ) ❡ h

            1 t❡♠♦s α 2δ

            h (τ, ξ) &gt; C (1 + τ )

            1

            

          1

            |ξ| ❡

            α 2δ

            h(τ, ξ) &gt; C 2 (1 + τ ) .

            |ξ| ❊♥tã♦ s❡❣✉❡ q✉❡

            2δ 2δ ′ α−1

            b (τ ) [h(τ, ξ)

            1 (τ, ξ)] µα(1 + τ ) [h(τ, ξ) 1 (τ, ξ)]

            |ξ| − h |ξ| − h

            6

            2α 4δ

            2h(τ, ξ)h (τ, ξ)

            2C C (1 + τ )

            1

            1

            2

            |ξ|

            −α−1 −2δ

            µα(1 + τ ) [h(τ, ξ) (τ, ξ)]

            1

            |ξ| − h = .

            2C C

            1

            2

            ❈♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❆✳✸✱ t❡♠♦s

            2δ ′

            b (τ ) |ξ|

            − h ≤ 2h (τ, ξ)

            1

            ❙✉❜st✐t✉✐♥❞♦ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛

            2δ 2δ ′

            ′

            b (τ ) [h(τ, ξ) (τ, ξ)] b (τ )

            1

            |ξ| − h |ξ|

            −α−1 −2δ

            . (1 + τ ) |ξ|

            2h(τ, ξ)h (τ, ξ) h (τ, ξ)

            1

            1 2δ α−1

            (1 + τ ) |ξ|

            −α−1 −2δ −(α+2) −2δ . (1 + τ ) = (1 + τ ) .

            |ξ| |ξ|

            α 2δ

            (1 + τ ) |ξ|

            (τ, ξ) ▲❡♠❛ ❆✳✺ P❛r❛ h(τ, ξ)✱ h

            1 ❡ b(τ) ❝♦♠♦ ❞❡✜♥✐❞♦s ❛♥t❡r✐♦r♠❡♥t❡

            ✈❛❧❡ q✉❡

            2δ 2δ ′

            b (τ ) h(τ, ξ) |ξ| − b(τ)|ξ|

            −(2α+1) 2(σ−2δ) −(α+2) −2δ

          • (1+τ ) |ξ| |ξ|

            2

            . (1+τ) 2h(τ, ξ)

            low

            ♣❛r❛ t♦❞♦ (τ, ξ) ∈ Z ✳

            ell

            ❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❡ q✉❡ q q

            2 4δ 2σ 4δ

            b(τ ) b(τ ) |ξ| − |ξ| ≤ |ξ|

            2δ

            (τ, ξ)

            1

            ⇒ −h ≥ −b(τ)|ξ|

            2δ (τ, ξ) .

            1

            ⇒ h(τ, ξ) − h ≥ h(τ, ξ) − b(τ)|ξ|

            ❙❡ h(τ, ξ) − b(τ)|ξ|

            − h(τ, ξ) . (1 + τ)

            α+1

            ◆♦t❡ q✉❡✱ ✉s❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s µ &gt; 0✱ 2δ &lt; σ(1 + α) ❡ (1 + τ )

            ✭❆✳✶✼✮ ❆ss✐♠✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡ ❛ ú❧t✐♠❛ ❡st✐♠❛t✐✈❛ é ✈❡r❞❛❞❡✐r❛✳

            2(σ−δ) .

            |ξ|

            

          −α

            2δ

            2δ

            ⇔ b(τ ) |ξ|

            2(σ−2δ)

            |ξ|

            −(2α+1)

            (1 + τ )

            4δ .

            |ξ|

            |ξ|

            ≥ N✱ t❡♠♦s c

            − h(τ, ξ) (1 + τ )

            = c

            2σ

            |ξ| 4σ(α+1)−2σ(α+1) α +1 ≤ c µ |ξ|

            − α +1

            N

            

          2

          C M

            |ξ| 4σ(α+1)−4δ α +1 ≤ c

            

          2

          N − α +1

            4(σ−δ)

            2

            |ξ| 4δα α +1 |ξ|

            

          2

          N − α +1

            ≤ c

            4(σ−δ)

            |ξ|

            −2α

            (1 + τ )

            2α

            2δ

            2δ

            b

            (τ, ξ)] 2h(τ, ξ)h

            1

            [h(τ, ξ) − h

            2δ

            (τ ) |ξ|

            ′

            2 .

            (τ, ξ) ❡ ♦ r❡s✉❧t❛❞♦ é ♦❜t✐❞♦ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ❧❡♠❛ ❛♥t❡r✐♦r✳

            2h(τ, ξ)

            2δ

            h(τ, ξ) − b(τ)|ξ|

            2δ

            (τ ) |ξ|

            ′

            ≥ 0 ❡♥tã♦ b

            1

            ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥s✐❞❡r❡ h(τ, ξ) − b(τ)|ξ|

            b(τ ) |ξ|

            . (1 + τ)

            2δ

            |ξ|

            α−1

            ⇔ (1 + τ )

            2(σ−2δ)

            |ξ|

            −(2α+1)

            2

            2δ

            2h(τ, ξ)

            

            h(τ, ξ) − b(τ)|ξ|

            2δ

            (τ ) |ξ|

            ′

            &lt; 0 ✳ ◆❡ss❡ ❝❛s♦ b

            , ♣❛r❛ c &gt; 0 ❡ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

          • 2cµ
          • µα(1 + τ )
          • c

            

            2δ

            P♦rt❛♥t♦✱ b(τ)|ξ|

            2 .

            = h(τ, ξ)

            2σ

            − |ξ|

            (τ ) |ξ|

            −α

            ′

            4δ

            |ξ|

            2

            ≤ b(τ)

            2σ

            − 2cµ |ξ|

            − c (1 + τ)

            |ξ|

            |ξ|

            ✐♠❡❞✐❛t❛✳ ❈❛s♦ ❝♦♥trár✐♦ b(τ)|ξ|

            low ell

            ≤ h(τ, ξ), ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳ Pr♦♣♦s✐çã♦ ❆✳✻ ❆ ❢✉♥çã♦ β(t, ξ) é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ Z

            2(σ−δ)

            |ξ|

            −α

            −c (1+τ)

            2δ

            &lt; 0 ✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é

            2(σ−δ)

            2(σ−δ)

            |ξ|

            

          −α

            − c (1 + τ)

            2δ

            ❆ss✐♠✱ s❡ b(τ)|ξ|

            ≤ h(τ, ξ), ♦ q✉❡ ♣r♦✈❛ ✭❆✳✶✼✮✳

            4(σ−δ)

            −2α

            ✱

            2σ

            |ξ|

            4δ

            |ξ|

            2α

            (1 + τ )

            2

            ≤ µ

            |ξ|

            α−1

            4(σ−δ)

            |ξ|

            2 −2α

            4δ

            |ξ|

            2 2α

            ❆ss✐♠✱ ♣❛r❛ cµ &gt; 1 t❡♠✲s❡

            2σ

            |ξ|

            (1 + τ )

            2

            

          2

            4δ

            |ξ|

            2α

            (1 + τ )

            2

            = µ

            i

            2δ .

            2(σ−δ)

            |ξ|

            −α

            − c (1 + τ)

            2δ

            |ξ|

            ▲♦❣♦ h b(τ )

          • b

            ❡

            −(α+1) −2δ −2α 2(σ−2δ)

          • (1 + t) |β(t, ξ) − 1| . (1 + t) |ξ| |ξ| q✉❛♥❞♦ t → ∞✳ ❉❡♠♦♥str❛çã♦✿

            ■♥✐❝✐❛❧♠❡♥t❡ q✉❡r❡♠♦s ♣r♦✈❛r q✉❡ β(t, ξ) é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛✱

            low

            ✐st♦ é✱ |β(t, ξ)| ≤ K, ∀ (t, ξ) ∈ Z ✳ ❚❡♠✲s❡ q✉❡

            ell

            Z

            ∞

            ∂ t h(τ, ξ) h(τ, ξ) (τ, ξ) dτ

            1

            |β(t, ξ)| = exp − h − 2h(τ, ξ)

            t

            Z

            ∞

            ∂ t h(τ, ξ) (τ, ξ) .

            1

            ≤ exp − h(τ, ξ) − h dτ 2h(τ, ξ)

            t

            ❆ ✐❞❡✐❛ ❞❛ ❞❡♠♦♥str❛çã♦ é s♦♠❛r ❡ s✉❜tr❛✐r ✉♠ t❡r♠♦ ❛❞✐❝✐♦♥❛❧ ♥♦ ✐♥t❡❣r❛♥❞♦ ❛❝✐♠❛ ❢❛❝✐❧✐t❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ❝❛❞❛ t❡r♠♦✳ ❆ss✐♠

            Z

            ∞ 2δ ′

            b (τ ) |ξ|

            1 (τ, ξ)

            |β(t, ξ)| ≤ exp − h(τ, ξ) − h dτ 2h

            1 (τ, ξ) t

            Z 2δ

            ∞ ′

            b (τ ) ∂ t h(τ, ξ) |ξ|

            ✭❆✳✶✽✮

          • − dτ

            2h (τ, ξ) 2h(τ, ξ)

            1 t

            Z Z

            ∞ ∞

            = exp P (τ, ξ) dτ exp Q(τ, ξ) dτ

            t t

            ♦♥❞❡

            2δ ′

            b (τ ) |ξ|

            P (τ, ξ) = (τ, ξ)

            1

            − h(τ, ξ) − h 2h (τ, ξ)

            1

            ❡

            2δ ′

            b (τ ) ∂ h(τ, ξ)

            t

            |ξ| Q(τ, ξ) =

            − . 2h (τ, ξ) 2h(τ, ξ)

            1

          • b

            ′

            −2δ .

            |ξ|

            −(α+2)

            (τ, ξ)h(τ, ξ) . (1 + τ )

            1

            2h

            1 (τ, ξ)

            h

            2δ

            (τ ) |ξ|

            h(τ, ξ) − b

            t h(τ, ξ)

            2δ

            (τ ) |ξ|

            

            2h(τ, ξ) = b

            2δ

            (τ ) |ξ|

            ′

            (τ, ξ) − b

            1

            2h

            ✭❆✳✷✵✮ P❛r❛ ♦ s❡❣✉♥❞♦ t❡r♠♦✱ ✉t✐❧✐③❛♠♦s ❛ ❡①♣r❡ssã♦ ♣❛r❛ ∂

            ❝❛❧❝✉✲ ❧❛❞❛ ❡♠ ✭✷✳✷✻✮✱ ♦ ▲❡♠❛ ❆✳✺ ❡ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ h(τ, ξ) ≈ (1+τ)

            (τ ) |ξ|

            (τ ) |ξ|

            4h(τ, ξ)

            2δ

            (τ ) |ξ|

            ′′

            4δ

            (τ ) |ξ|

            

            2b(τ )b

            2h(τ, ξ) −

            2δ

            ′

            α

            = b

            h(τ, ξ) 2h(τ, ξ)

            t

            ∂

            2h(τ, ξ) −

            2δ

            (τ ) |ξ|

            ′

            ❞❛❞❛ ❡♠ ✭✷✳✻✮✿ b

            2δ

            |ξ|

            2δ

            ′

            2

            h

            α

            [(1 + τ )

            4δ

            |ξ|

            2α−2

            (1 + τ )

            2 .

            2

            (τ, ξ)

            1

            2δ

            2δ

            (τ ) |ξ|

            ′

            (τ, ξ) b

            1

            6 Ch

            ♣r♦✈❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠♦s P (τ, ξ)

            

            ≈ b(τ)|ξ|

            1 (τ, ξ)

            ❱❛♠♦s ❡st✐♠❛r P (τ, ξ) ❡ Q(τ, ξ)✳ P❡❧♦ ▲❡♠❛ ❆✳✸✱ ❛ ❞❡✜♥✐çã♦ ❞❡ b(τ) ❡ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ h

            |ξ|

            ]

            P❛r❛ ♦ ♣r✐♠❡✐r♦ t❡r♠♦✱ ♣❡❧♦ ▲❡♠❛ ❆✳✹ t❡♠♦s b

            (τ ) |ξ|

            . ✭❆✳✶✾✮ ❊st✐♠❛♠♦s ❝❛❞❛ ♠ó❞✉❧♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ s❡♣❛r❛❞❛♠❡♥t❡✳

            h(τ, ξ) 2h(τ, ξ)

            t

            ∂

            2h(τ, ξ) −

            2δ

            (τ ) |ξ|

            ′

            2h(τ, ξ)

            

            ′

            3

            (τ, ξ) − b

            1

            2h

            2δ

            (τ ) |ξ|

            ′

            P♦r ♦✉tr♦ ❧❛❞♦ Q(τ, ξ) 6 b

            −2δ .

            |ξ|

            −(α+2)

            = (1 + τ )

          • b
          • b

          • (1 + τ )
          • (1 + τ )

            ∞ t

            |ξ|

            −(α+2)

            (1 + τ )

            ∞ t

            Z

            ✭❆✳✷✷✮ ❖❜s❡r✈❡ q✉❡ ♣❛r❛ t ✜①♦ ❡ T &gt; t t❡♠✲s❡

            2(σ−2δ) dτ .

            |ξ|

            −(2α+1)

            (1 + τ )

            dτ exp C Z

            dτ =

            −2δ

            |ξ|

            −(α+2)

            (1 + τ )

            ∞ t

            C Z

            ❯s❛♠♦s ❛s ❡st✐♠❛t✐✈❛s ♦❜t✐❞❛s ♣❛r❛ P (τ, ξ) ❡ Q(τ, ξ) ❡♠ ✭❆✳✶✽✮ s❡❣✉❡ q✉❡ |β(t, ξ)| ≤ exp

            −2δ .

            |ξ|

            −(α+2)

            

          −2δ

            |ξ|

            |ξ|

            −(α+1)

            −(α+1)

            − (1 + t)

            −(α+1)

            (1 + T )

            T →∞

            lim

            −2δ

            = |ξ|

            T t

            −(α + 1)

            (1 + τ )

            −2δ

            T →∞

            lim

            −2δ

            |ξ|

            dτ =

            −(α+2)

            (1 + τ )

            t

            Z T

            T →∞

            lim

            

          2(σ−2δ)

            −(2α+1)

            −(α + 1)

            4h(τ, ξ)

            ′′

            

          2

            ] 2h(τ, ξ)

            2δ

            [h(τ, ξ) − b(τ)|ξ|

            2δ

            (τ ) |ξ|

            ′

            ≤ b

            2

            2δ

            2δ

            (τ ) |ξ|

            ′′

            − b

            

          2

            ] 2h(τ, ξ)

            2δ

            [h(τ, ξ) − b(τ)|ξ|

            2δ

            (τ ) |ξ|

            ′

            = b

            (τ ) |ξ|

            4h(τ, ξ)

            Q(τ, ξ) . (1 + τ )

            2α

            ✭❆✳✷✶✮ ❈♦♥s✐❞❡r❛♥❞♦ ✭❆✳✷✵✮ ❡ ✭❆✳✷✶✮ ❡♠ ✭❆✳✶✾✮ ♦❜t❡♠♦s

            −2δ .

            |ξ|

            −(α+2)

            

          2(σ−2δ)

            |ξ|

            −(2α+1)

            . (1 + τ )

            4δ

            |ξ|

            (1 + τ )

            2 .

            2δ

            |ξ|

            α−2

            

          2

            ] 2h(τ, ξ)

            2δ

            [h(τ, ξ) − b(τ)|ξ|

            2δ

            (τ ) |ξ|

            ′

            b

          • (1 + τ )

            2(σ−2δ) .

            1 ✭❆✳✷✹✮ s❡♥❞♦ q✉❡ ♥❛ ú❧t✐♠❛ ❡st✐♠❛t✐✈❛ ❢♦✐ ✉s❛❞♦ q✉❡ 2δ &lt; σ(1 + α) ❡

            2(σ−2δ)

            lim

            T →∞

            (1 + T )

            −2α

            − (1 + t)

            −2α

            −2α =

            |ξ|

            2(σ−2δ)

            2α (1 + t)

            

          −2α

          .

            (1 + t)

            T t

            α+1

            |ξ|

            2δ

            ≥ N✳ P♦rt❛♥t♦✱ ✉s❛♥❞♦ ✭❆✳✷✸✮ ❡ ✭❆✳✷✹✮ ❡♠ ✭❆✳✷✷✮ ❝♦♥❝❧✉í♠♦s q✉❡

            |β(t, ξ)| ≤ exp (C) = K, s❡♥❞♦ K é ✉♠❛ ❝♦♥st❛♥t❡ ❡ N s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ P♦rt❛♥t♦✱ β(t, ξ) é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞♦ ❡♠ Z

            low ell

            ✳ ❘❡st❛ ♠♦str❛r q✉❡

            |β(t, ξ) − 1| . (1 + t)

            −(α+1)

            |ξ|

            −2δ

            −2α

            |ξ|

            = |ξ|

            −2α

            = |ξ|

            dτ =

            −2δ

            α + 1 (1 + t)

            −(α+1) .

            1 ✭❆✳✷✸✮

            ♣♦✐s (t, ξ) ∈ Z

            low ell

            ✳ ❆❧é♠ ❞✐ss♦✱ Z

            ∞ t

            (1 + τ )

            −(2α+1)

            |ξ|

            2(σ−2δ)

            |ξ|

            −2α

            2(σ−2δ)

            lim

            T →∞

            Z T

            

          t

            (1 + τ )

            −(2α+1)

            dτ =

            |ξ|

            2(σ−2δ)

            lim

            T →∞

            (1 + τ )

          • (1 + τ )
          P❛r❛ ✐ss♦ ♥♦t❡ q✉❡ Z

            ∞

            ∂ h(τ, ξ)

            t

            |β(t, ξ)−1| = − h − −exp (0) exp . 2h(τ, ξ)

            t x

            ❙❡❥❛ f(x) = e ❡ Z

            ∞

            ∂ h(τ, ξ)

            t

            a := h(τ, ξ)

            1 (τ, ξ) dτ &lt; − h − ∞.

            2h(τ, ξ)

            t

            P❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ s❡❣✉❡ q✉❡ Z

            ∞

            ∂ h(τ, ξ)

            t

            h(τ, ξ)

            1 (τ, ξ) dτ

            − h − − exp (0) exp 2h(τ, ξ)

            t

            Z

            ∞

            ∂ h(τ, ξ)

            t ′

            6 f (c)

            1 (τ, ξ)

            − h(τ, ξ) − h dτ, 2h(τ, ξ)

            t

            ♣❛r❛ ❛❧❣✉♠ c ❡♥tr❡ 0 ❡ a✳ ▲♦❣♦ Z

            ∞

            ∂ t h(τ, ξ) (τ, ξ)

            1

            |β(t, ξ) − 1| . − h(τ, ξ) − h dτ 2h(τ, ξ)

            t

            Z Z

            ∞ ∞

            6 P (τ, ξ) dτ + Q(τ, ξ) dτ

            t t −(α+1) −2δ −2α 2(σ−2δ)

            (1 + t) + (1 + t) .

            |ξ| |ξ| ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦✳

            ❉❡✜♥✐♠♦s ❛❣♦r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ Z t

            2(σ−δ) 2δ |ξ|

            γ(t, s, ξ) := exp h + (τ, ξ) dτ

            1

            − b(τ)|ξ| 2b(τ )

            s

            ❡ 

            

          1−3α

             (1 + t)

             s❡ α &lt; 1/3   

            ψ (t, s) =

            α

            log(e + t) s❡ α = 1/3      1−3α

            (1 + s) s❡ α &gt; 1/3.

            low

            Pr♦♣♦s✐çã♦ ❆✳✼ P❛r❛ t♦❞♦ t ≥ s ❡ (s, ξ) ∈ Z t❡♠✲s❡

            ell 2(2σ−3δ)

            ψ (t, s).

            α

            |γ(t, s, ξ) − 1| . |ξ| ❉❡♠♦♥str❛çã♦✿

            ■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ❝♦♠ r❡st♦ ❞❡ ▲❛✲ √ x (t, ξ)

            ❣r❛♥❣❡ ❞❛ ❢✉♥çã♦ f(x) = ♣❛r❛ ❡s❝r❡✈❡r♠♦s h

            1 ❞❡ ♠❛♥❡✐r❛ ♠❛✐s 2 4δ

            (t, ξ) ❝♦♥✈❡♥✐❡♥t❡✳ ❈♦♥s✐❞❡r❡ ❡♥tã♦ x = −m

            1 ❡ a = b(t) ✳ Pr♦❝❡✲

            |ξ| ❞❡♥❞♦ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥♦ ▲❡♠❛ ❆✳✸ t❡♠♦s q p

            1

            2 4δ 2 4δ

            (t, ξ) + = b(t) (t, ξ)

            1

            1

            −m |ξ| p −m − b(t) |ξ|

            2 4δ

            2 b(t) |ξ|

            2

            1

            2 4δ 1 (t, ξ)

            p − −m − b(t) |ξ|

            2 4δ

            

          3

            8 (b(t) ) |ξ|

            1

            3 2 4δ

          • 1

            (t, ξ)

            √ −m − b(t) |ξ|

            5

            16 a &lt; a

            ❝♦♠ x &lt; a ✳ ▲♦❣♦✱

            2σ 4σ 6σ 2δ |ξ| |ξ| |ξ|

            h (t, ξ) = b(t)

            1

            |ξ| − − − √

            2δ 3 6δ

            5

            2b(t) 8b(t) 16 a |ξ| |ξ|

            6σ 2(σ−δ) 2(2σ−3δ) 2δ |ξ| |ξ| |ξ| = b(t) .

            √ |ξ| − − −

            3

            5

            2b(t) 8b(t) 16 a

          • |ξ|

            3

            

          2(2σ−3δ)

            |ξ|

            s

            Z t

            dτ − 0 .

            5

            16 √ a

            6σ

            − |ξ|

            8b(τ )

            

          3

            2(2σ−3δ)

            − |ξ|

            s

            Z t

            c

            ❡ c ∈ R✱ a &lt; c &lt; 0 t❡♠♦s |γ(t, s, ξ) − 1| = e

            &lt; ∞

            5

            16 √ a

            b(τ )

            6σ

            − |ξ|

            , ∀ (s, ξ) ∈ Z

            2σ

            ≤ µ|ξ|

            4(σ−δ)

            |ξ|

            −2α

            (1 + τ )

            1

            . ◆❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❆✳✺ ♦❜t✐✈❡♠♦s ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ c

            low ell

            5

            2 √ a

            2 √ a

            6σ

            |ξ|

            3 .

            b(τ )

            2(2σ−3δ)

            |ξ|

            ✭❆✳✷✺✮ ❱❛♠♦s ♠♦str❛r q✉❡

            5 dτ.

            6σ

            

          3

            ,

            = exp Z

            8b(τ )

            2(2σ−3δ)

            |ξ|

            2b(τ ) −

            2(σ−δ)

            − |ξ|

            2δ

            b(τ ) |ξ|

            t s

            2b(τ ) − 1

            − |ξ|

            2(σ−δ)

            |ξ|

            2δ

            − b(τ)|ξ|

            t s

            |γ(t, s, ξ) − 1| = exp Z

            (t, ξ) ♦❜té♠✲s❡

            1

            ❯t✐❧✐③❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ♦❜t✐❞❛ ♣❛r❛ h

            3

            6σ

            8b(τ )

            − |ξ|

            2(2σ−3δ)

            − |ξ|

            s

            Z t

            ❊♥tã♦ ❛♣❧✐❝❛♥❞♦ ♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ♥❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ❝♦♠ a :=

            dτ − exp (0) .

            5

            16 √ a

            6σ

            3

            16 √ a

            8b(τ )

            2(2σ−3δ)

            − |ξ|

            t s

            = exp Z

            2b(τ ) dτ − 1

            2(σ−δ)

            2δ

            − b(τ)|ξ|

            5

          • |ξ|

            ❛ss✐♠✱

            2σ 2α 4δ

            |ξ| ≤ µ (1 + τ) |ξ|

            2 4δ −1

            = µ b(τ ) |ξ|

            4σ 4 8δ 2 b(τ ) ξ .

            ⇒ |ξ| ≤ c |

            3 &gt; 0

            P♦r ✭❆✳✶✺✮✱ ❡①✐st❡ c t❛❧ q✉❡

            2 4δ

            m (τ, ξ) b(τ )

            1

            3

            ≤ −c |ξ|

            2 4δ 3 b(τ ) 1 (τ, ξ).

            ⇒ c |ξ| ≤ −m &gt; x = (τ, ξ)

            ❉❛s ❞✉❛s ❡st✐♠❛t✐✈❛s ❛❝✐♠❛ ❡ ❧❡♠❜r❛♥❞♦ q✉❡ a

            1

            −m ♦❜t❡♠♦s

            6 4σ+12δ 10 20δ

            b(τ ) b(τ )

            2

            |ξ| ≤ c |ξ|

            5 2 4δ

            = c b(τ )

            2

            |ξ|

            5

            (τ, ξ)

            4

            1

            ≤ c − m

            5 a .

            4

            ≤ c P♦rt❛♥t♦✱ q

            

          3 2σ+6δ

            5

            b(τ ) a .

            |ξ| q

            3 6σ+6δ 4σ

            5

            b(τ ) . a ⇒ |ξ| |ξ|

            ⇒ |ξ|

            dτ =

            − |ξ|

            2(2σ−3δ)

            −3α + 1 (1 + s)

            −3α+1 .

            |ξ|

            2(2σ−3δ) ψ α (t, s).

            ❙❡ α = 1/3 ❡♥tã♦ |γ(t, s, ξ) − 1| . |ξ|

            2(2σ−3δ)

            Z t

            

          s

            (1 + τ )

            −1

            |ξ|

            −3α + 1 (1 + t)

            2(2σ−3δ)

            (ln (1 + τ ))

            t s

            = |ξ|

            2(2σ−3δ)

            ln (1 + t) − |ξ|

            2(2σ−3δ)

            ln (1 + s) . |ξ|

            2(2σ−3δ) ln (1 + t).

            ❆s ❞✉❛s ú❧t✐♠❛s ❡st✐♠❛t✐✈❛s ♣r♦✈❛♠ ❛ ♣r♦♣♦s✐çã♦✳ ▲❡♠❛ ❆✳✽ P❛r❛ t♦❞♦ t ≥ s ❡ (s, ξ) ∈ Z

            low ell

            ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❢✉♥✲ çã♦ a ell

            −3α+1

            2(2σ−3δ)

            6σ

            3

            2 p a

            5 .

            |ξ|

            4σ−6δ

            b(τ )

            3 .

            ❯s❛♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛ ❡♠ ✭❆✳✷✺✮ ♦❜t❡♠♦s ♣❛r❛ α 6= 1/3✿ |γ(t, s, ξ) − 1| .

            Z t

            s

            |ξ|

            2(2σ−3δ)

            b(τ )

            dτ . |ξ|

            = |ξ|

            2(2σ−3δ)

            Z t

            s

            (1 + τ )

            −3α

            dτ =

            |ξ|

            2(2σ−3δ)

            (1 + τ )

            −3α+1

            −3α + 1

            t s

            (t, s, ξ) ❞❡✜♥✐❞❛ ❡♠ ✭✷✳✷✺✮ ❡♠ ❢✉♥çã♦ ❞❡ γ(t, s, ξ) ❞❛ s❡❣✉✐♥t❡

            ♠❛♥❡✐r❛✿ β(s, ξ)

            ♯ 2(σ−δ)

            −|ξ| β(t, ξ)

            −1 −1

            N P (t, ξ)Q (t, s, ξ)N(s, ξ)P

            ell

            s❡♥❞♦ β(t, ξ) ❞❡✜♥✐❞♦ ❡♠ ✭❆✳✶✻✮✳ ❉❡♠♦♥str❛çã♦✿

            (t, s, ξ) (t, s, ξ) ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ a ell ✱ ❛ ❞❡✜♥✐çã♦ ❞❡ e ell ❞❛❞❛ ❡♠

            low

            (t, s, ξ) ✭✷✳✹✵✮ ❡ ❛ ❡①♣r❡ssã♦ E ❞❛❞❛ ❡♠ ✭✷✳✹✺✮ t❡♠♦s

            ell

            h(s, ξ)

            

          2δ low

            a (t, s, ξ) = exp Λ(t, s) E (t, s, ξ)

            ell

            −|ξ| ell h(t, ξ) h(s, ξ)

            2δ

          −1 −1

            N = exp Λ(t, s) P (t, ξ)e (t, s, ξ)Q (t, s, ξ)N(s, ξ)P

            ell ell

            −|ξ| h(t, ξ) 1 2 Z t h(s, ξ)

            2δ −1 −1

            N = P (t, ξ)exp b(τ ) dτ

            −|ξ| h(t, ξ)

            s

            Z t exp h(τ, ξ) dτ Q (t, s, ξ)N(s, ξ)P

            ell s 1 2 Z t h(s, ξ) 2δ

            = exp b(τ ) + h(τ, ξ) dτ −|ξ| h(t, ξ)

            s −1 −1

            N P (t, ξ)Q ell (t, s, ξ)N(s, ξ)P. ❱❛♠♦s r❡❡s❝r❡✈❡r ♦s ❞♦✐s ♣r✐♠❡✐r♦s t❡r♠♦s ❞♦ ♣r♦❞✉t♦ ❛❝✐♠❛ ❞❡ 2 1

            h(s,ξ)

            ✉♠❛ ♦✉tr❛ ❢♦r♠❛✳ ❖ ♣r✐♠❡✐r♦ t❡r♠♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

            h(t,ξ)

            ❛ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ❞❡ s ❛ t✱ ❡ ❛ss✐♠ ❛❣r✉♣❛r ❝♦♠ ❛ ♦✉tr❛

            ❡①♣♦♥❡♥❝✐❛❧✳ P❛r❛ ✐ss♦ ♥♦t❡ q✉❡ ∂ h(τ, ξ) ∂ (ln(h(τ, ξ)))

            t t

            2h(τ, ξ)

            2 ❆ss✐♠✱

            Z Z

            t t

            ∂ h(τ, ξ) ∂ (ln(h(τ, ξ)))

            t t

            exp dτ = exp dτ − −

            2h(τ, ξ)

            2

            s s

            Z t 1 1 t − − 2 2 = exp ∂ ln(h(τ, ξ)) dτ = exp ln(h(τ, ξ))

            t s 1 s 1 1 2 ! h(s, ξ) − − 2 2

            = exp ln(h(t, ξ)) = exp ln − ln((s, ξ)) 2 1 h(t, ξ) h(s, ξ)

            = . h(t, ξ)

            ❆❧é♠ ❞✐ss♦✱ Z t

            2δ

            exp b(τ ) + h(τ, ξ) dτ −|ξ|

            s

            Z t

            2(σ−δ) 2δ |ξ|

            = exp b(τ ) + h(τ, ξ) + h (τ, ξ) +

            1

            −|ξ| 2b(τ )

            s 2(σ−δ)

            |ξ| (τ, ξ) dτ

            1

            −h − 2b(τ )

            Z t = exp (h(τ, ξ) (τ, ξ)) dτ

            1

            − h

            s

            Z t

            2(σ−δ)

          2δ |ξ|

            exp h (τ, ξ) b(τ ) + dτ

            1

            − |ξ| 2b(τ )

            s

            Z t

            2(σ−δ)

            |ξ| exp dτ −

            2b(τ )

            s

            Z t

            ♯ 2(σ−δ)

            = exp (h(τ, ξ) (τ, ξ)) dτ γ(t, s, ξ) exp Λ (t, s) .

            1

            − h −|ξ|

            s P❡❧❛s ❞✉❛s ú❧t✐♠❛s ✐❣✉❛❧❞❛❞❡s t❡♠♦s h(s, ξ) h(t, ξ) 1 2 Z

            t s

            ∞ s

            Z

            h(τ, ξ) 2h(τ, ξ) dτ exp

            t

            ∂

            (τ, ξ) −

            

          1

            h(τ, ξ) − h

            Z

            h(τ, ξ) − h

            (t, s) = exp

            ♯

            Λ

            2(σ−δ)

            γ(t, s, ξ) exp −|ξ|

            h(τ, ξ) 2h(τ, ξ) dτ

            t

            ∞ t

            

          1

            1 (τ, ξ)

            β(s, ξ) β(t, ξ)

            ❞♦ ✐♥í❝✐♦ ❞❛ ❞❡♠♦♥str❛çã♦ ♦ ❧❡♠❛ ❡stá ❞❡♠♦♥str❛❞♦✳

            ell (t, s, ξ)

            ❙✉❜st✐t✉✐♥❞♦ ❡ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♥❛ ✐❣✉❛❧❞❛❞❡ ❞❡ a

            ♯ (t, s) .

            Λ

            2(σ−δ)

            γ(t, s, ξ) exp −|ξ|

            (t, s) =

            (τ, ξ) −

            ♯

            Λ

            2(σ−δ)

            γ(t, s, ξ) exp −|ξ|

            h(τ, ξ) 2h(τ, ξ) dτ

            t

            ∂

            − ∂

            h(τ, ξ) − h

            −|ξ|

            (τ, ξ)) dτ γ(t, s, ξ) exp

            s

            Z t

            (t, s) = exp

            ♯

            Λ

            2(σ−δ)

            −|ξ|

            1

            1

            (h(τ, ξ) − h

            t s

            2h(τ, ξ) dτ exp Z

            − ∂ t h(τ, ξ)

            t s

            = exp Z

            2δ

            h(τ, ξ) − h

            (τ, ξ) −

            ∞ t

            ∞ s

            − Z

            h(τ, ξ) 2h(τ, ξ) dτ

            t

            ∂

            (τ, ξ) −

            

          1

            h(τ, ξ) − h

            Z

            ∂

            (t, s) = exp

            ♯

            Λ

            2(σ−δ)

            γ(t, s, ξ) exp −|ξ|

            h(τ, ξ) 2h(τ, ξ) dτ

            t

            ❯s❛♥❞♦ ❛s Pr♦♣♦s✐çõ❡s ❆✳✶✱ ❆✳✻ ❡ ❆✳✼ ♣r♦✈❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧✲ t❛❞♦✳ Pr♦♣♦s✐çã♦ ❆✳✾ ❙❡❥❛

            −1

            ∞, s, ξ)N(s, ξ)P,

            low

            ❡♥tã♦ ♣❛r❛ t♦❞♦ t ≥ s ❡ (s, ξ) ∈ Z ✈❛❧❡ ❛ ❡st✐♠❛t✐✈❛✿ 2(σ−δ) ♯ ell Λ (t,s)

            −|ξ|

          ell (t, s, ξ)

            − a K(s, ξ)e 2(σ−δ) ♯

            Λ (t,s)

          −(α+1) −2δ 2(2σ−3δ) −|ξ|

            . (1 + t) + ψ (t, s) e .

            α

            |ξ| |ξ| ❉❡♠♦♥str❛çã♦✿

            ❱❛♠♦s r❡❡s❝r❡✈❡r ♦ t❡r♠♦ ❞❛ ❡sq✉❡r❞❛ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✉s❛♥❞♦

            ell (t, s, ξ)

            ❛ ❞❡✜♥✐çã♦ ❞❡ K(s, ξ) ❡ ❛ ✐❣✉❛❧❞❛❞❡ ❞❡ a ❞❛❞❛ ♥♦ ▲❡♠❛ ❆✳✽ ❞❡ ♠❛♥❡✐r❛ ❛ ❢❛❝✐❧✐t❛r ♥♦ss❛s ❡st✐♠❛t✐✈❛s✳ ❆ss✐♠✱ 2(σ−δ) ♯

            Λ (t,s) −|ξ| ell (t, s, ξ)

            − a K(s, ξ)e 2(σ−δ) ♯

            Λ (t,s) −1 −|ξ|

            = Q (

            ell

            ∞, s, ξ)N(s, ξ)P e β(s, ξ)P 2(σ−δ) ♯

            β(s, ξ)

            Λ (t,s) −|ξ| −1 −1

            N γ(t, s, ξ) e P (t, ξ)Q (t, s, ξ)N(s, ξ)P

            ell

            − β(t, ξ) 2(σ−δ) ♯

            Λ (t,s) −|ξ| −1

            = P β(s, ξ)e

            γ(t, s, ξ)

            

          −1

            N N Q ( (t, ξ)Q (t, s, ξ) (s, ξ)P

            ell ell ✭❆✳✷✻✮

            ∞, s, ξ) − .

            β(t, ξ) ❆❣♦r❛✱ ♥♦t❡ q✉❡ ♣❛r❛ ♣r♦✈❛r♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡✈❡♠♦s ❡st✐♠❛r ♦s t❡♠♦s ❞❡ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✳ ❖❜s❡r✈❡ q✉❡ ❼ P❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✻✱ β(s, ξ) é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞♦❀

            −1

            ❼ ❆s ♠❛tr✐③❡s P ❡ P sã♦ ❝♦♥st❛♥t❡s❀ ❼ ❆ ♠❛tr✐③ N(s, ξ) é ❧✐♠✐t❛❞❛❀

            ❈♦♠ ✐ss♦ r❡st❛✲♥♦s ❡st✐♠❛r ❞❡ ♠❛♥❡✐r❛ ❝♦♥✈❡♥✐❡♥t❡ ❛ ❡①♣r❡ssã♦ ❡♥tr❡ ❝♦❧❝❤❡t❡s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✳ ❚❡♠♦s q✉❡

            −1 −1

            N Q ell ( (t, ξ)Q ell (t, s, ξ)

            ∞, s, ξ) − γ(t, s, ξ)β(t, ξ) = Q ( ( (

            ell ell ell

            ∞, s, ξ) − γ(t, s, ξ)Q ∞, s, ξ) + γ(t, s, ξ)Q ∞, s, ξ)

            −1 −1

            N (t, ξ)Q ell (t, s, ξ)

            −γ(t, s, ξ)β(t, ξ) Q (

            γ(t, s, ξ) − 1 ell ≤ ∞, s, ξ)

            −1 −1

            N γ(t, s, ξ) Q + ell ( (t, ξ)Q ell (t, s, ξ)

            ∞, s, ξ) − β(t, ξ) = Q (

            γ(t, s, ξ) − 1 ell ∞, s, ξ)

            −1

            γ(t, s, ξ) Q ( Q + ell ell ( ∞, s, ξ) − β(t, ξ) ∞, s, ξ)

            −1 −1 −1

            N

          • β(t, ξ) Q ( (t, ξ)Q (t, s, ξ)

            ell ell

            ∞, s, ξ) − β(t, ξ)

            Q ell ( γ(t, s, ξ) Q ell ( γ(t, s, ξ) − 1 1 − β(t, ξ)

          • −1

            ≤ ∞, s, ξ) ∞, s, ξ)

            −1 −1

          • ell ell

            ( (t, ξ)Q (t, s, ξ) β(t, ξ) Q

            ∞, s, ξ) − N

            = Q ell ( γ(t, s, ξ) Q ell ( γ(t, s, ξ) − 1 1 − β(t, ξ)

          • −1

            ∞, s, ξ) ∞, s, ξ)

            −1 −1

          • ell ell

            ( (t, ξ)Q ( β(t, ξ) Q

            ∞, s, ξ) − N ∞, s, ξ)

            −1 −1

          • N (t, ξ)Q ell ( (t, ξ)Q ell (t, s, ξ)

            ∞, s, ξ) − N

            −1

            Q

            

          ell γ(t, s, ξ) Q ell

          • ( (

            ≤ ∞, s, ξ) ∞, s, ξ) 1 − β(t, ξ) γ(t, s, ξ) − 1

            −1 −1

            ell

            I − N ∞, s, ξ)

          • (t, ξ) β(t, ξ) Q (

            −1

          • ell ell . ✭❆✳✷✼✮

            N (t, ξ) ( (t, s, ξ) Q

            ∞, s, ξ) − Q

          • e

            −1

            −1

            (t, ξ) ❯t✐❧✐③❛♥❞♦ q✉❡ N = I + N

            1

            ❡ ❛ ❡st✐♠❛t✐✈❛ ♦❜t✐❞❛ ❡♠ ✭✷✳✸✵✮ t❡♠♦s✱ I − N

            −1

            (t, ξ) = N

            −1

            (t, ξ) − N

            −1

            (t, ξ)N(t, ξ) =

            N

            (t, ξ) I − N(t, ξ)

            −2δ .

            = N

            −1

            (t, ξ) N

            1

            (t, ξ) . N

            1 (t, ξ)

            . (1 + t)

            −(α+1)

            |ξ|

            −2δ .

            ❊st✐♠❛t✐✈❛ ✸✿ 1 − β(t, ξ)

            ❊st✐♠❛t✐✈❛ ✷✿ I − N

            |ξ|

            ❊st✐♠❛r❡♠♦s ❛s q✉❛tr♦ ❞✐❢❡r❡♥ç❛s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ s❡♣❛r❛❞❛✲ ♠❡♥t❡✳ ❊st✐♠❛t✐✈❛ ✶✿ Q ell

            ∞, s, ξ) − Q

            ( ∞, s, ξ) − Q

            ell

            (t, s, ξ) ❯t✐❧✐③❛♥❞♦ ❛s ❡st✐♠❛t✐✈❛s ✭❆✳✶✶✮ ❡ ✭❆✳✶✹✮ ❡ ❛ Pr♦♣♦s✐çã♦ ❆✳✶✱ q✉❡

            ♥♦s ❣❛r❛♥t❡ q✉❡ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ❞❡ Q ell (

            ∞, s, ξ) é ♥✉❧❛✱ ✐st♦ é✱ e

            1 T

            Q

            ell

            ( ∞, s, ξ) =

            (0, 0) ✱ s❡❣✉❡ q✉❡

            Q ell (

            ell

            −(α+1)

            (t, s, ξ) ≤ e

            T

            (Q ell ( ∞, s, ξ) − Q

            ell (t, s, ξ)) + e

            1 T

            (Q ell (t, s, ξ)) . (1 + t)

            −(α+1)

            |ξ|

            −2δ

            −c

          |ξ|

          Λ(t,0)

            . (1 + t)

            −1 P❡❧♦s ❝á❧❝✉❧♦s ❢❡✐t♦s ♥❛ Pr♦♣♦s✐çã♦ ❆✳✻ ♦❜té♠✲s❡ Z

            ∞

            ∂ h(τ, ξ)

            t −1

            |β(t, ξ) | = − − h − exp 2h(τ, ξ)

            t

            Z

            ∞

            ∂ h(τ, ξ)

            t 1 (τ, ξ)

            ≤ exp − h(τ, ξ) − h dτ 2h(τ, ξ)

            t ≤ K.

            ❆ss✐♠✱

            −1 −1 −1

            β(t, ξ) |1 − β(t, ξ) | = |β(t, ξ) − β(t, ξ) |

            −1

            = |β(t, ξ) | |β(t, ξ) − 1|

            −(α+1) −2δ −2α 2(σ−2δ) . (1 + t) + (1 + t) .

            |ξ| |ξ| ❊st✐♠❛t✐✈❛ ✹✿

            γ(t, s, ξ) − 1 P❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✼✿

            2(2σ−3δ) (t, s) .

          α

            |γ(t, s, ξ) − 1| . ψ |ξ| (

            ❯s❛♥❞♦ ❛s q✉❛tr♦ ❡st✐♠❛t✐✈❛s ❛❝✐♠❛ ❡ ♦ ❢❛t♦ ❞❛s ❢✉♥çõ❡s Q ell ∞, s, ξ)✱

            −1 −1

            γ(t, s, ξ) (t, ξ) ✱ β(t, ξ) ❡ N s❡r❡♠ ❧✐♠✐t❛❞❛s ❡♠ ✭❆✳✷✼✮ ♦❜t❡♠♦s

            −1 −1

            N Q ( (t, ξ)Q (t, s, ξ)

            ell ell

            ∞, s, ξ) − γ(t, s, ξ)β(t, ξ)

            2(2σ−3δ) −(α+1) −2δ −2α 2(σ−2δ) ψ (t, s) + (1 + t) + (1 + t) .

            . α |ξ| |ξ| |ξ| P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ s❡❣✉❡ q✉❡ (α+1) 1−3α

            −2α 2(σ−2δ) −δ 2σ−3δ

          2

          2

            |ξ| |ξ| |ξ|

            1

            −(α+1) −2δ 1−3α 2(2σ−3δ)

            (1 + t) + (1 + t) ≤ |ξ| |ξ|

            2

            1

            −(α+1) −2δ 2(2σ−3δ)

            (1 + t) + ψ α (t, s) ≤ |ξ| |ξ|

            2

            1−3α

            (t, s) ♣♦✐s (1 + t) α ✳

            ≤ ψ ❈♦♥s✐❞❡r❛♥❞♦ ❛s ❞✉❛s ú❧t✐♠❛s ❡st✐♠❛t✐✈❛s ❡♠ ✭❆✳✷✻✮ ❝♦♥❝❧✉í♠♦s q✉❡ 2(σ−δ) ♯

            Λ (t,s) −|ξ|

            (t, s, ξ)

            ell

            − a K(s, ξ)e 2(σ−δ) ♯

            Λ (t,s) 2(2σ−3δ) −(α+1) −2δ −|ξ|

            . ψ α (t, s) + (1 + t) e |ξ| |ξ|

            ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦✳

            ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

            ❬✶❪ ❘✳ ❆✳ ❆❞❛♠s✱ ❙♦❜♦❧❡✈ ❙♣❛❝❡s✳ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✱ ✶✾✼✺✳ ❬✷❪ ❍✳ ❇r❡③✐s✱ ❆♥á❧✐s✐s ❢✉♥❝✐♦♥❛❧ ❚❡♦r✐❛ ② ❛♣❧✐❝❛❝✐♦♥❡s✳ ❆❧✐❛♥③❛ ❊❞✐✲ t♦r✐❛❧✱ ▼❛❞r✐❞✱ ✶✾✽✸✳ ❬✸❪ ❘✳ ❈✳ ❈❤❛rã♦✱ ❈✳ ❘✳ ❞❛ ▲✉③✱ ❘✳ ■❦❡❤❛t❛✱ ◆❡✇ ❞❡❝❛② r❛t❡s ❢♦r ❛

            ♣r♦❜❧❡♠ ♦❢ ♣❧❛t❡ ❞②♥❛♠✐❝s ✇✐t❤ ❢r❛❝t✐♦♥❛❧ ❞❛♠♣✐♥❣✳ ❏✳ ❍②♣❡r❜♦❧✐❝ ❉✐✛❡r✳ ❊q♥s✳ ✭✷✵✶✸✮✱ ♥♦✳ ✸✱ ✺✻✸✕✺✼✺✳

            ❬✹❪ ❘✳ ❈♦✐♠❜r❛ ❈❤❛rã♦✱ ❈✳ ❘✳ ❞❛ ▲✉③✱ ❘✳ ■❦❡❤❛t❛✱ ❙❤❛r♣ ❞❡❝❛② r❛t❡s ❢♦r ✇❛✈❡ ❡q✉❛t✐♦♥s ✇✐t❤ ❛ ❢r❛❝t✐♦♥❛❧ ❞❛♠♣✐♥❣ ✈✐❛ ♥❡✇ ♠❡t❤♦❞ ✐♥ t❤❡ ❋♦✉r✐❡r s♣❛❝❡✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✭✷✵✶✸✮✱ ♥♦✳ ✶✱ ✷✹✼✕✷✺✺✳

            ❬✺❪ ▼✳ ❉✬❆❜❜✐❝❝♦✱ ❘✳ ❈♦✐♠❜r❛ ❈❤❛rã♦✱ ❈✳ ❘✳ ❞❛ ▲✉③✱ ❙❤❛r♣ t✐♠❡ ❞❡❝❛② r❛t❡s ♦♥ ❛ ❤②♣❡r❜♦❧✐❝ ♣❧❛t❡ ♠♦❞❡❧ ✉♥❞❡r ❡✛❡❝ts ♦❢ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ ❞❛♠♣✐♥❣ ✇✐t❤ ❛ t✐♠❡✲❞❡♣❡♥❞❡♥t ❝♦❡✣❝✐❡♥t✳ ❉✐s❝r❡t❡ ❈♦♥t✐♥✳ ❉②♥✳ ❙②st✳ ❙❡r✳ ❆ ✭✷✵✶✻✮✱ ✷✹✶✾✕✷✹✹✼✳

            ❬✻❪ ▼✳ ❉✬❆❜❜✐❝❝♦✱ ▼✳ ❘✳ ❊❜❡rt✱ ❆ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ str✉❝t✉r❛❧ ❞✐ss✐✲ ♣❛t✐♦♥s ❢♦r ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs✳ ▼❛t❤✳ ▼❡t❤♦❞s ✐♥ t❤❡ ❆♣♣❧✳ ❙❝✐✳✱ ✭✷✵✶✺✮✳ ❆✈❛✐❧❛❜❧❡ ❢r♦♠✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✶✵✵✷✴♠♠❛✳✸✼✶✸✳

            ❬✼❪ ❘✳ ❉❛✉tr❛②✱ ❏✳ ▲✳ ▲✐♦♥s✱ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ❱♦❧✉♠❡ ✷✿ ❋✉♥❝t✐♦♥❛❧ ❛♥❞ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ✶✾✽✽✳

            ❬✽❪ ▲✳ ❈✳ ❊✈❛♥s✱ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛✲ t✐❝❛❧ ❙♦❝✐❡t②✱ ✷✵✵✷✳ ❬✾❪ ●✳ ❑❛r❝❤✱ ❙❡❧❢s✐♠✐❧❛r ♣r♦✜❧❡s ✐♥ ❧❛r❣❡ t✐♠❡ ❛s②♠♣t♦t✐❝s ♦❢ s♦❧✉t✐♦♥s t♦ ❞❛♠♣❡❞ ✇❛✈❡ ❡q✉❛t✐♦♥s✳ ❙t✉❞✐❛ ▼❛t❤✳ ✶✹✸ ✭✷✵✵✵✮✱ ✶✼✺✕✶✾✼✳

            ❬✶✵❪ ❙✳ ❑❡s❛✈❛♥✱ ❚♦♣✐❝s ✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❲✐❧❡② ❊❛st❡r♥ ▲✐♠✐t❡❞✱ ❇❛♥❣❛❧♦r❡✱ ✶✾✽✾✳

            ❬✶✶❪ P✳ ▼❛rt✐♥❡③✱ ❆ ♥❡✇ ♠❡t❤♦❞ t♦ ♦❜t❛✐♥ ❞❡❝❛② r❛t❡ ❡st✐♠❛t❡s ❢♦r ❞✐s✲ s✐♣❛t✐✈❡ s②st❡♠s✳ ❊❙❆■▼✿ ❈♦♥tr♦❧✱ ❖♣t✐♠✐s❛t✐♦♥ ❛♥❞ ❈❛❧❝✉❧✉s ♦❢ ❱❛r✐❛t✐♦♥s✱ ✹ ✭✶✾✾✾✮✱ ♣♣✳ ✹✶✾✲✹✹✹✳

            ❬✶✷❪ ▲✳ ❆✳ ▼❡❞❡✐r♦s✱ P✳ ❍✳ ❘✐✈❡r❛✱ ■♥✐❝✐❛çã♦ ❛♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈✳ ■▼✲❯❋❘❏✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ✶✾✼✼✳

            ❬✶✸❪ ▲✳ ❆✳ ▼❡❞❡✐r♦s✱ P✳ ❍✳ ❘✐✈❡r❛✱ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❡ ❛♣❧✐❝❛çõ❡s às ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✳ ❚❡①t♦s ❞❡ ▼ét♦❞♦s ▼❛t❡♠át✐❝♦s ◆♦✳ ✾✱ ■▼✲❯❋❘❏✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳

            ❬✶✹❪ ▼✳ ❑✳ ▼❡③❛❞❡❦✱ ❙tr✉❝t✉r❛❧ ❞❛♠♣❡❞ σ✲❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs✳ ❉r✳❙❝ ❚❤❡s✐s ✱ ❚❯ ❇❡r❣❛❦❛❞❡♠✐❡ ❋r❡✐❜❡r❣✱ ●❡r♠❛♥②✱ ✷✵✶✹✳

            ❬✶✺❪ ▼✳ ❘❡✐ss✐❣✱ L p q ❉❡❝❛② ❊st✐♠❛t❡s ❢♦r ❲❛✈❡ ❊q✉❛t✐♦♥ ✇✐t❤ ❚✐♠❡✲ −L

            ❉❡♣❡♥❞❡♥t ❈♦❡✣❝✐❡♥ts✳ ❏✳ ◆♦♥❧✐♥❡❛r ▼❛t❤✳ P❤②s✐❝s ✭✷✵✵✹✮✱ ✺✸✹✕ ✺✹✽✳

            ❬✶✻❪ ▼✳ ❘❡✐ss✐❣✱ ❳✳▲✉✱ ❘❛t❡s ♦❢ ❞❡❝❛② ❢♦r str✉❝t✉r❛❧ ❞❛♠♣❡❞ ♠♦❞❡❧s ✇✐t❤ ❞❡❝r❡❛s✐♥❣ ✐♥ t✐♠❡ ❝♦❡✣❝✐❡♥ts✳ ■♥t✳ ❏✳ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❛♥❞ ❉✐✛✳ ❊q♥s✳ ✭✷✵✵✾✮✱ ✷✶✕✺✺✳

            ❬✶✼❪ ❏✳ ❲✐rt❤✱ ❆s②♠♣t♦t✐❝ ♣r♦♣❡rt✐❡s ♦❢ s♦❧✉t✐♦♥s t♦ ✇❛✈❡ ❡q✉❛t✐♦♥s ✇✐t❤ t✐♠❡✲❞❡♣❡♥❞❡♥t ❞✐ss✐♣❛t✐♦♥✳ ❉r✳❙❝ ❚❤❡s✐s✱ ❚❯ ❇❡r❣❛❦❛❞❡♠✐❡ ❋r❡✐✲ ❜❡r❣✱ ●❡r♠❛♥②✱ ✷✵✵✺✳

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