Taxas de Decaimento para a Energia Associada a um Sistema Semilinear de Ondas Elásticas em R

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

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

❚❛①❛s ❞❡ ❉❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❊♥❡r❣✐❛ ❆ss♦❝✐❛❞❛ ❛

✉♠ ❙✐st❡♠❛ ❙❡♠✐❧✐♥❡❛r ❞❡ ❖♥❞❛s ❊❧ást✐❝❛s ❡♠

R

n

❝♦♠ P♦t❡♥❝✐❛❧ ❞♦ ❚✐♣♦ ❉✐ss✐♣❛t✐✈♦

❏❛q✉❡❧✐♥❡ ▲✉✐③❛ ❍♦r❜❛❝❤

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

❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘✉② ❈♦✐♠❜r❛ ❈❤❛rã♦

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

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

❚❛①❛s ❞❡ ❉❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❊♥❡r❣✐❛ ❆ss♦❝✐❛❞❛ ❛

✉♠ ❙✐st❡♠❛ ❙❡♠✐❧✐♥❡❛r ❞❡ ❖♥❞❛s ❊❧ást✐❝❛s ❡♠

R

n

❝♦♠ P♦t❡♥❝✐❛❧ ❞♦ ❚✐♣♦ ❉✐ss✐♣❛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✐❝❛✱ ❝♦♠ ár❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦

❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s✳

❏❛q✉❡❧✐♥❡ ▲✉✐③❛ ❍♦r❜❛❝❤

❋❧♦r✐❛♥ó♣♦❧✐s

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❚❛①❛s ❞❡ ❉❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❊♥❡r❣✐❛ ❆ss♦❝✐❛❞❛ ❛ ✉♠ ❙✐st❡♠❛ ❙❡♠✐❧✐♥❡❛r

❞❡ ❖♥❞❛s ❊❧ást✐❝❛s ❡♠Rn❝♦♠ P♦t❡♥❝✐❛❧ ❞♦ ❚✐♣♦ ❉✐ss✐♣❛t✐✈♦ ♣♦r

❏❛q✉❡❧✐♥❡ ▲✉✐③❛ ❍♦r❜❛❝❤

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

▼❛t❡♠át✐❝❛✑✱ ár❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s✱ ❡

❛♣r♦✈❛❞❛ ❡♠ s✉❛ ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠

▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛

Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s

❈♦♦r❞❡♥❛❞♦r ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦

❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿

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

Pr♦❢✳ ❉r✳ ❘②♦ ■❦❡❤❛t❛ ✭❍✐r♦s❤✐♠❛ ❯♥✐✈❡rs✐t②✮

Pr♦❢✳ ❉r✳ ❈❧á✉❞✐♦ ❘✳ ➪✈✐❧❛ ❞❛ ❙✐❧✈❛ ❏✉♥✐♦r ✭■❋❚❘✲P❘✮

Pr♦❢✳ ❉r✳ ❏❛r❞❡❧ ▼♦r❛✐s P❡r❡✐r❛ ✭❯❋❙❈✮

❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸✳

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❆ ❉❡✉s✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❈❧❡✈❡rs♦♥✳

❆♦s ♠❡✉s ♣❛✐s ❱❛❧❝✐r ❡ ❙✉❡❧✐✳

❆♦ ♠❡✉ ✐r♠ã♦ ❏✉❧✐❛♥♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s✳

❆♦ ❈◆Pq ❡ ❛♦ ❘❊❯◆■✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛✲s❡ ❛ ❡①✐stê♥❝✐❛ ❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❣❧♦✲

❜❛✐s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧✱ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ s❡♠✐❧✐♥❡❛r ❞❡ ♦♥❞❛s

❡❧ást✐❝❛s ❡♠ ✉♠ ♠❡✐♦ ✐s♦tró♣✐❝♦✱ ❝♦♠ ✉♠❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ❞♦ t✐♣♦ ♥ã♦ ❛❜✲

s♦r✈❡♥t❡✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❞♦ t❡r♠♦ ❞✐ss✐♣❛t✐✈♦ é ✉♠ ♣♦t❡♥❝✐❛❧ ♥ã♦ ❝♦♥st❛♥t❡

❡♠Rn❡ ❡st✉❞❛♠✲s❡ ♦s ❝❛s♦s q✉❛♥❞♦ ❡ss❡ ♣♦t❡♥❝✐❛❧ ❞✐ss✐♣❛t✐✈♦ é ❞♦ t✐♣♦ ❝♦♥s✐❞❡r❛❞♦ ❝rít✐❝♦ ❡ ♥ã♦ ❝rít✐❝♦✳ ❚❛①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛ t♦t❛❧

t❛♠❜é♠ sã♦ ❡st✉❞❛❞❛s ♣❛r❛ ♦s ❝❛s♦s ❧✐♥❡❛r ❡ s❡♠✐❧✐♥❡❛r✳ ❆ ❡①✐stê♥❝✐❛ ❡ ♦

❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r sã♦ ♦❜t✐❞♦s ♠❡❞✐❛♥t❡ ❛ ❤✐♣ót❡s❡

❞❡ ❞❛❞♦s ✐♥✐❝✐❛✐s ♣❡q✉❡♥♦s✳ ◆❡st❡ tr❛❜❛❧❤♦ s❡❣✉✐♠♦s ✐❞é✐❛s ❞❡ ❈❤❛rã♦✲

■❦❡❤❛t❛ ❬✻❪ ❡ ❞❡ ■❦❡❤❛t❛✲❚♦❞♦r♦✈❛✲❨♦r❞❛♥♦✈ ❬✶✵❪✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❖♥❞❛s ❡❧ást✐❝❛s✱ s❡♠✐❣r✉♣♦s✱ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡

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

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❆❜str❛❝t

❲❡ st✉❞② t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❣❧♦❜❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ✐♥✐✲

t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ s❡♠✐✲❧✐♥❡❛r s②st❡♠ ♦❢ ❡❧❛st✐❝ ✇❛✈❡s

✐♥ ❛♥ ✐s♦tr♦♣✐❝ ♠❡❞✐✉♠ ✇✐t❤ ❛ ♥♦♥❧✐♥❡❛r✐t② ♦❢ t②♣❡ ♥♦♥❛❜s♦r♣t✐♦♥✳ ❚❤❡

❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ❞✐ss✐♣❛t✐✈❡ t❡r♠ ✐s ♥♦♥ ❝♦♥st❛♥t ♣♦t❡♥t✐❛❧ ✐♥Rn❛♥❞ ✇❡ st✉❞② t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ♣♦t❡♥t✐❛❧ t②♣❡ ♦❢ ❞❛♠♣✐♥❣ ✐s ❝♦♥s✐❞❡r❡❞ ❝r✐t✐❝❛❧

❛♥❞ ♥♦♥❝r✐t✐❝❛❧✳ ❉❡❝❛② r❛t❡s ♦❢ t❤❡ t♦t❛❧ ❡♥❡r❣② ❛r❡ ❛❧s♦ st✉❞✐❡❞ ❢♦r ❧✐♥❡❛r

❛♥❞ s❡♠✐✲❧✐♥❡❛r s②st❡♠✳ ❚❤❡ ❣❧♦❜❛❧ ❡①✐st❡♥❝❡ ❛♥❞ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r

♦❢ t❤❡ s❡♠✐✲❧✐♥❡❛r ♣r♦❜❧❡♠ ❛r❡ ♦❜t❛✐♥❡❞ ♦♥ t❤❡ ❤②♣♦t❤❡s✐s ♦❢ s♠❛❧❧ ✐♥✐t✐❛❧

❞❛t❛✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ❢❛❧❧♦✇ ✐❞❡❛s ♦❢ ❈❤❛rã♦✲■❦❡❤❛t❛ ❬✻❪ ❛♥❞ ■❦❡❤❛t❛✲

❚♦❞♦r♦✈❛✲❨♦r❞❛♥♦✈ ❬✶✵❪✳

❑❡②✇♦r❞s✿ ❊❧❛st✐❝ ✇❛✈❡s✱ s❡♠✐❣r♦✉♣s✱ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉✲

t✐♦♥✱ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ ♠❡t❤♦❞ ♦❢ ♠✉❧t✐♣❧✐❡rs✳

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❙✉♠ár✐♦

✶ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s ✻

✶✳✶ ◆♦t❛çõ❡s ❡ ✐❞❡♥t✐❞❛❞❡s ✈❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✷ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✶✳✸ ❊s♣❛ç♦sLp(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✹ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✶✳✺ ❉❡s✐❣✉❛❧❞❛❞❡s ✐♠♣♦rt❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

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

✶✳✼ ❖♣❡r❛❞♦r❡s ❡❧í♣t✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✶✳✽ ❚❡♦r❡♠❛ ❞❡ ▲❛①✲▼✐❧❣r❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✶✳✾ ❙❡♠✐❣r✉♣♦s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

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✷ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s ♣❛r❛ ♦ ❙✐st❡♠❛ ❞❡ ❖♥✲

❞❛s ❊❧ást✐❝❛s ▲✐♥❡❛r ✸✹

✸ ❚❛①❛s ❞❡ ❉❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❊♥❡r❣✐❛ ❚♦t❛❧ ❞♦ ❙✐st❡♠❛

▲✐♥❡❛r ✹✾

✸✳✶ ❋✉♥çã♦ ♣♦t❡♥❝✐❛❧ V(x)≥ C0

1 +|x| ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✷ ❋✉♥çã♦ ♣♦t❡♥❝✐❛❧ V(t, x)≥ C0

1 +|x|+t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✸✳✸ ❋✉♥çã♦ ♣♦t❡♥❝✐❛❧ V(x)≥ C0

(1 +|x|)α ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

✸✳✹ ❋✉♥çã♦ ♣♦t❡♥❝✐❛❧ ❞❡ ❊✉❧❡r✲P♦✐ss♦♥ ✲❉❛r❜♦✉① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✹ ❙✐st❡♠❛ ❙❡♠✐❧✐♥❡❛r ❞❡ ❖♥❞❛s ❊❧ást✐❝❛s ✽✷

✹✳✶ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❧♦❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

✹✳✷ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❣❧♦❜❛✐s ❡ t❛①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛

❛ ❡♥❡r❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

❇✐❜❧✐♦❣r❛✜❛ ✶✵✾

(9)

■♥tr♦❞✉çã♦

◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❡①✐stê♥❝✐❛✱ ✉♥✐❝✐❞❛❞❡ ❡ ❝♦♠✲

♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ❛ss♦✲

❝✐❛❞♦ ❛♦ s✐st❡♠❛ ❞❡ ♦♥❞❛s ❡❧ást✐❝❛s s♦❜ ❡❢❡✐t♦s ❞❡ ✉♠ t❡r♠♦ s❡♠✐❧✐♥❡❛r✱ ❞♦

t✐♣♦ ♥ã♦ ❛❜s♦r✈❡♥t❡✱ ❡ ❞❡ ✉♠ t❡r♠♦ ❞✐ss✐♣❛t✐✈♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ♣♦t❡♥❝✐❛❧

❞♦ t✐♣♦ ✬❞❛♠♣✐♥❣✬✱V(t, x)∈L∞(R+×Rn)✱ ❛ s❛❜❡r

utt−a2∆u−(b2−a2)∇❞✐✈u+V(t, x)ut=|u|p−1u u(0, x) =u0(x)

ut(0, x) =u1(x)

(10)

s❡♥❞♦(t, x)∈R+×Rn❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ▲❛♠éa >0✱b >0s❛t✐s❢❛③❡♠ 0< a2< b2

◆♦ ♣r♦❜❧❡♠❛ ❛❝✐♠❛u=u(t, x) = (u1(t, x),· · ·, un(t, x))é ✉♠❛ ❢✉♥çã♦

✈❡t♦r✐❛❧ q✉❡ r❡♣r❡s❡♥t❛ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞❛ ♦♥❞❛ ♥♦ ♣♦♥t♦x❡ ♥♦ ✐♥st❛♥t❡

❞❡ t❡♠♣♦t✳ ❖ ❡①♣♦❡♥t❡ p♥♦ t❡r♠♦ ♥ã♦ ❧✐♥❡❛r é ❝♦♥st❛♥t❡ ❡ ❛ss✉♠✐♠♦s

q✉❡

s❡n= 2 ❡♥tã♦ 1 + 4

n−1< p <∞

s❡n≥3 ❡♥tã♦ 1 + 4

n−1< p <

n+ 2

n−2·

❖s ❞❛❞♦s ✐♥✐❝✐❛✐su0 ❡u1sã♦ ❡s❝♦❧❤✐❞♦s ♦ ♠❛✐s ❢r❛❝♦ ♣♦ssí✈❡❧ ♣❛r❛ q✉❡

❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛ ❡st❡❥❛ ❜❡♠ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ ❡s❝♦❧❤❡♠♦s

[u0, u1]∈(H1(Rn))n×(L2(Rn))n.

◆❡st❡ tr❛❜❛❧❤♦ ♠♦str❛♠♦s ❡st✐♠❛t✐✈❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♥♦ t❡♠♣♦ ♣❛r❛

❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛ ❞❡ ♦♥❞❛s ❡❧ást✐❝❛s ❛❝✐♠❛✳ ◆♦ ❝❛s♦ ❞♦ s✐st❡♠❛

❞❡ ♦♥❞❛s ❡❧ást✐❝❛s ❧✐♥❡❛r ❡ ♥♦ ❝❛s♦ s❡♠✐❧✐♥❡❛r ♥ã♦ ❛❜s♦r✈❡♥t❡✱ ❛s t❛①❛s q✉❡

❡♥❝♦♥tr❛♠♦s sã♦ ♣♦❧✐♥♦♠✐❛✐s✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥♦ ❛rt✐❣♦ ❞❡ ❈❤❛rã♦ ❡

■❦❡❤❛t❛ ❬✻❪ q✉❡ ❡st✉❞❛ ♦ ❝❛s♦ s❡♠✐❧✐♥❡❛r ❛❜s♦r✈❡♥t❡✳ ❆❧é♠ ❞❡ ❝♦♥s❡❣✉✐r✲

♠♦s ♦❜t❡r t♦❞♦s ♦s r❡s✉❧t❛❞♦s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ ❬✻❪✱ t❛♠❜é♠ ❡st✉❞❛♠♦s ♦

❝❛s♦ ❞♦ ♣♦t❡♥❝✐❛❧V(t, x) =V(x)✱ ♥ã♦ ❡st✉❞❛❞♦ ❡♠ ❬✻❪✱ s❛t✐s❢❛③❡♥❞♦ ✉♠❛

(11)

❝♦♥❞✐çã♦ ❞♦ t✐♣♦

V(x)≥ C0

(1 +|x|)α

❝♦♠α✉♠❛ ❝♦♥st❛♥t❡ s❛t✐s❢❛③❡♥❞♦0< α <1✳

❖ ❝❛s♦ ❞❡ ✉♠ ♣♦t❡♥❝✐❛❧ ❞♦ t✐♣♦ ❊✉❧❡r✲P♦✐ss♦♥✲❉❛r❜♦✉① q✉❡ ❢♦✐ ♠❡♥❝✐✲

♦♥❛❞♦ r❛♣✐❞❛♠❡♥t❡ ♥♦ ❛rt✐❣♦ ❞❡ ❈❤❛rã♦✲■❦❡❤❛t❛ ❬✻❪ é ❢❡✐t♦ ♥❡st❡ tr❛❜❛❧❤♦

❝♦♠ t♦❞♦s ♦s ❞❡t❛❧❤❡s q✉❡ ♥ã♦ ❛♣❛r❡❝❡♠ ❡♠ ❬✻❪✳

❖s r❡s✉❧t❛❞♦s q✉❡ ♦❜t❡♠♦s ♣❛r❛ ♦ s✐st❡♠❛ ❞❡ ♦♥❞❛s ❡❧ást✐❝❛s ❝♦♠

t❡r♠♦ s❡♠✐❧✐♥❡❛r ♥ã♦ ❛❜s♦r✈❡♥t❡ ❛✐♥❞❛ ♥ã♦ ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s ❡♠ r❡✈✐st❛

✐♥❞❡①❛❞❛✱ ♠❛s ❛♣❛r❡❝❡♠ ❡♠ ✉♠ ♣r❡♣r✐♥t ❡ ❡♠ ✉♠ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦

♣♦r ❘✉② ❈❤❛rã♦ ❡ ❘②♦ ■❦❡❤❛t❛ ♥♦ ❱■■ ❲♦r❦s❤♦♣ ♦❢ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧

❊q✉❛t✐♦♥s ✭▲◆❈❈❱✴❯❋❘❏✮❬✼❪✳

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

❞❛r ♣r✐♠❡✐r♦ ♦ ❝❛s♦ ❧✐♥❡❛r ❞❡ ♠❛♥❡✐r❛ ❝♦♠♣❧❡t❛✳ ❉❡♣♦✐s ♦❜t❡r r❡s✉❧t❛❞♦s

♣❛r❛ ♦ ❝❛s♦ s❡♠✐❧✐♥❡❛r ♥ã♦ ❛❜s♦r✈❡♥t❡ ✉s❛♥❞♦ ❛s ❡st✐♠❛t✐✈❛s ❞♦ ♣r♦❜❧❡♠❛

❧✐♥❡❛r✱ ❝♦♠❜✐♥❛❞❛s ❝♦♠ ❡st✐♠❛t✐✈❛s ❡ ✐❞❡✐❛s ♠❛✐s ❛✈❛♥ç❛❞❛s ✉t✐❧✐③❛❞❛s

❡♠ tr❛❜❛❧❤♦s ❛♥t❡r✐♦r❡s ❞❡ ❚♦❞♦r♦✈❛✲❨♦r❞❛♥♦✈ ❬✶✾❪ ❡ ■❦❡❤❛t❛✲❚♦❞♦r♦✈❛✲

❨♦r❞❛♥♦✈ ❬✶✵❪✳

(12)

P❛r❛ ♦ ❝❛s♦ ❞❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❝♦♠ ❞✐ss✐♣❛çã♦ ❞♦ t✐♣♦ ♣♦t❡♥❝✐❛❧

utt(t, x)−∆u(t, x) +V(t, x)ut(t, x) +η|u(t, x)|p−1u(t, x) = 0,

❡①✐st❡♠ ✈ár✐♦s tr❛❜❛❧❤♦s ❝♦♠ r❡s✉❧t❛❞♦s s♦❜r❡ ♦ ❞❡❝❛✐♠❡♥t♦ ❡ ♥ã♦ ❞❡❝❛✐✲

♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛ t♦t❛❧✱ ♣♦❞❡♠♦s ❝✐t❛r ♣♦r ❡①❡♠♣❧♦ ❬✶✷❪✱ ❬✶✺❪✱ ❬✶✻❪✱ ❬✷✵❪

♣❛r❛ ♦ ❝❛s♦η= 0✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ❚♦❞♦r♦✈❛✲❨♦r❞❛♥♦✈ ❬✶✾❪ ♦❜t✐✈❡r❛♠ t❛✲ ①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❝♦♥s✐❞❡r❛♥❞♦ V(x)≈(1 +|x|)−γ

❝♦♠0≤γ <1❡ η= 0✳ ❊❧❡s t❛♠❜é♠ ❡st✉❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r

❝♦♠η= 1✭✈❡r ❬✶✽❪✮✳

P❛r❛ ♦ s✐st❡♠❛ ❞❡ ♦♥❞❛s ❡❧ást✐❝❛s ❡♠ ❞♦♠í♥✐♦s ❡①t❡r✐♦r❡s✱ ❝♦♠ ✉♠❛

❞✐ss✐♣❛çã♦ ❧♦❝❛❧✐③❛❞❛ ♣ró①✐♠♦ ❛♦ ✐♥✜♥✐t♦✱ ❈❤❛rã♦✲■❦❡❤❛t❛ ❬✹❪ ♦❜t✐✈❡r❛♠

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

♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ▲❛♠é✿ b2<4a2

❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❡♠ ✹ ❝❛♣ít✉❧♦s✳ ◆♦ ❈❛♣ít✉❧♦ ✶ é ❛♣r❡s❡♥t❛❞♦

r❡s✉❧t❛❞♦s t❡ór✐❝♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ ❆

❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ♣❛r❛ ♦ ❝❛s♦ ❧✐♥❡❛r ❝♦♠ V(t, x) =V(x)✱ ✈✐❛ t❡♦r✐❛

❞❡ s❡♠✐❣r✉♣♦s✱ é ❡st✉❞❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷✳

◆♦ ❈❛♣ít✉❧♦ ✸ é ❡st✉❞❛❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞♦ ♣r♦❜❧❡♠❛

❧✐♥❡❛r ♣❛r❛ q✉❛tr♦ ❞✐❢❡r❡♥t❡s ♣♦t❡♥❝✐❛✐s q✉❡ ❢♦r❛♠ ❞✐✈✐❞✐❞♦s ❡♠ ✹ s❡çõ❡s✳

(13)

◆❛ ❙❡çã♦ ✸✳✶ ❡st✉❞❛♠♦s ♦ ❝❛s♦ ❝rít✐❝♦ ♦♥❞❡ ♦ ♣♦t❡♥❝✐❛❧V(x) ≥ C0

1 +|x|✱ ❥á ♥❛ ❙❡çã♦ ✸✳✸ ♦ ♣♦t❡♥❝✐❛❧ ❡st✉❞❛❞♦ s❛t✐s❢❛③ V(x) ≥ C0

(1 +|x|)α✱ ♦♥❞❡

0 < α < 1✳ ❊ss❡s ❞♦✐s ♣♦t❡♥❝✐❛✐s ❢♦r❛♠ ❡st✉❞❛❞♦s s❡♣❛r❛❞❛♠❡♥t❡ ♣♦✐s

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

❙❡çã♦ ✸✳✸✳ ◆❛s ❙❡çõ❡s ✸✳✷ ❡ ✸✳✹ ♦s ♣♦t❡♥❝✐❛✐s ❡st✉❞❛❞♦s ❞❡♣❡♥❞❡♠ ❞❡x❡ ❞❡

t✱ sã♦ ❡❧❡s✿ V(t, x)≥ C0

1 +|x|+t ❡ ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ❊✉❧❡r✲P♦✐ss♦♥✲❉❛r❜♦✉①✳

◆❛s três ♣r✐♠❡✐r❛s s❡çõ❡s ❞❡ss❡ ❝❛♣ít✉❧♦ ♣r❡❝✐s❛♠♦s ❞❛ ❝♦♥❞✐çã♦ ❞❡ q✉❡

♦s ❞❛❞♦s ✐♥✐❝✐❛✐s t❡♥❤❛♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ t❛❧ ❝♦♥❞✐çã♦ ♥ã♦ é ♥❡❝❡ssár✐❛

q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ❊✉❧❡r✲P♦✐ss♦♥✲❉❛r❜♦✉①✳

◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞♦ ❈❛♣ít✉❧♦ ✹ é ❡st✉❞❛❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s

❧♦❝❛✐s ♣❛r❛ ♦ s✐st❡♠❛ ❞❡ ♦♥❞❛s ❡❧ást✐❝❛s s❡♠✐❧✐♥❡❛r✳ ◆❛ ❙❡çã♦ ✹✳✷ é ♦❜t✐❞❛✱

♣❛r❛ ❞❛❞♦s ✐♥✐❝✐❛✐s ♣❡q✉❡♥♦s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❣❧♦❜❛✐s ❡ t❛①❛s ❞❡

❞❡❝❛✐♠❡♥t♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s

❣❧♦❜❛✐s ❡ ♦ ❞❡❝❛✐♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛✱ ❝♦♠ t❛①❛s ♣♦❧✐♥♦♠✐❛✐s✱ sã♦ ♦❜t✐❞❛s

s✐♠✉❧t❛♥❡❛♠❡♥t❡✳

(14)

❈❛♣ít✉❧♦ ✶

❘❡s✉❧t❛❞♦s ❇ás✐❝♦s

◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s q✉❡

s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳ ❆s ❞❡♠♦♥str❛çõ❡s sã♦ ♦♠✐✲

t✐❞❛s ♣♦r s❡ tr❛t❛r❡♠ ❞❡ r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s✱ ♠❛s ❝✐t❛♠♦s ❛s s❡❣✉✐♥t❡s

r❡❢❡rê♥❝✐❛s✱ ❆❞❛♠s ❬✶❪✱ ❆❣♠♦♥✲❉♦✉❣❧✐s✲◆✐r❡♥❜❡r❣ ❬✷❪✱ ❇r❡③✐s ❬✸❪✱ ❆❧✈❡r❝✐♦

❬✾❪✱ ❑❡s❛✈❛♥ ❬✶✶❪ ❡ ▼❡❞❡✐r♦s✲❘✐✈❡r❛ ❬✶✸❪✱ ❬✶✹❪✱ P❛③② ❬✶✼❪✱ ♦♥❞❡ t❛✐s r❡s✉❧✲

t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s✳

❊♠ t♦❞♦ ❡st❡ tr❛❜❛❧❤♦✱ ♦ sí♠❜♦❧♦ Ω r❡♣r❡s❡♥t❛rá ✉♠ s✉❜❝♦♥❥✉♥t♦

❛❜❡rt♦ ❞♦ ❡s♣❛ç♦Rn✱ q✉❡ ❡✈❡♥t✉❛❧♠❡♥t❡ ♣♦❞❡rá s❡r t♦❞♦Rn✳

(15)

✶✳✶ ◆♦t❛çõ❡s ❡ ✐❞❡♥t✐❞❛❞❡s ✈❡t♦r✐❛✐s

✶✳ K✐♥❞✐❝❛ ♦ ❝♦r♣♦R♦✉C✳

✷✳ || · ||r❡♣r❡s❡♥t❛ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠L2(Rn)

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

✹✳ Dαu= ∂|α|u ∂xα1

1 ... ∂x

αn

n

, α = (α1,· · ·, αn) ∈ Nn✳

✺✳ ❙❡f: Ω⊂RnRé ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❡♥tã♦ ♦ ❣r❛❞✐❡♥t❡ ❞❡f✱ q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r∇f✱ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ✈❡t♦r ❞♦Rn❞❛❞♦ ♣♦r

∇f=

∂f ∂x1

, . . . , ∂f ∂xn

.

✻✳ ❙❡ F(x) = (f1(x), . . . , fn(x)) é ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❞❡ ❝❧❛ss❡ C1✱

❞❡✜♥✐♠♦s ♦ ❞✐✈❡r❣❡♥t❡ ❞❡F(x)✱ ❞❡♥♦t❛❞♦ ♣♦r ❞✐✈(F)✱ ❝♦♠♦

❞✐✈(F) =∇ ·F=

n X i=1 ∂fi ∂xi ,

♦♥❞❡∇é ♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❝♦♠♦∇=

∂ ∂x1 , ∂ ∂x2 , . . . , ∂ ∂xn ✳

✼✳ ❖ ❧❛♣❧❛❝✐❛♥♦ ❞❡ ✉♠❛ ❢✉♥çã♦fé ❞❡✜♥✐❞♦ ❝♦♠♦

❞✐✈(∇f) =∇ · ∇f=

n

X

i=1

∂2f ∂x2

(16)

❡ é ❞❡♥♦t❛❞♦ ♣♦r∆f✳

✽✳ ❙❡F(x) = (f1(x),· · ·, fn(x))❡♥tã♦

∆F(x) = (∆f1(x),· · ·,∆fn(x)) ❡

|∇F(x)|2 =

n

X

i=1

|∇fi(x)|2.

■❞❡♥t✐❞❛❞❡s út❡✐s

❙❡f, gsã♦ ❢✉♥çõ❡s ❡s❝❛❧❛r❡s ❞❡ ❝❧❛ss❡C1(Ω),Ω ⊆ Rn✱cé ✉♠❛ ❝♦♥s✲ t❛♥t❡ r❡❛❧ ❡F ❡ Gsã♦ ❝❛♠♣♦s ✈❡t♦r✐❛✐s t❛♠❜é♠ ❞❡ ❝❧❛ss❡C1(Ω)✱ ❡♥tã♦

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

✶✳ ∇(f+g) =∇f+∇g

✷✳ ∇(cf) =c∇f

✸✳ ∇(f g) =f∇g+g∇f

✹✳ ❞✐✈(F+G) =❞✐✈(F) +❞✐✈(G)

✺✳ ❞✐✈(f F) =f❞✐✈(F) +▽f·F

❖ ♣♦♥t♦·✐♥❞✐❝❛ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠Rn✳

(17)

P❛r❛ t♦❞♦ ✈❡t♦r u = (u1, u2,· · ·, un) v = (v1, v2,· · ·, vn) ✈❛♠♦s

❞❡✜♥✐r ♦ ✈❡t♦r✿

v:∇u=v1∇u1+v2∇u2+· · ·+vn∇un=

n

X

i=1

vi∇ui.

▲❡♠❛ ✶✳✶ ❙❡❥❛ u = u(t, x) ❝♦♠ t ∈ R+ ❡ x ∈ Rn✱ ✉♠❛ ❢✉♥çã♦ ✈❡t♦✲ r✐❛❧ ❝♦♠ n ❝♦♠♣♦♥❡♥t❡s✳ ❙❡ u é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ C2 ❛s s❡❣✉✐♥t❡s

✐❞❡♥t✐❞❛❞❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿

❛✮ div(u:∇u) = (u·∆u) +|∇u|2;

❜✮ div(ut:∇u) = (ut·∆u) +

1 2

d dt|∇u|

2;

❝✮ div(u❞✐✈u) = (❞✐✈u)2+ (u· ∇❞✐✈u);

❞✮ div(ut❞✐✈u) = 1

2

d dt(❞✐✈u)

2+ (u

t· ∇❞✐✈u).

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ u, v :R+×Rn Rn ❢✉♥çõ❡s ❞❡ ❝❧❛ss❡C2 t❛❧

q✉❡ u(t, x) = (u1(t, x), ..., un(t, x))v(t, x) = (v1(t, x), ..., vn(t, x)).❈♦♥✲

s✐❞❡r❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡v:∇u❞❛❞❛ ❛❝✐♠❛ t❡♠✲s❡ q✉❡

div(v:∇u) = div

n

X

i=1

(vi∇ui)

(18)

=

n

X

i=1

div(vi∇ui) =

n

X

i=1

div(viuix1,· · ·, v

iui xn)

= n X i=1 ∂ ∂x1

(viuix1) +· · ·+

∂ ∂xn

(viuixn)

=

n

X

i=1

(vi(uix1x1+· · ·+u

i

xnxn) +v

i x1u

i

x1+· · ·+v

i xnu

i xn)

=

n

X

i=1

(vi∆ui+ (∇vi· ∇ui)) = (v·∆u) +

n

X

i=1

(∇vi· ∇ui).

❈♦♥s✐❞❡r❛♥❞♦v=u♥❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ t❡♠✲s❡ q✉❡

div(u:∇u) = (u·∆u) +

n

X

i=1

(∇ui· ∇ui) = (u·∆u) +|∇u|2.

P♦r ♦✉tr♦ ❧❛❞♦✱ t♦♠❛♥❞♦v=utt❡♠✲s❡

div(ut:∇u) = (ut·∆u) + n

X

i=1

(∇uit· ∇ui)

= (ut·∆u) + n X i=1 1 2 d dt|∇u

i|2

= (ut·∆u) +

1 2

d dt|∇u|

2.

(19)

❆s ❞✉❛s ú❧t✐♠❛s ✐❣✉❛❧❞❛❞❡s ♣r♦✈❛♠ ♦s ✐t❡♥s ❛✮ ❡ ❜✮ ❞♦ ❧❡♠❛✳

P❛r❛ ❞❡♠♦♥str❛r ♦s ✐t❡♥s ❝✮ ❡ ❞✮ ✈❛♠♦s ✉s❛r ❛ ✐❞❡♥t✐❞❛❞❡

❞✐✈(F f) =f❞✐✈(F) + (F· ∇f),

♦♥❞❡f é ✉♠❛ ❢✉♥çã♦ ❡s❝❛❧❛r ❡F é ✉♠❛ ❢✉♥çã♦ ✈❡t♦r✐❛❧✳

❈♦♥s✐❞❡r❛♥❞♦f=❞✐✈u ❡ F =u t❡♠♦s

div(u❞✐✈u) = (❞✐✈u)2+ (u· ∇❞✐✈u)

❡ s❡ ❝♦♥s✐❞❡r❛r♠♦sf=❞✐✈u ❡ F =ut t❡♠♦s

div(ut❞✐✈u) =❞✐✈u❞✐✈ut+ (ut· ∇❞✐✈u)

=1 2

d dt(❞✐✈u)

2

+ (ut· ∇❞✐✈u).

❆ss✐♠✱ ♦ ❧❡♠❛ ❡stá ❞❡♠♦♥str❛❞♦✳

(20)

✶✳✷ ❉✐str✐❜✉✐çõ❡s

❙❡❥❛ u ✉♠❛ ❢✉♥çã♦ ♥✉♠ér✐❝❛ ❞❡✜♥✐❞❛ ❡♠ Ω✱ u ♠❡♥s✉rá✈❡❧✱ ❡ s❡❥❛

(Ki)i∈I❛ ❢❛♠í❧✐❛ ❞❡ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❛❜❡rt♦sKi❞❡Ωt❛✐s q✉❡u= 0

q✉❛s❡ s❡♠♣r❡ ❡♠Ki✳ ❈♦♥s✐❞❡r❛✲s❡ ♦ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦K=

[

i∈I Ki✳ ❊♥✲

tã♦

u= 0 quase sempre em 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 ♣♦r C∞

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∞

0 (Ω)é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡K❝♦♠ ❛s ♦♣❡r❛çõ❡s

(21)

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

◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠

C

0

(Ω)

❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠{ϕk}k∈N✉♠❛ s❡q✉ê♥❝✐❛ ❡♠C0∞(Ω)❡ϕ ∈ C0∞(Ω)✳

❉✐③❡♠♦s q✉❡ϕk→ϕs❡✿

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

✐✐✮ P❛r❛ ❝❛❞❛ α ∈ Nn, Dαϕk(x) Dαϕ(x) ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x

Ω✳

❉❡✜♥✐çã♦ ✶✳✸ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ C0∞(Ω)❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛

❞❡✜♥✐❞❛ ❛❝✐♠❛ é ❞❡♥♦t❛❞♦ ♣♦r D(Ω)❡ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s✳

❉❡✜♥✐çã♦ ✶✳✹ ❯♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ Ωé ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❞❡✜♥✐❞♦

❡♠ D(Ω) ❡ ❝♦♥tí♥✉♦ ❡♠ r❡❧❛çã♦ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡✜♥✐❞❛ ❡♠

D(Ω)✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❞✐str✐❜✉✐çõ❡s s♦❜r❡Ωé ❞❡♥♦t❛❞♦ ♣♦rD′(Ω)✳

❉❡ss❡ ♠♦❞♦✱

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

(22)

❖❜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✳

◆♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠

D

(Ω)

❉❡✜♥✐çã♦ ✶✳✺ ❉✐③❡♠♦s q✉❡Tk→T ❡♠D ′

(Ω)s❡

hTk, ϕi → hT, ϕi✱ ♣❛r❛ t♦❞❛ϕ ∈ D(Ω)✳

✶✳✸ ❊s♣❛ç♦s

L

p

(Ω)

◆❡st❡ tr❛❜❛❧❤♦ ❛s ✐♥t❡❣r❛✐s r❡❛❧✐③❛❞❛s s♦❜r❡ Ω sã♦ ♥♦ s❡♥t✐❞♦ ❞❡ ▲❡✲

❜❡s❣✉❡✱ ❛ss✐♠ ❝♦♠♦ ❛ ♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ❞❛s ❢✉♥çõ❡s ❡♥✈♦❧✈✐❞❛s✳

❉❡✜♥✐çã♦ ✶✳✻ ❙❡❥❛♠ Ω✉♠ ❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧ ❡ 1≤p≤ ∞✳ ■♥❞✐❝❛✲

♠♦s ♣♦r Lp(Ω)♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f : Ω →K t❛✐s q✉❡ kfkLp(Ω)<∞♦♥❞❡✿

kfkLp(Ω)=

Z

|f(x)|pdx

1/p

, s❡1≤p <∞

(23)

kfkL∞(Ω) = supessx|f(x)|

= inf{C∈R+ / med{x / |f(x)|> C}= 0}

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

♦♥❞❡ med{A}s✐❣♥✐✜❝❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧A✳

❖❜s❡r✈❛çã♦ ✶✳✶ ❆s ❢✉♥çõ❡s k · 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)1p≤ ∞✱ 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)✮ ❙❡❥❛♠ 1p < q

(24)

❞✐ss♦✱

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

r =α

1

p+ (1−α)

1

q✳

❊s♣❛ç♦s

L

ploc

(Ω)

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

❝❛♠♦s ♣♦r Lploc(Ω) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f : Ω→Rt❛✐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♠♦♥❞✮ ❙❡❥❛ u∈L1

loc(Ω)✳ ❊♥tã♦Tu= 0s❡ ❡

s♦♠❡♥t❡ s❡u= 0 q✉❛s❡ s❡♠♣r❡ ❡♠Ω✳

(25)

❆ ❛♣❧✐❝❛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)t❡♠♦s q✉❡

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

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

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

T✱ ❞❡♥♦t❛❞❛ ♣♦r DαT✱ é ❞❡✜♥✐❞❛ ♣♦r

hDαT, ϕi = (−1)|α|hT, Dαϕi, ♣❛r❛ t♦❞❛ϕ∈ D(Ω),

T ❝❤❛♠❛✲s❡ ❛ ❞❡r✐✈❛❞❛ ❞✐str✐❜✉❝✐♦♥❛❧ ❞❡T

❈♦♠ ❡st❛ ❞❡✜♥✐çã♦ t❡♠✲s❡ q✉❡ s❡u ∈ Ck(Ω) ❡♥tã♦DαT

u = TDαu✱ ♣❛r❛ t♦❞♦ |α| ≤ k, ♦♥❞❡ Dαu ✐♥❞✐❝❛ ❛ ❞❡r✐✈❛❞❛ ❝❧áss✐❝❛ ❞❡ u✳ ❊✱ s❡ T ∈ D′(Ω) ❡♥tã♦ DαT ∈ D′(Ω) ♣❛r❛ t♦❞♦ α∈Nn.

(26)

✶✳✹ ❊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♦ ✶✳✾ ❙❡❥❛♠ m∈N❡1≤p≤ ∞✳ ■♥❞✐❝❛r❡♠♦s ♣♦r Wm,p(Ω)

♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s u ❞❡ Lp(Ω) t❛✐s q✉❡ ♣❛r❛ t♦❞♦ |α| 6 m, Dαu ♣❡rt❡♥❝❡ ❛ Lp(Ω)✱ s❡♥❞♦ Dαu ❛ ❞❡r✐✈❛❞❛ ❞✐str✐❜✉❝✐♦♥❛❧ ❞❡ u✳ Wm,p(Ω)é ❝❤❛♠❛❞♦ ❞❡ ❊s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ ♦r❞❡♠mr❡❧❛t✐✈♦ ❛♦ ❡s♣❛ç♦ Lp(Ω)

❘❡s✉♠✐❞❛♠❡♥t❡✱

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

◆♦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

|(Dαu)(x)|pdx

 

1/p

, p∈[1,∞)

(27)

kukm,∞=

X

|α|6m

kDαukL∞(Ω), p=∞,

❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ s♦❜r❡Wm,p(Ω)✳

❖❜s❡r✈❛çõ❡s✿

✶✳ (Wm,p(Ω),k · k

m,p)é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

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

❞❡ ❍✐❧❜❡rt ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❛❞♦ ♣♦r

(u, v)m,2=

X

|α|6m

(Dαu, Dαv)L2(Ω) u, v∈Wm,2(Ω). ✸✳ ❉❡♥♦t❛✲s❡ Wm,2(Ω) ♣♦r Hm(Ω)

✹✳ Hm(Ω)é r❡✢❡①✐✈♦ ❡ s❡♣❛rá✈❡❧✳

✺✳ ❙❡❥❛v= (v1,· · ·, vn)(Hm(Ω))n ❡♥tã♦

||v||2(Hm(Ω))n=

n

X

i=1

||vi||2Hm(Ω).

(28)

❖ ❊s♣❛ç♦

W

0m,p

(Ω)

❉❡✜♥✐çã♦ ✶✳✶✵ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ W0m,p(Ω) ❝♦♠♦ s❡♥❞♦ ♦ ❢❡❝❤♦ ❞❡

C∞

0 (Ω)❡♠Wm,p(Ω)✳

❖❜s❡r✈❛çõ❡s✿

✶✳ ◗✉❛♥❞♦p= 2✱ ❡s❝r❡✈❡✲s❡Hm

0 (Ω)❡♠ ❧✉❣❛r ❞❡W0m,p(Ω)✳

✷✳ ❙❡ W0m,p(Ω) =Wm,p(Ω), ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡❡♠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✉❡ 1p + 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❡ ♣♦rH−m(Ω)✳

■♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈

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

✐✮ ❙❡ 1

p− m

n >0❡♥tã♦W m,p(Ω)

⊂Lq(Ω), 1 q =

1

p− m

n❀

(29)

✐✐✮ ❙❡ 1

p− m

n = 0❡♥tã♦W

m,p(Ω)Lq(Ω), 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♦ab6

1

pa p+1

qb q

❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r

❙❡❥❛♠ f ∈ Lp(Ω) g Lq(Ω) ❝♦♠ 1 < p < 1 p +

1

q = 1 ♦✉ q= 1❡p=∞♦✉q=∞❡p= 1✳ ❊♥tã♦f g∈L1(Ω)❡

Z

|f(x)g(x)|dx6kfkLp(Ω)kgkLq(Ω).

❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ P♦✐♥❝❛ré

❙❡❥❛Ω✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❞♦Rn✳ ❊♥tã♦

(30)

♣❛r❛ t♦❞♦ u∈H1

0(Ω)✱ s❡♥❞♦ C >0✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ❞❡♣❡♥❞❡ ❞♦ ❞✐â✲

♠❡tr♦ ❞❡Ω✳

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ u ∈ (H1(Rn))n ❡ s✉♣♣(u) ❡stá ❝♦♥t✐❞♦ ♥❛ ❜♦❧❛ |x| ≤R, R >0✱ ✈❛❧❡ q✉❡

||u||2≤CR2||∇u||2,

❝♦♠C >0✉♠❛ ❝♦♥st❛♥t❡✳

❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●❛❣❧✐❛r❞♦✲◆✐r❡♥❜❡r❣

❙❡❥❛Ω⊂Rn❛❜❡rt♦ ❡1≤r < z≤ ∞✱1≤q≤z ❡0≤k≤m✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡C˜g>0t❛❧ q✉❡

||u||Wk,z(Ω)≤C˜g||Dmu||θLq(Ω)||u||1L−r(Ω)θ , ♣❛r❛ t♦❞♦u∈W0m,q(Ω)∩Lr(Ω)♦♥❞❡

||Dmu||Lq(Ω)=

X

|α|=m

||Dαu||Lq(Ω), ❡ θ=

k n+

1

r−

1

z m

n +

1

r −

1

q

−1

(31)

❖❜s❡r✈❛çã♦ ✶✳✸ ◆♦ ❝❛♣ít✉❧♦ ✹ ❛♣❧✐❝❛r❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●❛❣❧✐❛r❞♦✲

◆✐r❡♥❜❡r❣ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ ✈❡t♦r✐❛❧ u = (u1,· · ·, un)✳ ❖❜s❡r✈❡♠♦s q✉❡

♣❛r❛z >2t❡♠♦s

Z

|u(x)|zdx=

Z

(|u(x)|2)z/2dx

=

Z

(|u1(x)|2+· · ·+|un(x)|2)z/2dx

≤Cz

Z

(|u1(x)|z+· · ·+|un(x)|z)dx

=Cz n

X

i=1

Z

|ui(x)|zdx

=Cz n

X

i=1

||ui||zLz.

❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●❛❣❧✐❛r❞♦✲◆✐r❡♥❜❡r❣ ♣❛r❛ ♦ ❝❛s♦ ♣❛rt✐❝✉✲

❧❛r m= 1✱k= 0❡ r=q= 2 t❡♠♦s

Z

|u(x)|zdx≤Cz n

X

i=1

˜

Cg||∇ui||θz||ui||(1−θ)z

≤CzC˜g||∇u||θz||u||(1−θ)z

=Cg||∇u||θz||u||(1−θ)z,

❝♦♠ Cg ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ q✉❡ ❞❡♣❡♥❞❡ ❞❡ z, n ❡ ❞❛ ❝♦♥st❛♥t❡ q✉❡

(32)

❛♣❛r❡❝❡ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●❛❣❧✐❛r❞♦✲◆✐r❡♥❜❡r❣✳

✶✳✻ ❚❡♦r❡♠❛ ❞❛ ❉✐✈❡r❣ê♥❝✐❛ ❡ ❋ór♠✉❧❛s

❞❡ ●r❡❡♥

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

✐✳

Z

(divF)(x)dx=

Z

Γ

F(x)·η(x)dΓ, F∈(H1(Ω))n;

✐✐✳

Z

v(x)△u(x)dx=−

Z

∇v(x)·∇u(x)dx, v∈H01(Ω), u∈H2(Ω);

✐✐✐✳

Z

v(x)△u(x)dx=

Z

△v(x)u(x)dx, v∈H02(Ω), u∈H2(Ω)

s❡♥❞♦ Ωé ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❞♦ Rn ❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡ C2 ❡ η(x) ❞❡♥♦t❛ ❛ ♥♦r♠❛❧ ❡①t❡r✐♦r ✉♥✐tár✐❛ ♥♦ ♣♦♥t♦ x ∈ Γ = ∂Ω✳ ❆ ❢✉♥çã♦ F

✐♥t❡❣r❛❞❛ s♦❜r❡∂Ωé ♥♦ s❡♥t✐❞♦ ❞❛ ❢✉♥çã♦ tr❛ç♦✳

(33)

✶✳✼ ❖♣❡r❛❞♦r❡s ❡❧í♣t✐❝♦s

❉❡✜♥✐çã♦ ✶✳✶✷ ❯♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♦r❞❡♠2m, m∈N,❞❛ ❢♦r♠❛

Lu= X

|α|≤m

Cα(x)D2αu, x∈Ω

é ❝❤❛♠❛❞♦ ❞❡ ♦♣❡r❛❞♦r ❡❧í♣t✐❝♦ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C >0t❛❧ q✉❡

X

|α|≤m

Cα(x)ξ2α

≥C|ξ|

2m

♣❛r❛ t♦❞♦ξ∈Rn ❡ ♣❛r❛ t♦❞♦x∈Ω✳

Pr♦♣♦s✐çã♦ ✶✳✶ ❖ ♦♣❡r❛❞♦rL(u) =−a2u(b2a2)∇❞✐✈u+ 2ué ✉♠

♦♣❡r❛❞♦r ❡❧í♣t✐❝♦ ❞❡ ♦r❞❡♠ ✷✳

❚❡♦r❡♠❛ ✶✳✹ ✭❚❡♦r❡♠❛ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❡❧í♣t✐❝❛✮ ❙❡❥❛♠L✉♠ ♦♣❡✲

r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❡❧í♣t✐❝♦ ❞❡ ♦r❞❡♠ 2m, m∈N✱ ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❛❜❡rt♦ ❞♦ Rn ❡u∈ D(Ω)✳ ❙❡ué s♦❧✉çã♦ ❞❡Lu=f✱ ♥♦ s❡♥t✐❞♦ ❞❛s ❞✐str✐❜✉✐✲ çõ❡s✱ ❝♦♠f∈L2(Ω)❡♥tã♦uH2m(Ω).

❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❆❣♠♦♥✲❉♦✉❣❧✐s✲

◆✐r❡♥❜❡r❣ ❬✷❪✳

(34)

✶✳✽ ❚❡♦r❡♠❛ ❞❡ ▲❛①✲▼✐❧❣r❛♠

❉❡✜♥✐çã♦ ✶✳✶✸ ❙❡❥❛H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt r❡❛❧✳ ❯♠❛ ❛♣❧✐❝❛çã♦

B:H×H→R

é ❝❤❛♠❛❞❛ ❞❡ ❢♦r♠❛ ❜✐❧✐♥❡❛r s❡B(., y)é ❧✐♥❡❛r ♣❛r❛ ❝❛❞❛ y∈H ❡B(x, .)

é ❧✐♥❡❛r ♣❛r❛ ❝❛❞❛ x∈H✳

Bé ❝❤❛♠❛❞❛ ❞❡ ❧✐♠✐t❛❞❛ ✭❝♦♥tí♥✉❛✮ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡Ct❛❧ q✉❡

|B(x, y)| ≤CkxkH kykH, ∀x, y∈H.

B é ❝❤❛♠❛❞❛ ❝♦❡r❝✐✈❛ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡δ >0t❛❧ q✉❡

B(x, x)≥δkxk2H, ∀x∈H.

❚❡♦r❡♠❛ ✶✳✺ ✭▲❛①✲▼✐❧❣r❛♠✮ ❙❡❥❛ B ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r✱ ❧✐♠✐t❛❞❛ ❡

❝♦❡r❝✐✈❛ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r

❝♦♥tí♥✉♦F ❡♠H✱ ❡①✐st❡ ✉♠ ú♥✐❝♦u∈H t❛❧ q✉❡

B(x, u) =F(x), ∀x∈H.

(35)

❆s ❞❡✜♥✐çõ❡s ❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▲❛①✲▼✐❧❣r❛♠ ♣♦❞❡♠

s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❇r❡③✐s ❬✸❪✳

✶✳✾ ❙❡♠✐❣r✉♣♦s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s

P❛r❛ ❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❝✐t❛♠♦s ❝♦♠♦ r❡✲

❢❡rê♥❝✐❛s ❆❧✈❡r❝✐♦ ❬✾❪✱ ❇r❡③✐s ❬✸❪ ❡ P❛③② ❬✶✼❪✳

❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡L(X)❛ á❧❣❡❜r❛ ❞♦s ♦♣❡✲

r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦s ❞❡X✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦

S:R+→ L(X)

é ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦s ❡♠ X s❡✿

■✲ S(0) =I✱ ♦♥❞❡I é ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ❞❡L(X)❀

■■✲ S(t+s) =S(t)S(s), ∀t, s∈R+✳

❉✐③✲s❡ q✉❡ ♦ s❡♠✐❣r✉♣♦S é ❞❡ ❝❧❛ss❡C0 s❡

■■■✲ lim

t→0+k(S(t)−I)xkX= 0, ∀x∈X.

Pr♦♣♦s✐çã♦ ✶✳✷ ❚♦❞♦ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0 é ❢♦rt❡♠❡♥t❡ ❝♦♥tí♥✉♦ ❡♠

(36)

R+✱ ✐st♦ é✱ s❡ t∈R+ ❡♥tã♦

lim

s→tS(s)x=S(t)x, ∀x∈X.

❉❡✜♥✐çã♦ ✶✳✶✺ ❙❡ kS(t)kL(X) ≤ 1, ∀t ≥ 0✱ S é ❞✐t♦ s❡♠✐❣r✉♣♦ ❞❡

❝♦♥tr❛çõ❡s ❞❡ ❝❧❛ss❡C0✳

❉❡✜♥✐çã♦ ✶✳✶✻ ❖ ♦♣❡r❛❞♦rA:D(A)→X ❞❡✜♥✐❞♦ ♣♦r

D(A) =

x∈X lim

h→0+

S(h)−I

h x existe

A(x) = lim

h→0+

S(h)−I

h x, ∀x∈D(A)

é ❞✐t♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞♦ s❡♠✐❣r✉♣♦S✳

Pr♦♣♦s✐çã♦ ✶✳✸ ❖ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0 é

✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡ ❢❡❝❤❛❞♦ ❡ s❡✉ ❞♦♠í♥✐♦ é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡♥s♦

❡♠X✳

Pr♦♣♦s✐çã♦ ✶✳✹ ❙❡❥❛S ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0 ❡ A♦ ❣❡r❛❞♦r ✐♥✜✲

(37)

♥✐t❡s✐♠❛❧ ❞❡S✳ ❙❡x∈D(A)✱ ❡♥tã♦S(t)x∈D(A), ∀t≥0,❡ d

dtS(t)x=A S(t)x=S(t)Ax.

❉❡✜♥✐çã♦ ✶✳✶✼ ❙❡❥❛ S ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0 ❡ A s❡✉ ❣❡r❛❞♦r ✐♥✲

✜♥✐t❡s✐♠❛❧✳ P♦♥❤❛♠♦s A0 = I A1 = A ❡✱ s✉♣♦♥❞♦ q✉❡ Ak−1 ❡st❡❥❛

❞❡✜♥✐❞♦✱ ✈❛♠♦s ❞❡✜♥✐rAk ♣♦♥❞♦

D(Ak) =x xD(Ak−1) e Ak−1xD(A)

Akx=A(Ak−1x), ∀x∈D(Ak).

Pr♦♣♦s✐çã♦ ✶✳✺ ❙❡❥❛S ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0 ❡A s❡✉ ❣❡r❛❞♦r ✐♥✲

✜♥✐t❡s✐♠❛❧✳ ❊♥tã♦✿

■✲ D(Ak)é ✉♠ s✉❜❡s♣❛ç♦ ❞❡X Aké ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❞❡ ❳❀

■■✲ ❙❡ x∈D(Ak)❡♥tã♦S(t)xD(Ak), t0, dk

dtkS(t)x=A

kS(t)x=S(t)Akx,

∀k∈N;

■■■✲ \

k

D(Ak) é ❞❡♥s♦ ❡♠ ❳✳

▲❡♠❛ ✶✳✸ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❢❡❝❤❛❞♦ ❞❡ X✳ P♦♥❞♦✱ ♣❛r❛ ❝❛❞❛

(38)

x∈D(Ak)

|x|k= k

X

j=0

kAjxkX ✭✶✳✶✮

♦ ❢✉♥❝✐♦♥❛❧ | · |k é ✉♠❛ ♥♦r♠❛ ❡♠ D(Ak) ♠✉♥✐❞♦ ❞❛ q✉❛❧D(Ak) é ✉♠

❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

❉❡✜♥✐çã♦ ✶✳✶✽ ❆ ♥♦r♠❛ ✭✶✳✶✮ é ❞✐t❛ ♥♦r♠❛ ❞♦ ❣rá✜❝♦✳ ❖ ❡s♣❛ç♦ ❞❡

❇❛♥❛❝❤ q✉❡ s❡ ♦❜té♠ ♠✉♥✐♥❞♦D(Ak)❞❛ ♥♦r♠❛ ✭✶✳✶✮ s❡rá r❡♣r❡s❡♥t❛❞♦

♣♦r [D(Ak)]

❚❡♦r❡♠❛ ▲✉♠❡r✲P❤✐❧❧✐♣s

❉❡✜♥✐çã♦ ✶✳✶✾ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❞❡ X✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s λ ∈

C ♣❛r❛ ♦s q✉❛✐s ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r λI −A é ✐♥✈❡rsí✈❡❧ ❡ s❡✉ ✐♥✈❡rs♦ é

❧✐♠✐t❛❞♦ ❡ t❡♠ ❞♦♠í♥✐♦ ❞❡♥s♦ ❡♠ ❳✱ é ❞✐t♦ ❝♦♥❥✉♥t♦ r❡s♦❧✈❡♥t❡ ❞❡ A❡ é

r❡♣r❡s❡♥t❛❞♦ ♣♦rρ(A)✳

❖ ♦♣❡r❛❞♦r ❧✐♥❡❛r(λI−A)−1✱ r❡♣r❡s❡♥t❛❞♦ ♣♦rR(λ, A)✱ é ❞✐t♦ r❡s♦❧✲

✈❡♥t❡ ❞❡A✳

❙❡❥❛X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ X∗ ♦ ❞✉❛❧ ❞❡X ❡ h ·,· i❛ ❞✉❛❧✐❞❛❞❡

❡♥tr❡X ❡X∗✳ P♦♥❤❛♠♦s✱ ♣❛r❛ ❝❛❞❛xX

(39)

J(x) =x∗∈X∗ hx, x∗i=kxk2X =kx∗k2X∗ .

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤✱J(x)6=∅, ∀x∈X✳ ❯♠❛ ❛♣❧✐❝❛çã♦

❞✉❛❧✐❞❛❞❡ é ✉♠❛ ❛♣❧✐❝❛çã♦j:X−→X∗t❛❧ q✉❡j(x)∈J(x), ∀x∈X✳

■♠❡❞✐❛t❛♠❡♥t❡ s❡ ✈ê q✉❡kj(x)kX∗ =kxkX.

❉❡✜♥✐çã♦ ✶✳✷✵ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❉✐③✲s❡ q✉❡ ♦ ♦♣❡r❛❞♦r

❧✐♥❡❛r A:D(A)⊂X→X é ❞✐ss✐♣❛t✐✈♦ s❡✱ ♣❛r❛ ❛❧❣✉♠❛ ❛♣❧✐❝❛çã♦ ❞✉❛❧✐✲

❞❛❞❡✱j✱

RehAx, j(x)i ≤0, ∀x∈D(A).

❊♠ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ♦♣❡r❛❞♦r ❞✐ss✐♣❛t✐✈♦ é✿

❉❡✜♥✐çã♦ ✶✳✷✶ ❙❡❥❛ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❉✐③✲s❡ q✉❡ ♦ ♦♣❡r❛❞♦r

❧✐♥❡❛rA:D(A)⊂H→H é ❞✐ss✐♣❛t✐✈♦ s❡✱

RehAx, xi ≤0, ∀x∈D(A).

❚❡♦r❡♠❛ ✶✳✻ ✭▲✉♠❡r✲P❤✐❧❧✐♣s✮ ❙❡ A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠

s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çõ❡s ❞❡ ❝❧❛ss❡C0 ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤X ❡♥tã♦✿

✐✲ Aé ❞✐ss✐♣❛t✐✈♦❀

(40)

✐✐✲ Im(λ−A) =X, λ >0 (Im(λ−A) = imagem de λI−A)✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡

✐✲ D(A)é ❞❡♥s♦ ❡♠X❀

✐✐✲ Aé ❞✐ss✐♣❛t✐✈♦❀

✐✐✐✲ Im(λ0−A) =X, para algum λ0>0,❡♥tã♦Aé ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡✲

s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝♦♥tr❛çõ❡s ❞❡ ❝❧❛ss❡C0✳

❉❡✜♥✐çã♦ ✶✳✷✷ ❖ ♦♣❡r❛❞♦rAé ❞✐t♦ s❡rm✲❞✐ss✐♣❛t✐✈♦ s❡Aé ❞✐ss✐♣❛t✐✈♦

❡ Im(λ0−A) =X✱ ♣❛r❛ ❛❧❣✉♠λ0✳

Pr♦♣♦s✐çã♦ ✶✳✻ ❙❡ A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡

❝❧❛ss❡ C0 ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ X ❡ B é ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡ ❧✐♠✐✲

t❛❞♦ ❡♥tã♦A+B é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0

❡♠X✳

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Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❆❜str❛t♦

❙❡❥❛X✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡A✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❞❡X✳ ❈♦♥s✐❞❡r❡

♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❛❜str❛t♦

dU

dt =AU(t)

U(0) =U0 ✭✶✳✷✮

♦♥❞❡U0∈X ❡t≥0✳

❉❡✜♥✐çã♦ ✶✳✷✸ ❯♠❛ ❢✉♥çã♦u:R+X,❝♦♥tí♥✉❛ ♣❛r❛t0✱ ❝♦♥t✐♥✉✲ ❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ t♦❞♦t >0✱ t❛❧ q✉❡u(t)∈D(A) ♣❛r❛ t♦❞♦t >0

❡ q✉❡ s❛t✐s❢❛③ ✭✶✳✷✮ é ❞✐t❛ s♦❧✉çã♦ ❢♦rt❡ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✷✮✳

❚❡♦r❡♠❛ ✶✳✼ ❙❡A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡

C0 ❡♥tã♦✱ ♣❛r❛ ❝❛❞❛ U0∈D(A)♦ ♣r♦❜❧❡♠❛ ✭✶✳✷✮ t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦

❢♦rt❡

U(t) =S(t)U0∈C(R+, D(A)),

♦♥❞❡ S é ♦ s❡♠✐❣r✉♣♦ ❣❡r❛❞♦ ♣♦rA✳

❙❡ U0 ∈ X ❡♥tã♦ ❞✐③❡♠♦s q✉❡ U(t) = S(t)U0 ∈ C(R+, X) é ✉♠❛

s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✷✮✳

(42)

❈❛♣ít✉❧♦ ✷

❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡

❙♦❧✉çõ❡s ♣❛r❛ ♦ ❙✐st❡♠❛

❞❡ ❖♥❞❛s ❊❧ást✐❝❛s ▲✐♥❡❛r

◆❡st❡ ❝❛♣ít✉❧♦✱ ✉s❛♥❞♦ ❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s✱ ✈❛♠♦s ♠♦str❛r ❛ ❡①✐s✲

tê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ♦♥❞❛s ❡❧ást✐❝❛s✿

(43)

utt−a2∆u−(b2−a2)∇❞✐✈u+V(x)ut= 0 ✭✷✳✶✮

u(0, x) =u0(x) ✭✷✳✷✮

ut(0, x) =u1(x) ✭✷✳✸✮

♦♥❞❡ (t, x) ∈R+×Rn✱ u=u(t, x) = (u1(t, x), ..., un(t, x)) é ♦ ✈❡t♦r ❞❡

❞❡❢♦r♠❛çõ❡s ❝♦♠n❝♦♠♣♦♥❡♥t❡s✱ ♦s ❝♦❡✜❝✐❡♥t❡sa >0 ❡ b >0s❛t✐s❢❛③❡♠ 0< a2< b2❡ ❛ ❢✉♥çã♦ ♣♦t❡♥❝✐❛❧V(x)L(Rn)✳ ❉❡ ❢❛t♦✱ ♦s ❝♦❡✜❝✐❡♥t❡s a ❡ b ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ▲❛♠éλ ❡ µ ❞❛ s❡❣✉✐♥t❡

❢♦r♠❛✿ b2=λ+ 2µ ❡ a2=λ+µ✳

❋❛③❡♥❞♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧ ❡♠(L2(Rn))n❞❛ ❡q✉❛çã♦ ✭✷✳✶✮ ❝♦♠

ut t❡♠♦s q✉❡

Z

Rn

ut· utt−a2∆u−(b2−a2)∇❞✐✈u+V(x)ut

dx= 0,

q✉❡✱ ❢♦r♠❛❧♠❡♥t❡✱ ✐♠♣❧✐❝❛ ❡♠

1 2

d dt

||ut||2+a2||∇u||2+ (b2−a2)||❞✐✈u||2 +

Z

Rn

(44)

❆ ✐❞❡♥t✐❞❛❞❡ ❛♥t❡r✐♦r ♠♦t✐✈❛ ❞❡✜♥✐r ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛ ✭✷✳✶✮

♣♦r✿

E(t) = 1 2 ||ut||

2+a2||∇u||2+ (b2a2)||❞✐✈u||2.

◆♦t❛✲s❡ ❞❛ r❡❢❡r✐❞❛ ✐❞❡♥t✐❞❛❞❡ ❛❝✐♠❛ q✉❡ ❛ ❡♥❡r❣✐❛E(t) é ✉♠❛ ❢✉♥✲

çã♦ ❞❡❝r❡s❝❡♥t❡ ❞♦ t❡♠♣♦t✱ ♦ q✉❡ ❝❛r❛❝t❡r✐③❛ ♦ s✐st❡♠❛ ✭✷✳✶✮✲✭✷✳✸✮ ❝♦♠♦

❞✐ss✐♣❛t✐✈♦✳

◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♠♦s ♦ s❡❣✉✐♥t❡ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✿

X= (H1(Rn))n×(L2(Rn))n ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡✜♥✐❞♦ ♣♦r

(U, V)X=

Z

Rn

u1·v1dx+a2

Z

Rn

n

X

i=1

∇ui1· ∇v1idx

+ (b2−a2)

Z

Rn

❞✐✈(u1)❞✐✈(v1)dx+

Z

Rn

u2·v2dx

♣❛r❛ t♦❞♦U= (u1, u2)❡V = (v1, v2)❡♠X✳

❖❜s❡r✈❛çã♦ ✷✳✶ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ♥♦r♠❛ ❡♠ (H1(Ω))n ❝♦♠♦ s❡♥❞♦

||u||(H1(Ω))n=

Z

Rn

|u|2dx+a2

Z

Rn

|∇u|2dx+ (b2−a2)

Z

Rn|❞✐✈

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❞❛❞♦ ♣❡❧❛s três ♣r✐♠❡✐r❛s ✐♥t❡❣r❛✐s ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ X ❞❡✜♥✐❞♦

❛❝✐♠❛✳

❊s❝r❡✈❡♥❞♦ ♦ s✐st❡♠❛ ✭✷✳✶✮✲✭✷✳✸✮ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ t❡♠♦s

       d

dtU(t) =AU(t) +BU(t) U(0) =U0

✭✷✳✹✮

♦♥❞❡ U(t) =

   

u(t)

v(t)

  

 ∈ X, U0 =

    u0 u1   

 ∈ X, ♦ ♦♣❡r❛❞♦r

A:D(A)→X é ❞❛❞♦ ♣♦r

A=     0 I

a2∆ + (b2−a2)∇❞✐✈−I 0

  

 ✭✷✳✺✮

❝♦♠D(A) = (H2(Rn))n×(H1(Rn))n ❡ ♦ ♦♣❡r❛❞♦r B :X X é ❞❛❞♦

♣♦r B=     0 0

I −V(x)

   .

◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦U(t) q✉❡ s❛t✐s❢❛③ ♦ ♣r♦❜❧❡♠❛

(46)

✭✷✳✹✮✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦u=u(t, x)q✉❡ s❛t✐s❢❛③

♦ s✐st❡♠❛ ✭✷✳✶✮✲✭✷✳✸✮✱ ♦✉ s❡❥❛✱ q✉❡r❡♠♦s ♠♦str❛r ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿

❚❡♦r❡♠❛ ✷✳✶ ❈♦♥s✐❞❡r❡ ♦ s✐st❡♠❛ ✭✷✳✶✮✲✭✷✳✸✮ ❝♦♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s

a >0 ❡b >0t❛❧ q✉❡0< a2< b2 ❡ ❛ ❢✉♥çã♦ ♣♦t❡♥❝✐❛❧ V(x)∈L∞(Rn)✳ ❊♥tã♦ s❡(u0, u1)∈X ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛u(t, x) t❛❧ q✉❡

u∈C(R+,(H1(Rn)n)C1(R+,(L2(Rn)n).

P❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ✷✳✶ ✈❛♠♦s ♠♦str❛r q✉❡A+B é ❣❡r❛❞♦r ✐♥✜✲

♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0✳ P❡❧❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ✐ss♦

♦❝♦rr❡ s❡Bé ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦ ❡Aé ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦

❞❡ ❝❧❛ss❡C0✳

▲❡♠❛ ✷✳✶ ❖ ♦♣❡r❛❞♦rB :X →X ❞❡✜♥✐❞♦ ❛❝✐♠❛ é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✳

❉❡♠♦♥str❛çã♦✿ ❉❛❞♦s U = (u1, u2) ❡ V = (v1, v2) ♥♦ ❡s♣❛ç♦ ❡♥❡r❣✐❛

X✱ t❡♠✲s❡ q✉❡

(47)

B(U+V) =     0 0

I −V(x)

       

u1+v1

u2+v2

    =     0

u1+v1−V(x)u2−V(x)v2

    =     0

u1−V(x)u2

   +     0

v1−V(x)v2

   

=BU+BV,

♦ q✉❡ ♠♦str❛ q✉❡Bé ❧✐♥❡❛r✳

P❛r❛ ♠♦str❛r q✉❡ B é ❧✐♠✐t❛❞♦ ✈❛♠♦s ❡st✐♠❛r ❛ ♥♦r♠❛ ❞❡ BU ❡♠

X = (H1(Rn))n×(L2(Rn))n✳ ❙❡❥❛U = (u

1, u2)❡♥tã♦

||BU||2X =||(0, u1−V(x)u2)||2X

=||u1−V(x)u2||2

≤2||u1||2+ 2||V||2L∞||u2||2

≤max2,2||V||2L∞ ||U||2X.

(48)

▲♦❣♦✱||B||L(X)≤max{2,2||V||2L∞}1/2✳

P❛r❛ ♠♦str❛r q✉❡ A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❞❡

❝❧❛ss❡C0✈❛♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❡ ▲✉♠❡r✲P❤✐❧❧✐♣s✱ ♦✉ s❡❥❛✱ ✈❛♠♦s ♠♦str❛r

q✉❡Aém✲❞✐ss✐♣❛t✐✈♦ ❡ ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦✳

❈♦♠♦(C∞

0 (Rn))né ❞❡♥s♦ ❡♠(H1(Rn))n ❡

(C0∞(Rn))n⊂(H2(Rn))n⊂(H1(Rn))n

t❡♠♦s q✉❡ (H2(Rn))n é ❞❡♥s♦ ❡♠ (H1(Rn))n✳ ❚❛♠❜é♠ (C

0 (Rn))n é

❞❡♥s♦ ❡♠(L2(Rn))n❡

(C0∞(Rn))n⊂(H1(Rn))n⊂(L2(Rn))n.

▲♦❣♦✱(H1(Rn))né ❞❡♥s♦ ❡♠(L2(Rn))n

P♦rt❛♥t♦✱ ♦ ❞♦♠í♥✐♦ ❞❡A✱D(A)✱ é ❞❡♥s♦ ♥♦ ❡s♣❛ç♦ ❡♥❡r❣✐❛X✳ ▲❡♠❛ ✷✳✷ ❖ ♦♣❡r❛❞♦r A❞❡✜♥✐❞♦ ❡♠ ✭✷✳✺✮ ém✲❞✐ss✐♣❛t✐✈♦✳

❉❡♠♦♥str❛çã♦✿ P❛r❛ q✉❡As❡❥❛ ✉♠ ♦♣❡r❛❞♦rm✲❞✐ss✐♣❛t✐✈♦ é s✉✜❝✐❡♥t❡

♠♦str❛r q✉❡Aé ❞✐ss✐♣❛t✐✈♦ ❡ q✉❡Im(I−A) =X✳

(49)

❙❡❥❛U = (u, v)∈D(A)❡♥tã♦

(AU, U)X=

*   

v

a2u+ (b2a2)∇❞✐✈uu

   ,     u v     + X = Z Rn

v·u dx+a2

Z

Rn

n

X

i=1

∇vi· ∇uidx

+ (b2−a2)

Z

Rn❞✐✈

v❞✐✈u dx+a2

Z

Rn

∆u·v dx

+ (b2−a2)

Z

Rn

∇❞✐✈u·v dx−

Z

Rn

u·v dx,

♣❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠X ❡(H1(Rn))n

❯s❛♥❞♦ ♦ ▲❡♠❛ ✶✳✶✱ ♦ ❚❡♦r❡♠❛ ❞❛ ❉✐✈❡r❣ê♥❝✐❛ ❡ ❛ ✐❞❡♥t✐❞❛❞❡

❞✐✈(v❞✐✈u) = (∇❞✐✈u·v) +❞✐✈v❞✐✈u t❡♠♦s

(AU, U)X=a2

Z

Rn

n

X

i=1

∇vi· ∇uidx+ (b2−a2)

Z

Rn❞✐✈

v❞✐✈u dx

+a2

Z

Rn❞✐✈

(v:∇u)dx−a2

Z

Rn

n

X

i=1

∇vi· ∇uidx

+ (b2−a2)

Z

Rn❞✐✈

(v❞✐✈u)dx−(b2−a2)

Z

Rn❞✐✈

v❞✐✈u dx= 0,

♦ q✉❡ ♠♦str❛ q✉❡A:D(A)→X é ❞✐ss✐♣❛t✐✈♦✳

❘❡st❛ ♣r♦✈❛r q✉❡Im(I−A) =X.

Figure

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