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

165 

Full text

(1)

❯♥✐✈❡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

(2)
(3)

❯♥✐✈❡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❡✳

(4)

Ficha de identificação da obra elaborada pelo autor,

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

Ribeiro, Jéssika

Estudo assintótico para um modelo de evolução com operadores fracionários e coeficiente dependendo do tempo / Jéssika Ribeiro ; orientador, Cleverson Roberto da Luz -Florianópolis, SC, 2016.

165 p.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Matemática Pura e Aplicada. Inclui referências

1. Matemática Pura e Aplicada. 2. Equação de Evolução. 3. Dissipação com Coeficiente Dependendo do Tempo. 4.

(5)

❊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❊❙

(6)
(7)

❖ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛ é ♦ ♠❛✐s ✐♥❞✐❝❛❞♦ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛s ❢❛❝✉❧❞❛❞❡s✱ ❢♦rt❛❧❡❝❡r ♦ r❛❝✐♦❝í♥✐♦ ❡ ✐❧✉♠✐♥❛r ♦ ❡s♣ír✐t♦✳ ✭❙ó❝r❛t❡s✱ ❋✐❧ós♦❢♦ ●r❡❣♦✮

(8)
(9)

❆❣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❛♠

(10)

❝✉✐❞❛❞♦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✲

(11)

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✐♠♦ ❛♥♦✳

(12)

❘❡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♦ ❞❡ ❋♦✉r✐❡r✱ Rn

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

❞✐❢❡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❛ ❡♠ ❬✸❪ ❡ ❬✹❪✳

(13)

❆❜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❡❞ ❈❛✉✲ ❝❤② ♣r♦❜❧❡♠✱ ✇❡ s♣❧✐t t❤❡ ❋♦✉r✐❡r s♣❛❝❡✱ Rn

ξ✱ ✐♥ t✇♦ r❡❣✐♦♥s✿ ❤✐❣❤

❛♥❞ ❧♦✇ ❢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❛ ✐♥ ❬✸❪ ❛♥❞ ❬✹❪✳

(14)
(15)

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ①✐✈

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

✶✳✶ ◆♦t❛çõ❡s ❡ Pr✐♠❡✐r♦s ❈♦♥❝❡✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❊s♣❛ç♦sLp(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✹ ❊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❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

(16)

✷✳✷ ❩♦♥❛ ❊❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

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

✹ ❘❡s✉❧t❛❞♦ Pr✐♥❝✐♣❛❧ ✼✽

❆ ❘❡s✉❧t❛❞♦s ❆❞✐❝✐♦♥❛✐s ✶✵✷

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✹✼

(17)

■♥tr♦❞✉çã♦

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ♣r♦♣r✐❡❞❛❞❡s ❞❛ s♦✲ ❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ σ✲❡✈♦❧✉çã♦ ❡♠ Rn✱ ❝♦♠ ♦ ❝♦❡✜❝✐❡♥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❛ ❛ ♥♦r♠❛L2 ❞❛ s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ ♠♦❞❡❧♦✿

utt(t, x) + (−∆)σu(t, x) + 2b(t)(−∆)δut(t, x) = 0, t>0, x∈Rn ✭✶✮

u(0, x) =u0(x), ut(0, x) =u1(x), x∈Rn

♦♥❞❡σ > 0✱ δ(0, σ)✱ 2δ < σ(1 +α) ❡ b(t) =µ(1 +t)α ✉♠❛ ❢✉♥çã♦

♣♦s✐t✐✈❛ ❝♦♠ µ >0 ❡ 0< α <1.

(18)

❖ t❡r♠♦ 2b(t)(−∆)δu

t r❡♣r❡s❡♥t❛ ♦ ❛♠♦rt❡❝✐♠❡♥t♦ ❡str✉t✉r❛❧ ❞❡✲

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

(∆)δv(x) := F−1(

|

b

v(ξ))(x), ♣❛r❛ t♦❞♦ v H2δ(Rn)✱ ♦♥❞❡ F ❞❡✲

♥♦t❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❝♦♠ r❡s♣❡✐t♦ ❛ ✈❛r✐á✈❡❧x❡ r❡♣r❡s❡♥t❛✲

♠♦s ♣♦rbv=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❛

b(t) =µ(1 +t)α ♣❛r❛ ❛❧❣✉♠ µ >0 ❡ α∈(0,1).

❚❡♠♦s q✉❡ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❡s♣❡❝í✜❝❛ ❞❡ ❝♦❡✜❝✐❡♥t❡s b(t) q✉❡ ❞❡✲ ♣❡♥❞❡♠ ❞♦s ❡①♣♦❡♥t❡s ❢r❛❝✐♦♥ár✐♦sσ❡δ✱ ♦ ♣❡r✜❧ ❛ss✐♥tót✐❝♦ ❞❛ s♦❧✉çã♦

❞❡ ✭✶✮✱ q✉❛♥❞♦t→ ∞✱ é ♦ ♠❡s♠♦ ❞♦ ♣r♦❜❧❡♠❛ vt(t, x) +

1 2b(t)(−∆)

σ−δv(t, x) = 0, v(0, x) =v0(x), ✭✷✮

♣❛r❛ ✉♠❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞❡v0✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡u0, u1, b(t), σ ❡δ. ❖ ♣r♦❜❧❡♠❛ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞❡ ♣r♦❜❧❡♠❛ ❞❡ ❞✐❢✉sã♦ ❛♥ô♠❛❧❛✱ q✉❡ s✐❣♥✐✜❝❛ q✉❡✱ ❡♠ ❣❡r❛❧✱ σ−δ 6= 1✱ ❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r✱ q✉❛♥❞♦σδ= 1.

(19)

❈♦♥s✐❞❡r❛♥❞♦ ♦✉tr❛ ❝❧❛ss❡ ❞❡ ❝♦❡✜❝✐❡♥t❡sb(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❛❞♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ♦♣❡r❛❞♦r ♣s❡✉❞♦✲❞✐❢❡r❡♥❝✐❛❧ exp−(−∆)δRt

0b(τ)dτ

à s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❞❛ ❡q✉❛çã♦ ❞❡ ❡✈♦❧✉çã♦ ❧✐✈r❡ utt+

(∆)σu= 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❛−1α <0✳ P❛r❛α <

−1t❡♠♦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

utt(t, x)−∆u(t, x) + 2b(t)ut(t, x) = 0 (u, ut)(0, x) = (u0, u1)(x), ♣r♦✈❛♥❞♦ q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ss❛ ❡q✉❛çã♦ t❡♠ ♠✉❞❛♥✲ ç❛s ♣r♦❢✉♥❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ b(t)✳

(20)

❘❡❝❡♥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♦Lp−Lq♣❛r❛

♠♦❞❡❧♦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

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

u(0, x) =u0(x), ut(0, x) =u1(x),

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

−∆ é ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡✱(∆)σu

t♦ t❡r♠♦ ❞❡ ❞✐ss✐♣❛çã♦ ❡b(t)♦ ❝♦❡✜❝✐❡♥t❡

❞❡❝r❡s❝❡♥t❡ q✉❡ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦✳

(21)

▼✳ ❑✳ ▼❡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✉❞❛❞♦ ❢♦✐

utt(t, x) + (−∆)σu(t, x) +b(t)(−∆)δut(t, x) = 0

u(0, x) =u0(x), ut(0, x) =u1(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)✱ ✉s❛♠♦s ❛ ❝✉r✈❛ ΓN(t, ξ)✱ ♣❛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❡✳

(22)

❖ 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✱ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s s♦❜r❡ ❊s♣❛ç♦s Lp✱ ❚❡♦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❛❞❛ ♣♦rZpd ❡ ❡❧í♣t✐❝❛ ❞❡♥♦✲

t❛❞❛ ♣♦rZlow 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❛✳

(23)

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❛

¯

θ > 1+α:

k(−∆)k2u(t)k

L2(Rn) ≤ Ct−k01ku0kLp(Rn)+Ct−k11ku1kLp(Rn) + ku0kHk(Rn)+ku1kHk−σ(Rn)e−C(1+t)

1−α

❝♦♠θ¯ k01 k11 ❞❡✜♥✐❞♦s ♥♦ ❚❡♦r❡♠❛ ✹✳✷✳ ❚❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s r❡✲ s✉❧t❛❞♦s ♣❛r❛ θ < 1+α ❡ θ= 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❡♠❛ ✷ ❞❛❞♦ ❡♠ ❬✻❪✳

(24)

❈❛♣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❛❧✐❞❛❞❡✳

(25)

✶✳✶ ◆♦t❛çõ❡s ❡ Pr✐♠❡✐r♦s ❈♦♥❝❡✐t♦s

✶✳ K✐♥❞✐❝❛ ♦ ❝♦r♣♦R♦✉C

✷✳ x= (x1, x2, x3,· · ·, xn) ♣♦♥t♦ ♥♦ ❡s♣❛ç♦ Rn❀

✸✳ | · | ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ ❡♠Rn

✹✳ |α| = α1 +α2 +· · ·+αn ♣❛r❛ α = (α1,· · ·, αn) ∈ Nn, n∈N❀

✺✳ L2(Rn) é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s u:Rn

→R✱ ♠❡♥s✉rá✈❡✐s t❛✐s q✉❡

Z

Rn|

u(x)|2 dx <+; ✻✳ ❙❡ u L2(Rn) ❡♥tã♦

kukL2(Rn) =

Z

Rn|

u(x)|2 dx 1/2

❞❡✜♥❡ ♥♦r♠❛❀

✼✳ ut=

∂u

∂t ❞❡r✐✈❛❞❛ ❞❡ u ❡♠ r❡❧❛çã♦ ❛ t❀

✽✳ utt=

∂2u

∂t2 s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞❡ u ❡♠ r❡❧❛çã♦ ❛ t❀

✾✳ Dαu= ∂|α|u

∂xα1

1 ... ∂xαnn, α = (α1,· · ·, αn) ∈

Nn

✶✵✳ ▽u =grad u=

∂u

∂x1

, ∂u ∂x2

, ∂u ∂x3

,· · · , ∂u ∂xn

r❡♣r❡s❡♥t❛ ♦ ❣r❛❞✐✲ ❡♥t❡ ❞❛ ❢✉♥çã♦ u❀

✶✶✳ ❙❡ u = (u1, u2, u3, ..., un) ❡♥tã♦ div u = n

X

i=1

∂ui

∂xi r❡♣r❡s❡♥t❛ ♦

❞✐✈❡r❣❡♥t❡ ❞❛ ❢✉♥çã♦ u❀

✶✷✳ △u=

n

X

i=1

∂2u

∂x2

i

r❡♣r❡s❡♥t❛ ♦ ❧❛♣❧❛❝✐❛♥♦ ❞❛ ❢✉♥çã♦ u❀

✶✸✳ ❙❡ ξRn ❡♥tã♦ ξ= (ξ

1, ξ2, ξ3,· · · , ξn);

(26)

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

✶✺✳ F−1r❡♣r❡s❡♥t❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✐♥✈❡rs❛❀

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

✶✼✳ ∂k

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

✶✽✳ ❙❡❥❛♠f, g: ΩR❞✉❛s ❢✉♥çõ❡s✳ ❯s❛✲s❡ ❛ ♥♦t❛çã♦f .gs❡ ❡①✐t❡

✉♠❛ ❝♦♥st❛♥t❡C1>0t❛❧ q✉❡f(x)6C1g(x)✱ ♣❛r❛ t♦❞♦ x∈Ω❀ ✶✾✳ ❙❡❥❛♠f, g: ΩR❞✉❛s ❢✉♥çõ❡s✳ ❯s❛✲s❡ ❛ ♥♦t❛çã♦f gs❡f .g

❡g.f❀

✷✵✳ ❙❡❥❛R∈M2 ✉♠❛ ♠❛tr✐③2×2✳ ❉❡♥♦t❛✲s❡ ♣♦r|R|=maxi,j|Rij|✳

■❞❡♥t✐❞❛❞❡s út❡✐s

❙❡f, gsã♦ ❢✉♥çõ❡s ❡s❝❛❧❛r❡s ❞❡ ❝❧❛ss❡C1✱cé ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧

❡F ❡Gsã♦ ❝❛♠♣♦s ✈❡t♦r✐❛✐s t❛♠❜é♠ ❞❡ ❝❧❛ss❡C1✱ ❡♥tã♦ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r♦✈❛❞❛s✳

✶✳ ∇(f+g) =∇f+∇g; ✷✳ ∇(cf) =cf;

✸✳ ∇(f g) =fg+gf;

✹✳ ❞✐✈(F+G) =❞✐✈(F) +❞✐✈(G);

✺✳ ❞✐✈(f F) =f❞✐✈(F) +f ·F;

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

(27)

✶✳✷ ❉✐str✐❜✉✐çõ❡s

❈♦♥s✐❞❡r❡Ω✉♠ ❛❜❡rt♦ ❞❡ Rn✳ ◆❡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: ΩK✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡ s❡❥❛(Ki)iI ❛ ❢❛♠í❧✐❛ ❞❡

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

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

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.

❉❡✜♥✐çã♦ ✶✳✶ ❘❡♣r❡s❡♥t❛♠♦s ♣♦rC∞0 (Ω) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s

u: Ω→K,

❝✉❥❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡ t♦❞❛s ❛s ♦r❞❡♥s sã♦ ❝♦♥tí♥✉❛s ❡ ❝✉❥♦ s✉♣♦rt❡ é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❞❡ Ω✳ ❖s ❡❧❡♠❡♥t♦s ❞❡C∞0 (Ω) sã♦ ❝❤❛♠❛❞♦s

❞❡ ❢✉♥çõ❡s t❡st❡s✳

◆❛t✉r❛❧♠❡♥t❡✱C∞

(28)

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

◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠

C

0

(Ω)

❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠{ϕk}k∈N✉♠❛ s❡q✉ê♥❝✐❛ ❡♠C∞0 (Ω)❡ϕ ∈ C∞0 (Ω)✳

❉✐③❡♠♦s q✉❡ϕk→ϕ s❡✿

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

✐✐✮ P❛r❛ ❝❛❞❛αNn, Dαϕ

k(x)→Dαϕ(x) ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠x∈Ω✳

❉❡✜♥✐çã♦ ✶✳✸ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧C∞0 (Ω) ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛

❞❡✜♥✐❞❛ ❛❝✐♠❛ é ❞❡♥♦t❛❞♦ ♣♦rD(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❡é ❞❡♥♦t❛❞♦ ♣♦r D′(Ω)✳

❉❡ss❡ ♠♦❞♦✱

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

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

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

❛♣❧✐❝❛❞♦ ♥♦ ❡❧❡♠❡♥t♦ϕ✳

(29)

◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠

D

(Ω)

❉❡✜♥✐çã♦ ✶✳✺ ❉✐③❡♠♦s q✉❡Tk→T ❡♠D

(Ω) s❡

hTk, ϕi → hT, ϕi, ∀ϕ ∈ D(Ω).

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

❞❡T✱ ❞❡♥♦t❛❞❛ ♣♦r DαT✱ é ❞❡✜♥✐❞❛ ♣♦r

hDαT, ϕi = (1)|α|hT, Dαϕi, ♣❛r❛ t♦❞❛ϕD(Ω).

❈♦♠ ❡st❛ ❞❡✜♥✐çã♦ t❡♠✲s❡ q✉❡ s❡ u Ck(Ω) ❡♥tã♦ DαTu = TDαu✱ ♣❛r❛ t♦❞♦ |α| ≤ k, ♦♥❞❡ Dαu ✐♥❞✐❝❛ ❛ ❞❡r✐✈❛❞❛ ❝❧áss✐❝❛ ❞❡ u✳ ❊✱ s❡

T D′(Ω) ❡♥tã♦ DαTD′(Ω) ♣❛r❛ t♦❞♦ αNn.

✶✳✸ ❊s♣❛ç♦s

L

p

(Ω)

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

■♥❞✐❝❛♠♦s ♣♦rµ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡Lp(Ω)♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s

♠❡♥s✉rá✈❡✐s f : Ω→Kt❛✐s q✉❡ kfkLp(Ω)<∞♦♥❞❡✿

kfkLp(Ω)=

Z

Ω|

f(x)|p

1/p

, s❡1≤p <∞

(30)

kfkL∞(Ω) = sup ess

x∈Ω|f(x)| = inf{C∈R+ / µ

{x∈Ω / |f(x)|> C}= 0}

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

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

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

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

❖s ❡s♣❛ç♦sLp(Ω)✱ 1

≤p≤ ∞✱ sã♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ s❡♥❞♦L2(Ω) ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧ ❞❛ ✐♥t❡❣r❛❧✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛1< p <✱ Lp(Ω)é r❡✢❡①✐✈♦✳

❚❡♦r❡♠❛ ✶✳✾ C∞

0 (Ω) é ❞❡♥s♦ ❡♠ Lp(Ω)✱ 1≤p <+∞✳ ❚❡♦r❡♠❛ ✶✳✶✵ ✭■♥t❡r♣♦❧❛çã♦ ❞♦s ❡s♣❛ç♦sLp(Ω)✮ ❙❡❥❛♠

1 ≤ p < q ≤ ∞✳ ❙❡ f ∈ Lp(Ω)Lq(Ω) ❡♥tã♦ f Lr(Ω) ♣❛r❛ t♦❞♦

r[p, q]✳ ❆❧é♠ ❞✐ss♦✱

kfkLr(Ω)≤ kfkαLp(Ω)kfk1L−q(Ω)α ❝♦♠ α[0,1]t❛❧ q✉❡ 1

r =α

1

p+ (1−α)

1

q✳

(31)

❊s♣❛ç♦s

L

p

loc

(Ω)

❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛♠ Ω ✉♠ ❛❜❡rt♦ ❞♦ ❡s♣❛ç♦ Rn 1

≤ p <

■♥❞✐❝❛♠♦s ♣♦r Lploc(Ω) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐sf : ΩR

t❛✐s q✉❡ f χK ∈ Lp(Ω)✱ ♣❛r❛ t♦❞♦ K ❝♦♠♣❛❝t♦ ❞❡ Ω✱ ♦♥❞❡ χK é ❛

❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ K✳

❖❜s❡r✈❛çã♦ ✶✳✶✷ L1

loc(Ω)é ❝❤❛♠❛❞♦ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡

✐♥t❡❣rá✈❡✐s✳

P❛r❛ u∈ L1

loc(Ω) ❝♦♥s✐❞❡r❡♠♦s ♦ ❢✉♥❝✐♦♥❛❧ T = Tu : D(Ω) → K

❞❡✜♥✐❞♦ ♣♦r

hT, ϕi = hTu, ϕi =

Z

u(x)ϕ(x)dx.

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

▲❡♠❛ ✶✳✶✸ ✭❉✉ ❇♦✐s ❘❡②♠♦♥❞✮ ❙❡❥❛ uL1

loc(Ω)✳ ❊♥tã♦Tu = 0

s❡ ❡ s♦♠❡♥t❡ s❡ u= 0 q✉❛s❡ s❡♠♣r❡ ❡♠Ω✳

❆ ❛♣❧✐❝❛çã♦

L1

loc(Ω) −→ D

(Ω)

u 7−→ Tu

é ❧✐♥❡❛r✱ ❝♦♥tí♥✉❛ ❡ ✐♥❥❡t✐✈❛ ✭❞❡✈✐❞♦ ❛♦ ▲❡♠❛ ✶✳✶✸✮✳ ❊♠ ❞❡❝♦rrê♥❝✐❛ ❞✐ss♦ é ❝♦♠✉♠ ✐❞❡♥t✐✜❝❛r ❛ ❞✐str✐❜✉✐çã♦ Tu ❝♦♠ ❛ ❢✉♥çã♦ u∈L1loc(Ω)✳

◆❡ss❡ s❡♥t✐❞♦ t❡♠✲s❡ q✉❡ L1

loc(Ω) ⊂ D

(Ω)✳ ❈♦♠♦ Lp(Ω)

⊂ L1

loc(Ω)

(32)

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

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

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

♣❡r❛❞❛s

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

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

❝❡♥t❡ ♥♦ ✐♥✜♥✐t♦ q✉❛♥❞♦ ♣❛r❛ ❝❛❞❛kN t❡♠✲s❡

pk(ϕ) = max

|α|≤kxsup∈Rn

(1 +|x|2)k|Dαϕ(x)|<,

♦♥❞❡αNn✱ ♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r

lim

|x|→∞P(x)D

αϕ(x) = 0,

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

❈♦♥s✐❞❡r❡♠♦s S(Rn)♦ ❡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✿

pk(ϕ) = max

α≤kxsupRn(1 +|x| 2)k

|Dαϕ(x)|.

(33)

◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠

S

(

R

n

)

❯♠❛ s❡q✉ê♥❝✐❛{ϕn}n∈N⊂S(Rn)❝♦♥✈❡r❣❡ ♣❛r❛ϕ❡♠S(Rn)s❡

pk(ϕn−ϕ)❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦ ❡♠K✱ ♣❛r❛ t♦❞♦k∈N✳

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

Pr♦♣♦s✐çã♦ ✶✳✶✻ ❚❡♠✲s❡ q✉❡S(Rn)

⊂Lp(Rn)❞❡♥s❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦

1≤p≤ ∞✳

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

♥✐❞❛ ❛❝✐♠❛✳ ❙❡T :S(Rn)

→Ké ❧✐♥❡❛r ❡ ❝♦♥tí♥✉❛✱ ❞✐③✲s❡ q✉❡T é ✉♠❛

❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳

❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s ❝♦♠ ❛ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ♣♦♥t✉❛❧ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦rS′(Rn)✳

✶✳✺ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r

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

❉❡✜♥✐çã♦ ✶✳✶✽ ❙❡❥❛ϕ∈L1(Rn)✱ ❞❡✜♥✐♠♦s s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉✲

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

(Fϕ)(x) = 1

(2π)n/2

Z

Rn

e−i x·ξϕ(ξ)dξ.

(34)

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

Pr♦♣♦s✐çã♦ ✶✳✷✵ P❛r❛uL1(Rn)✱ ❡①✐st❡ C >0 t❛❧ q✉❡

||ub||2

L∞ ≤C||u||2

L1.

❖❜s❡r✈❛çã♦ ✶✳✷✶ ❆ ❛♣❧✐❝❛çã♦Fe❞❛❞❛ ♣♦r(Feϕ)(x) = (Fϕ)(x), x

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

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

❈♦♠♦S(Rn)

⊂L1(Rn)✱ ❡♥tã♦FϕFeϕ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ♣❛r❛ϕ

S(Rn)✳ ❆❧é♠ ❞✐ss♦✱ ❛♠❜❛s sã♦ r❛♣✐❞❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡s ❞♦ ✐♥✜♥✐t♦✳

Pr♦♣♦s✐çã♦ ✶✳✷✷ ❆s ❛♣❧✐❝❛çõ❡s

F:S(Rn)S(Rn) ❡ Fe:S(Rn)S(Rn)

sã♦ ✐s♦♠♦r✜s♠♦s ❝♦♥tí♥✉♦s ❡F−1=Fe✳

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

✐✮ F(Dαϕ) =i|α|xαFϕ, αNn

✐✐✮ Dα(Fϕ) =F(

−i|α|xαϕ),

∀αNn

(35)

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

tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

hFT, ϕi=hT,Fϕi, ϕS(Rn),

hFeT, ϕi=hT,Feϕi, ϕS(Rn).

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

S(Rn)✱ t❡♠♦s q✉❡FT FeT sã♦ ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s✳

Pr♦♣♦s✐çã♦ ✶✳✷✻ ❆s ❛♣❧✐❝❛çõ❡s

F:S′(Rn)

→S′(Rn) Fe:S(Rn)

→S′(Rn)

sã♦ ✐s♦♠♦r✜s♠♦s ❝♦♥tí♥✉♦s ❡ F−1=Fe✳

P❛r❛ ϕ L2(Rn) ❞❡✜♥✐♠♦s ϕ

k = ϕχBk(0), k ∈ N✱ ♦♥❞❡ χBk(0) é ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ❝♦♥❥✉♥t♦ {x∈Rn;|x| ≤ k}✳ ❆ss✐♠✱ Fϕ

k é

❞❛❞❛ ♣♦r

(Fϕk)(x) = 1

(2π)n/2

Z

|ξ|≤k

e−i x·ξϕ(ξ)dξ, xRn.

➱ ♣♦ssí✈❡❧ ♣r♦✈❛r q✉❡FϕkL2(Rn)❡ q✉❡

{Fϕk}kNé ✉♠❛ s❡q✉ê♥✲

❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠L2(Rn)✳ ❈♦♠♦ ❡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❛❞❛✮

(36)

❝♦✐♥❝✐❞❡♠ ♣❛r❛ ϕ ∈ L2(Rn)✳ ❆ss✐♠ ✜❝❛ ❞❡✜♥✐❞❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡

❋♦✉r✐❡r ♥♦ ❡s♣❛ç♦L2(Rn)✳

❚❡♦r❡♠❛ ✶✳✷✼ ❙❡ u, vL2(Rn) ❡♥tã♦ ❡①✐st❡♠ C

1, C2∈R+ t❛❧ q✉❡ ✐✮ Ddαu= (iy)α

b

u ♣❛r❛ ❝❛❞❛ ♠✉❧t✐✲í♥❞✐❝❡ αt❛❧ q✉❡ Dαu

∈L2(Rn)✳

✐✐✮ (\u

∗v) =C1ubvb✳ ✐✐✐✮ u vc =C2(ub∗bv)✳ ✐✈✮ u=F−1(bu)✳

❚❡♦r❡♠❛ ✶✳✷✽ ✭❚❡♦r❡♠❛ ❞❡ P❧❛♥❝❤❡r❡❧✮ ❆s ❛♣❧✐❝❛çõ❡s

F:L2(Rn)L2(Rn) ❡ Fe:L2(Rn)L2(Rn)

sã♦ ✐s♦♠♦r✜s♠♦s ❞❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt t❛✐s q✉❡

hFϕ,FψiL2 =hϕ, ψiL2 =hFeϕ,FeψiL2

♣❛r❛ t♦❞♦ ♣❛rϕ, ψ∈L2(Rn)✳

❈♦r♦❧ár✐♦ ✶✳✷✾ ❙❡ϕ∈L2(Rn)✱ ❡♥tã♦

||ϕ||=||Fϕ||

Pr♦♣♦s✐çã♦ ✶✳✸✵ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍❛✉s❞♦r✛✲❨♦✉♥❣✮ ❙❡ f Lp(Rn) ❝♦♠ 1 6 p 6 2 p, q ❝♦♥❥✉❣❛❞♦s✱ ✐st♦ é✱ 1

p +

1

q = 1✱ t❡♠♦s

q✉❡

kfˆkq6kfkp.

(37)

❊①❡♠♣❧♦s

✶✳ F(∆ϕ)(x) =−|x|2F(ϕ)(x)✿

❉❛ Pr♦♣♦s✐çã♦ ✶✳✷✸✱F(Dαϕ) =i|α|xαFϕ✱ ❧♦❣♦ ♣❛r❛ ❝❛❞❛

j= 1,2, . . . , nt❡♠✲s❡

F ∂

2ϕ

∂x2

j

!

(x) =i2x2jF(ϕ)(x) =−x2jF(ϕ)(x).

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

F(∆ϕ)(x) = F  

n

X

j=1

∂2ϕ

∂x2

j

 (x) =

n

X

j=1

F ∂

2ϕ

∂x2

j

!

(x)

=

n

X

j=1

(x2jF(ϕ)(x)) =−|x|2F(ϕ)(x).

✷✳ F(∆2ϕ)(x) =|x|4F(ϕ)(x)✿ ❯s❛♥❞♦ ♦ ❊①❡♠♣❧♦ ✶✱ t❡♠♦s

F(∆2ϕ)(x) = F(∆(∆ϕ))(x) =−|x|2F(∆ϕ)(x)

= −|x2|(−|x|2F(ϕ)(x)) =

|x|4F(ϕ)(x). ✸✳ F(ϕ)(x) =ixF(ϕ)(x)✿

❚❛♠❜é♠ ✉s❛♥❞♦ q✉❡F(Dαϕ) =i|α|xαFϕ✱ t❡♠♦s

F

∂ϕ ∂xj

(x) =ixjF(ϕ)(x), ∀j= 1,2, . . . , n.

(38)

❈♦♠ ✐ss♦✱

F(ϕ)(x) = F      

∂ϕ ∂x1

✳✳✳

∂ϕ ∂xn

     (x) =

     

F∂ϕ

∂x1

(x) ✳✳✳

F∂ϕ

∂xn

(x)

     

=

     

ix1F(ϕ)(x) ✳✳✳

ixnF(ϕ)(x)

     =ix

F(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⊆Rn ❛❜❡rt♦✱mN1p≤ ∞✳ ■♥❞✐❝❛r❡✲

♠♦s ♣♦r Wm,p(Ω) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡su❞❡Lp(Ω) t❛✐s q✉❡

♣❛r❛ t♦❞♦|α|6m, Dαu♣❡rt❡♥❝❡ ❛ Lp(Ω)✱ s❡♥❞♦Dαu❛ ❞❡r✐✈❛❞❛ ❞✐s✲

tr✐❜✉❝✐♦♥❛❧ ❞❡u✳ Wm,p(Ω)é ❝❤❛♠❛❞♦ ❞❡ ❊s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ ♦r❞❡♠

mr❡❧❛t✐✈♦ ❛♦ ❡s♣❛ç♦Lp(Ω)✳

❘❡s✉♠✐❞❛♠❡♥t❡✱

Wm,p(Ω) ={uLp(Ω) t❛❧ q✉❡ DαuLp(Ω) ♣❛r❛ t♦❞♦ |α|6m}.

(39)

◆♦r♠❛ ❡♠

W

m,p

(Ω)

P❛r❛ ❝❛❞❛u∈Wm,p(Ω) t❡♠✲s❡ q✉❡

kukm,p =

 X

|α|6m

kDαukpLp(Ω)

 

1/p

=

 X

|α|6m

Z

Ω|

(Dαu)(x)|pdx

 

1/p

,

❝♦♠p[1,)✱ ❡

kukm,∞=

X

|α|6m

kDαu

kL∞

(Ω),

❝♦♠p=∞✱ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ s♦❜r❡Wm,p(Ω)✳

❖❜s❡r✈❛çõ❡s✿

✶✳ (Wm,p(Ω),

k · km,p)é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

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

hu, vim,2=

X

|α|6m

hDαu, DαviL2(Ω), u, v∈Wm,2(Ω).

✸✳ ❉❡♥♦t❛✲s❡ Wm,2(Ω) ♣♦r Hm(Ω)✳

✹✳ Hm(Ω) é r❡✢❡①✐✈♦ ❡ s❡♣❛rá✈❡❧✳

(40)

✺✳ ❆ ♥♦r♠❛ ✉s✉❛❧ ❡♠H2(Rn)é ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ❞❛❞❛ ♣♦r

||u||H2=||u||2+||∆u||2.

❖ ❊s♣❛ç♦

W

0m,p

(Ω)

❉❡✜♥✐çã♦ ✶✳✸✷ ❙❡❥❛ΩRn✉♠ ❛❜❡rt♦✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦Wm,p

0 (Ω)

❝♦♠♦ s❡♥❞♦ ♦ ❢❡❝❤♦ ❞❡C∞

0 (Ω) ❡♠ Wm,p(Ω)✳

❖❜s❡r✈❛çõ❡s✿

✶✳ ◗✉❛♥❞♦p= 2✱ ❡s❝r❡✈❡✲s❡Hm

0 (Ω) ❡♠ ❧✉❣❛r ❞❡W

m,p

0 (Ω)✳

✷✳ ❙❡ W0m,p(Ω) =Wm,p(Ω), ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡Ω❡♠Rn ♣♦ss✉✐ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ✐❣✉❛❧ ❛ ③❡r♦✳

✸✳ ❱❛❧❡ q✉❡ W0m,p(Rn) =Wm,p(Rn)✳

❖ ❊s♣❛ç♦

W

−m,q

(Ω)

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

p +

1

q = 1✳

❘❡♣r❡s❡♥t❛✲s❡ ♣♦rW−m,q(Ω) ♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞❡Wm,p

0 (Ω)✳

❖ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞❡Hm

0 (Ω)r❡♣r❡s❡♥t❛✲s❡ ♣♦r H−m(Ω)✳

(41)

■♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈

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

p− m

n >0 ❡♥tã♦ W

m,p(Ω)

⊂Lq(Ω)✱ ♣❛r❛q s❛t✐s❢❛③❡♥❞♦

1

q =

1

p− m

n❀

✐✐✮ ❙❡ 1

p− m

n = 0❡♥tã♦ W

m,p(Ω)Lq(Ω), ♣❛r❛q[p,)❀

✐✐✐✮ ❙❡ 1

p− m

n <0 ❡♥tã♦ W

m,p(Ω)

⊂L∞(Ω);

s❡♥❞♦ ❛s ✐♠❡rsõ❡s ❛❝✐♠❛ ❝♦♥tí♥✉❛s✳

✶✳✼ ❉❡s✐❣✉❛❧❞❛❞❡s ■♠♣♦rt❛♥t❡s

❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣

❙❡a≥0 ❡b≥0❡1< p, q <∞❝♦♠ 1

p+

1

q = 1❡♥tã♦

ab≤a

p

p + bq

q.

❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r

❙❡❥❛♠ f Lp(Ω) g

∈ Lq(Ω) ❝♦♠ 1 < p <

∞ ❡ 1p+ 1

q = 1 ♦✉ q= 1❡p=♦✉q=❡p= 1✳ ❊♥tã♦f gL1(Ω)

Z

Ω|

f(x)g(x)|dx6kfkLp(Ω)kgkLq(Ω).

(42)

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

●r❡❡♥

❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s ♣❛r❛ ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ Ω ❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡C2✿

✐✳ P❛r❛F (H1(Ω))n

Z

(divF)(x)dx=

Z

Γ

F(x)·η(x)dΓ, Γ =∂Ω.

✐✐✳ P❛r❛vH1

0(Ω), u∈H2(Ω)✿

Z

v(x)∆u(x)dx=−

Z

Ω∇

v(x)· ∇u(x)dx.

✐✐✐✳ P❛r❛uH2(Ω)✱ v

∈H2 0(Ω)✿

Z

v(x)∆u(x)dx=

Z

∆v(x)u(x)dx.

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

❆ ❢ór♠✉❧❛ ❞❡ P❡❛♥♦✲❇❛❦❡r

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

d

dtu=A(t)u, u(0) =u0∈C

n

Figure

Updating...

References

Updating...

Download now (165 página)