UNIVERSIDADE FEDERAL DE SÃO CARLOS

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P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❛tr❛t♦r❡s ♣❛r❛ ♣r♦❜❧❡♠❛s ♣❛r❛❜ó❧✐❝♦s

s❡♠✐❧✐♥❡❛r❡s s♦❜ ♣❡rt✉r❜❛çõ❡s ❞♦ ❞♦♠í♥✐♦

❘♦❞r✐❣♦ ❆♥t♦♥✐♦ ❙❛♠♣r♦❣♥❛

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P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❛tr❛t♦r❡s ♣❛r❛ ♣r♦❜❧❡♠❛s ♣❛r❛❜ó❧✐❝♦s

s❡♠✐❧✐♥❡❛r❡s s♦❜ ♣❡rt✉r❜❛çõ❡s ❞♦ ❞♦♠í♥✐♦

❘♦❞r✐❣♦ ❆♥t♦♥✐♦ ❙❛♠♣r♦❣♥❛

❇♦❧s✐st❛ ❈❆P❊❙

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ❑❛r✐♥❛ ❙❝❤✐❛❜❡❧ ❙✐❧✈❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar

S192ca

Samprogna, Rodrigo Antonio.

Continuidade de atratores para problemas parabólicos semilineares sob perturbações do domínio / Rodrigo Antonio Samprogna. -- São Carlos : UFSCar, 2013.

73 f.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2013.

1. Equações diferenciais parciais. 2. Continuidade de atratores. 3. Operador laplaciano. 4. Pertubação (Matemática). I. Título.

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ t♦❞❛ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ♣❛✐s✱ ❆♥t♦♥✐♦ ❡ ❚❡r❡③✐♥❤❛✱ ❡ ♠✐♥❤❛ ✐r♠ã✱ P❛✉❧❛✱ ♣❡❧❛ ❛♣♦✐♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❞✉r❛♥t❡ ♠❡✉s ❡st✉❞♦s ❡ ❡♠ t♦❞♦s ♦s ❛s♣❡❝t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

➚ ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❇❡t✐♥❛ ❡ s✉❛ ❢❛♠í❧✐❛✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ❡ ♠❡ ❛❥✉❞❛r❛♠ ❝♦♠ ♠✉✐t♦ ❝❛r✐♥❤♦ ❡ ♣❛❝✐ê♥❝✐❛✳

❆♦s Pr♦❢❡ss♦r❡s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r✱ ■✈♦ ❡ ❉❛♥✐❡❧✱ q✉❡ ♠❡ ❛♣r❡s❡♥t❛r❛♠ ❛ ♣❡sq✉✐s❛ ♠❛t❡♠át✐❝❛ ❡ s❡♠♣r❡ ❡st✐✈❡r❛♠ ❞✐s♣♦st♦s ❛ ♠❡ ♦r✐❡♥t❛r ❡♠ ✈ár✐♦s ❛s♣❡❝t♦s✱ ❝♦♠ s✐♥❝❡r✐❞❛❞❡ ❡ ❝♦♠♣r❡❡♥sã♦✳ ❊ ❛✐♥❞❛ ❛♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ❢✉♥❝✐♦♥ár✐♦s ❡ ❞♦❝❡♥t❡s ❞♦ ❉▼✲❯❋❙❈❛r✳

❆♦ ♠❡✉ ❛♠✐❣♦ ▲❡♦♥❛r❞♦ P✐r❡s✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ♥♦s ❡st✉❞♦s ❞✉r❛♥t❡ ❛ ❣r❛❞✉✲ ❛çã♦ ❡ ♦ ♠❡str❛❞♦✳ ❆♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ❙ã♦ ❏♦sé ❞♦s ❈❛♠♣♦s✱ q✉❡ ♠❛♥t✐✈❡r❛♠ ❝♦♥t❛t♦ ❝♦♠✐❣♦✱ ♠❡ ❛♣♦✐❛♥❞♦ ❡ s❡♠♣r❡ ❞✐s♣♦st♦s ❛ ♠❡ ❛❥✉❞❛r ❝♦♠♦ ♣✉❞❡r❛♠✳

➚ Pr♦❢❡ss♦r❛ ❑❛r✐♥❛ ❙❝❤✐❛❜❡❧✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦✱ ♣❛❝✐ê♥❝✐❛✱ ✐♥❝❡♥t✐✈♦✱ s✐♥❝❡r✐❞❛❞❡ ❡ ❞✐s♣♦s✐çã♦ ♥❛ ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❊✱ ♣♦r ✜♠✱ ❛ ❉❡✉s✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ♦❜t❡r❡♠♦s ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❛tr❛t♦r❡s ♣❛r❛ ♣r♦❜❧❡♠❛s ♣❛r❛❜ó❧✐❝♦s s❡♠✐❧✐♥❡❛r❡s ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ◆❡✉♠❛♥♥ r❡❧❛t✐✈❛♠❡♥t❡ ❛ ♣❡rt✉r❜❛çõ❡s ❞♦ ❞♦♠í♥✐♦✳ ▼♦str❛r❡♠♦s q✉❡✱ s❡ ❛s ♣❡rt✉r❜❛çõ❡s ❞♦ ❞♦♠í♥✐♦ sã♦ t❛✐s q✉❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦❢✉♥çõ❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ◆❡✉♠❛♥♥ ❡stã♦ ❣❛r❛♥t✐❞❛s✱ ❡♥tã♦ ✈❛❧❡ ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❞♦s ❛tr❛t♦r❡s✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱ t♦❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ♣r♦❜❧❡♠❛ ♥ã♦ ♣❡rt✉r❜❛❞♦ é ❤✐♣❡r❜ó❧✐❝♦✱ ✈❛❧❡ t❛♠❜é♠ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❛tr❛t♦r❡s✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ ♦❜t❛✐♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ❛ttr❛❝t♦rs ❢♦r s❡♠✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ✇✐t❤ ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s r❡❧❛t✐✈❡❧② t♦ ♣❡rt✉r❜❛t✐♦♥s ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t✱ ✐❢ t❤❡ ♣❡rt✉r❜❛t✐♦♥s ♦♥ t❤❡ ❞♦♠❛✐♥ ❛r❡ s✉❝❤ t❤❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡✐♥❣❡♥✈❛❧✉❡s ❛♥❞ ❡✐♥❣❡♥❢✉♥❝t✐♦♥s ♦❢ t❤❡ ◆❡✉♠❛♥♥ ▲❛♣❧❛❝✐❛♥ ✐s ❣r❛♥t❡❞ t❤❡♥ ✇❡ ♦❜t❛✐♥ t❤❡ ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❛ttr❛❝t♦rs✳ ■❢✱ ♠♦r❡♦✈❡r✱ ❡✈❡r② ❡q✉✐❧✐❜r✐✉♠ ♦❢ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♣r♦❜❧❡♠ ✐s ❤②♣❡r❜♦❧✐❝ ✇❡ ❛❧s♦ ♦❜t❛✐♥ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ❛ttr❛❝t♦rs✳

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❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✈

❘❡s✉♠♦ ✈✐✐

❆❜str❛❝t ✐①

■♥tr♦❞✉çã♦ ✶

✶ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✺

✶✳✶ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❚❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✶ ❖♣❡r❛❞♦r❡s ❙❡t♦r✐❛✐s ❡ P♦tê♥❝✐❛s ❋r❛❝✐♦♥ár✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✷ P❡rt✉r❜❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✸ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❛❜str❛t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸ ❆tr❛t♦r❡s ♣❛r❛ s✐st❡♠❛s ♣❛r❛❜ó❧✐❝♦s ❝♦♠ ❡str✉t✉r❛ ❣r❛❞✐❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✶ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ❛tr❛t♦r❡s ♣❛r❛ s✐st❡♠❛s ❣r❛❞✐❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸✳✷ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❛tr❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷ ❈♦♥✈❡r❣ê♥❝✐❛ ❡s♣❡❝tr❛❧ ❡ ❞✐♥â♠✐❝❛ ❧✐♥❡❛r ✷✺

✷✳✶ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❡s♣❡❝tr❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❈♦♥✈❡r❣ê♥❝✐❛ ❞♦s ♦♣❡r❛❞♦r❡s r❡s♦❧✈❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ ❈♦♥✈❡r❣ê♥❝✐❛ ❞♦s s❡♠✐❣r✉♣♦s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸ ❈♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❛tr❛t♦r❡s ✹✺

✸✳✶ ❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❞♦s ❛tr❛t♦r❡s ❡ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐✲ ❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷ ❈♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ✈❛r✐❡❞❛❞❡s ✐♥stá✈❡✐s ❡ ❛tr❛t♦r❡s ✳ ✹✾

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✸✳✷✳✶ ❈♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷✳✷ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❛s ✈❛r✐❡❞❛❞❡s ✐♥stá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✷✳✸ ❈♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❛tr❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✸ ❊①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✳✸✳✶ ❯♠❛ C0 _{♣❡rt✉r❜❛çã♦ ❞❡ ❞♦♠í♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽}

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✶

(14)

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡q✉❛çõ❡s ❞❡ r❡❛çã♦✲❞✐❢✉sã♦ ❞❡♣❡♥✲ ❞❡♥t❡s ❞❡ ✉♠ ♣❛râ♠❡tr♦ ε >0✱ ❞❛❞❛ ♣♦r

  

ut−∆u=f(x, u) ❡♠ Ωε, ∂u

∂n = 0 ❡♠ ∂Ωε,

✭✶✮

♦♥❞❡ Ωε sã♦ ❞♦♠í♥✐♦s ❞❡ ▲✐♣s❝❤✐t③ ❧✐♠✐t❛❞♦s ♥♦ RN✱ N ≥2✱ ♣❛r❛ 0≤ε≤ε0✳

❖ ♦❜❥❡t✐✈♦ é ❢❛③❡r ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡ ❬✷❪✱ ❡♠ q✉❡ ♦s ❛✉t♦r❡s ✐♠♣õ❡♠ ❝♦♥❞✐çõ❡s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ❞♦♠í♥✐♦sΩεq✉❛♥❞♦ε→0✱ ❛ ✜♠ ❞❡ ❣❛r❛♥t✐r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢❛♠í❧✐❛ ❞❡ ❛tr❛t♦r❡s{Aε :ε∈[0, ε0]}✱ q✉❛♥❞♦ε→0✱ ❡♠ ✉♠ ❡s♣❛ç♦ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡♥✐❡♥t❡✳ ❆ss✉♠✐♠♦s q✉❡ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡f :RN_×R_→Ré ❝♦♥tí♥✉❛ ❡♠ ❛♠❜❛s ❛s ✈❛r✐á✈❡✐s

(x, u) ❡✱ ♣❛r❛x_∈RN _✜①♦✱_f₍_x,_·₎_{∈ C}2₍_R₎_{✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❢✉♥çã♦}_f _{s❛t✐s❢❛③} lim sup

|s|→∞

f(x, s)

s <0, ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ x∈R. ✭✷✮

❈♦♠ ❡ss❛s ❝♦♥❞✐çõ❡s✱ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬✸❪ q✉❡ ♦ ♣r♦❜❧❡♠❛ ✭✶✮ t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❡♠ W1,q_(Ω

ε), q > N✱ s❡♠ r❡str✐çõ❡s s♦❜r❡ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ f✳ ❆❧é♠ ❞✐ss♦✱ ❛ss✉♠✐♥❞♦ ❛ ❤✐♣ót❡s❡ ✭✷✮✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✮ ♣♦ss✉✐ ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ Aε✱ ♦ q✉❛❧ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ q ❡ ♦s

❛tr❛t♦r❡s Aε sã♦ ❧✐♠✐t❛❞♦s ❡♠ L∞(Ωε)✱ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ ε✳

❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ f(x,_·) :R _→ R

é C2₍_R₎ _{s❛t✐s❢❛③❡♥❞♦ ✭✷✮ ❡}

∂f ∂u(x, u)

≤

cf,

∂2_f

∂u2(x, u)

≤

˜

cf ∀(x, u)∈RN ×R, ✭✸✮ ❝♦♠cf ❡˜cf ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s✳ ❆❣♦r❛ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ é ❣❧♦❜❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❡♠u✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❡♠ H1_(Ω

ε)✳ ❆ss✐♠✱ ♦s ❛tr❛t♦r❡s ❡st❛rã♦ ❡♠ ❡s♣❛ç♦s ♠❛✐s r❡❣✉❧❛r❡s✱ ❝♦♠♦ W1,q_(Ω

ε) ♣❛r❛ 2 < q < ∞✱ ♠❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ s❡rã♦ ❛♥❛❧✐s❛❞❛s ♥❛ t♦♣♦❧♦❣✐❛ ❞❡ H1_✳

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❈♦♥s✐❞❡r❛♠♦sΩε❝♦♠♦ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❞♦ ❞♦♠í♥✐♦Ω0✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s

  

♣❛r❛ ❝❛❞❛ 0_≤ε_≤ε0, Ωε é ❧✐♠✐t❛❞♦✱ ▲✐♣s❝❤✐t③ ❡

♣❛r❛ t♦❞♦ K ⊂⊂Ω0, ❡①✐st❡ ε(K)t❛❧ q✉❡ K ⊂Ωε, 0< ε≤ε(K).

✭✹✮

◆♦t❡ q✉❡ ❛ ♣r✐♦r✐ ♥ã♦ ❡①✐❣✐♠♦s q✉❡

|Ωε\Ω0| →0 q✉❛♥❞♦ ε→0.

❆ ❣r❛♥❞❡ ❞✐✜❝✉❧❞❛❞❡ q✉❡ s✉r❣❡ q✉❛♥❞♦ ♣❡rt✉r❜❛♠♦s ♦ ❞♦♠í♥✐♦ ❞♦ ♣r♦❜❧❡♠❛ é q✉❡ ❛s s♦❧✉çõ❡s✱ ♣❛r❛ ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❞❡ ε✱ ❡stã♦ ❡♠ ❡s♣❛ç♦s ❞✐❢❡r❡♥t❡s✱ ❡ ♣♦rt❛♥t♦ ❡①♣r❡ssõ❡s

❝♦♠♦uε−u0✭♦♥❞❡uε ∈H1(Ωε)❡u0 ∈H1(Ω0)✮ ♥ã♦ ❢❛③❡♠ s❡♥t✐❞♦ s❡♠ ❛ ❞❡✈✐❞❛ ❛❞❛♣t❛çã♦✳ ❈♦♠ ❡st❡ ✐♥t✉✐t♦✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ♦ ❡s♣❛ç♦

Hε1 =H1(Ωε∩Ω0)⊕H1(Ωε\Ω0)⊕H1(Ω0\Ωε),

♠✉♥✐❞♦ ❞❡ ✉♠❛ ♥♦r♠❛ ❛♣r♦♣r✐❛❞❛✳

P❛r❛ ♦❜t❡r♠♦s ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢❛♠í❧✐❛ ❞❡ ❛tr❛t♦r❡s✱ ❡st✉❞❛r❡♠♦s ❞❡t❛❧❤❛❞❛♠❡♥t❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♣❛rt❡ ❧✐♥❡❛r ♣❛r❛✱ ♥❛ s❡q✉ê♥❝✐❛✱ ✉t✐❧✐③❛r♠♦s ♦ r♦t❡✐r♦ ✉s✉❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞❛s s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r ❞♦s ❛tr❛t♦r❡s✿

✭✐✮ ❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❞♦s ❛tr❛t♦r❡s Aε ❡♠ Hε1✱ q✉❡ é ♦❜t✐❞❛ ♣♦r ♠❡✐♦ ❞❛ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ❡s♣❡❝tr❛❧ ❡♠ H1

ε ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ◆❡✉♠❛♥♥ ❝♦♠ ε →0✱ ♦✉ s❡❥❛✱ ❞♦ ❢❛t♦ q✉❡ ♦s ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦❢✉♥çõ❡s ❞♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ❤♦♠♦❣ê♥❡❛s ♥❛ ❢r♦♥t❡✐r❛ ❝♦♠♣♦rt❛♠✲s❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❡♠ H1

ε q✉❛♥❞♦ ε→0✳ ✭✐✐✮ ❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❞♦s ❛tr❛t♦r❡s Aε ❡♠ Hε1✳ ❯♠❛ ✈❡③ q✉❡ t❡♠♦s ❛ s❡♠✐❝♦♥✲

t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r✱ ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❡♠ H1

ε é ♦❜t✐❞❛ ❡①✐❣✐♥❞♦ q✉❡ t♦❞❛ s♦❧✉çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ ♣r♦❜❧❡♠❛ ♥ã♦ ♣❡rt✉r❜❛❞♦ s❡❥❛ ❤✐♣❡r❜ó❧✐❝❛✳

(16)

❧✐♥❡❛r❡s ❣❡r❛❞♦s ♣❡❧♦ ♣r♦❜❧❡♠❛ ✭✶✮✳ ❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❛tr❛t♦r❡s é ♦❜t✐❞❛ ♥♦ ❈❛♣ít✉❧♦ ✸✳ ❯s❛♥❞♦ ❛ ❋ór♠✉❧❛ ❞❛ ❱❛r✐❛çã♦ ❞❛s ❈♦♥st❛♥t❡s ❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s s❡♠✐❣r✉♣♦s ❧✐♥❡❛r❡s✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐❣r✉♣♦s ♥ã♦ ❧✐♥❡❛r❡s {Tε(t) : t ≥ 0} ❝♦♠♣♦rt❛✲s❡ ❝♦♥t✐♥✲ ✉❛♠❡♥t❡ ❡♠ ε = 0✱ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ ❝♦♠♣❛❝t♦s ❞❡ (0,_∞)✱ ♦ q✉❡ ❥á é s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r✳ P❛r❛ ♠♦str❛r♠♦s ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❞♦s ❛tr❛t♦r❡s✱ ❞❡✈❡♠♦s ❛✐♥❞❛ ♦❜t❡r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛s ✈❛r✲ ✐❡❞❛❞❡s ✐♥stá✈❡✐s ❧♦❝❛✐s✱ ✉♠❛ ✈❡③ q✉❡ ♦s s❡♠✐❣r✉♣♦s ❛ss♦❝✐❛❞♦s tê♠ ❡str✉t✉r❛ ❣r❛❞✐❡♥t❡✳ ❋✐♥❛❧✐③❛♥❞♦✱ ❡①✐❜✐r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ♥♦ q✉❛❧ t❛❧ t❡♦r✐❛ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛✳

(17)

(18)

❈❛♣ít✉❧♦ ✶

❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s

✶✳✶ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

❊st❛ s❡çã♦ ❢❛③ ♣❛rt❡ ❞❡ ✉♠ ❢❡rr❛♠❡♥t❛❧ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛✲ ❜❛❧❤♦✳ ❯♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ❡ ❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✺❪✳

❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❞♦ RN_{✳ ❙✉♣♦♥❤❛ q✉❡} _u _∈ _C1_(Ω) _{s❡❥❛ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❝♦♥t✐♥✉✲} ❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❙❡ ϕ _∈ C∞

0 (Ω) é ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❡♠ Ω✱ s❡❣✉❡ ❞❛ ❢ór♠✉❧❛ ❞❡ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s q✉❡

Z

Ω

u∂ϕ ∂xi

dx=₋

Z

Ω

∂u ∂xi

ϕdx

♣❛r❛ i= 1, ..., n✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ Ω _⊂ RN _{✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡} _u _∈ _Lp

loc(Ω) ♣❛r❛ 1 ≤ p ≤ ∞✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ vi ∈Lp_loc(Ω) é ✉♠❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ u s❡

Z

Ω

u∂ϕ ∂xi

dx=₋

Z

Ω

viϕdx,

♣❛r❛ t♦❞❛ ϕ _∈C∞

0 (Ω)✳ ❙❡ ❡st❡ ❢♦r ♦ ❝❛s♦✱ ❞❡♥♦t❛♠♦s

vi =

∂u ∂xi

.

❉✐③❡♠♦s q✉❡ué ❢r❛❝❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ s❡ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❢r❛❝❛s ❞❡ ♣r✐♠❡✐r❛

♦r❞❡♠ ❞❡ u ❡①✐st❡♠✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❛s ❢✉♥çõ❡s ❢r❛❝❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s é ❞❡♥♦t❛❞♦

♣♦r W1,p_(Ω)_✳

(19)

◗✉❛♥❞♦ ❡①✐st❡✱ vi é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ❛ ♠❡♥♦s ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ♠❡❞✐❞❛ ♥✉❧❛✳ ❖❜s❡r✈❡ q✉❡ C1_(Ω)_⊂_W1,p_(Ω)_{✱ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❢r❛❝❛ é ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❝♦♥❝❡✐t♦} ❝❧áss✐❝♦ ❞❡ ❞❡r✐✈❛❞❛✱ q✉❡ ♠❛♥té♠ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❢ór♠✉❧❛ ❞❡ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❞♦ RN✳ ❉❡✜♥✐♠♦s ♣❛r❛ _m _≥₂

Wm,p(Ω) =

u_∈Lp(Ω) : ∂u

∂xi ∈

Wm−1,p(Ω) para todo i= 1, ..., n

. =   

u_∈Lp_(Ω)

tal que ∀α com |α| ≤m, ∃gα ∈Lp(Ω) tal que

R

ΩuD

α_ϕ_{= (}₋₁₎|α|R

Ωgαϕ ∀ϕ∈Cc∞(Ω)

  

,

♦♥❞❡ α = (α1, α2, ..., αN) ♣❛r❛ αi ≥0 ✉♠ ✐♥t❡✐r♦✱

|α|= N

X

i=1

αi e Dαϕ =

∂|α|_ϕ

∂xα1₁ ∂xα2₂ _{· · ·}∂xαN N

.

Wm,p_(Ω) _{é ❝❧❛r❛♠❡♥t❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ t❛❧ ❡s♣❛ç♦ é ❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈✳} ❉❡♥♦t❛♠♦s✱ q✉❛♥❞♦ ❡①✐st❡♠✱ gα =Dαu✳

◆♦t❛çã♦ ✶✳✸✳ ❖ ❡s♣❛ç♦ W1,2(Ω) é ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛❞♦ ♣♦r H1(Ω)✳ P♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ♥♦r♠❛ ❡♠ Wm,p_(Ω) _{❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛}

ku_kWm,p =

X

0≤|α|≤m

kDαu_kp

❖❜s❡r✈❛çã♦ ✶✳✹✳ ❙❡ Ω é s✉❛✈❡ ❡ ❝♦♠ ❢r♦♥t❡✐r❛ ❧✐♠✐t❛❞❛✱ ❡♥tã♦ ❛ ♥♦r♠❛ ❞❡ Wm,p_(Ω) _é ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛

ku_kp +

X

|α|=m kDα_u

kp.

❉❡✜♥✐♠♦s t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦ W₀1,p ❝♦♠♦ ♦ ❢❡❝❤♦ ❞♦ ❝♦♥❥✉♥t♦C1

0(Ω) ❡♠ W1,p(Ω)✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛H1_(Ω) _{t❡♠♦s ❛ s❡❣✉✐♥t❡ ♥♦r♠❛}

kukW1,2_(Ω) =

Z

Ω|

u|2₊ N X i=1 Z Ω ∂u ∂xi

2!12

.

❖ ❝♦♥❥✉♥t♦W₀1,2(Ω)é ♦ ❢❡❝❤♦ ❞❡ C1

0(Ω) ❡♠W1,2(Ω)✳ ❊♠ ❛♠❜♦s ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s W1,2_(Ω) _❡_W1,2

0 (Ω) ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

hu, v_i=

Z Ω uv+ N X i=1 ∂u ∂xi ∂v ∂xi

=_hu, v_i_L2_(Ω)+ N X i=1 ∂u ∂xi , ∂v ∂xi

L2_(Ω)

.

(20)

❉❡st❛ ❢♦r♠❛✱ ❛ ♥♦r♠❛ ❞❡✜♥✐❞❛ ❛❝✐♠❛ é ❞❡r✐✈❛❞❛ ❞❡st❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ❊❧❛ t❛♠❜é♠ é ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛

ku_kW1,2_(Ω) =

Z

Ω|

u_|2

1₂

+ N

X

i=1

Z

Ω

∂u ∂xi

2!1₂

=_ku_kL2_(Ω)+ N

X

i=1

∂u ∂xi

L2_(Ω)

.

❚❡♦r❡♠❛ ✶✳✺✳ W1,2_(Ω) _{é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❊♠ ♣❛rt✐❝✉❧❛r✱} _W1,2

0 (Ω) t❛♠❜é♠ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳

❚❡♦r❡♠❛ ✶✳✻✳ C∞_(Ω)_∩_W1,2_(Ω) _{é ❞❡♥s♦ ❡♠} _W1,2_(Ω)_{✳ ❙❡} _Ω _{é ✉♠ ❛❜❡rt♦ ❝♦♠ ❢r♦♥t❡✐r❛} ❞❡ ❝❧❛ss❡ C1_{✱ ❡♥tã♦} _C∞_(Ω)_∩_W1,2_(Ω) _{é ❞❡♥s♦ ❡♠} _W1,2_(Ω)_✳

❖s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ❝❛r❛❝t❡r✐③❛♠ ♦ ❡s♣❛ç♦ W₀1,2(Ω)✿

❚❡♦r❡♠❛ ✶✳✼✳ ❙❡ u_∈W1,2_(Ω) _{❡ s❛t✐s❢❛③ s✉♣♣}₍_u₎_⊂⊂_Ω_{✱ ❡♥tã♦} _u_∈_W1,2

0 (Ω)✳ ❙❡ Ω⊂RN é ✉♠ ❛❜❡rt♦ ❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡ C1 _❡ _u _∈ _W1,2_(Ω)_∩_C_(Ω)_{✱ ❡♥tã♦} _u _∈ _W1,2

0 (Ω) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u= 0 ❡♠ ∂Ω✳

❆s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛ ❞♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ sã♦ ❛s q✉❡ ❧❤❡ ❝♦♥❢❡r❡♠ ❛ s✉❛ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡✳ ❘❡❝♦r❞❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ❡ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛✿ ❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ E ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ ❞❡ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ F ✭♦✉

s❡❥❛✱ ❛ ♥♦r♠❛ ❞❡ E ♥ã♦ ♣r❡❝✐s❛ ♥❡❝❡ss❛r✐❛♠❡♥t❡ s❡r ❛ ♥♦r♠❛ ✐♥❞✉③✐❞❛ ❞❡ F✮✳ ❉✐③❡♠♦s

q✉❡ ❛ ✐♥❝❧✉sã♦ E ⊂ F é ✉♠❛ ✐♠❡rsã♦ ✭❝♦♥tí♥✉❛✮ s❡ ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ Id : E → F

❞❡✜♥✐❞❛ ♣♦r Id(x) = x ❢♦r ❝♦♥tí♥✉❛✳ ❉❡♥♦t❛♠♦s ❡st❡ ❢❛t♦ ♣♦r

E ֒_→F.

❙❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ ❢♦r ❝♦♠♣❛❝t❛✱ ❞✐③❡♠♦s q✉❡ ❛ ✐♠❡rsã♦ E ֒_→ F é

❝♦♠♣❛❝t❛✳ ❉❡♥♦t❛♠♦s ❛ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ E ❡♠ ✉♠

❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ F ♣♦r

E ֒→c F.

❈♦♠♦ ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ é ❧✐♥❡❛r✱ ♦ ❢❛t♦ ❞❡ ❡①✐st✐r ✉♠❛ ✐♠❡rsã♦ E ֒_→F é ❡q✉✐✈❛✲

❧❡♥t❡ à ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡

kxkF ≤CkxkE ♣❛r❛ t♦❞♦ x∈E.

(21)

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡(xn)é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠E✱ ❡♥tã♦(xn)t❛♠❜é♠ é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ F❀ ❧♦❣♦✱ s❡xn →x❡♠ E✱ ❡♥tã♦xn →x❡♠ F t❛♠❜é♠✳ ➱ ❝❧❛r♦ q✉❡ s❡E t❡♠ ❛ ♥♦r♠❛ ✐♥❞✉③✐❞❛ ❞❡F✱ ❡♥tã♦ ❛ ✐♥❝❧✉sã♦E ⊂F é ✉♠❛ ✐♠❡rsã♦✱ ❝♦♠C = 1✳ ◗✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ✐♠❡rsã♦ E ֒_→ F✱ ❞✐③❡r q✉❡ ❡❧❛ é ❝♦♠♣❛❝t❛ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡

s❡q✉ê♥❝✐❛s ❧✐♠✐t❛❞❛s ❞❡ (E,k · kE)♣♦ss✉❡♠ s✉❜s❡q✉ê♥❝✐❛s ❝♦♥✈❡r❣❡♥t❡s ❡♠ (F,k · kF)✳ ❚❡♦r❡♠❛ ✶✳✾ ✭❚❡♦r❡♠❛ ❞❛ ■♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈✮✳ ❙❡❥❛ Ω⊂RN ✉♠ ❛❜❡rt♦✳ ❊♥tã♦

W1,2(Ω) ֒_→L2(Ω),

W₀1,2(Ω) ֒_→L2(Ω).

❖❜s❡r✈❡ q✉❡✱

kukW1,2_(Ω) =kuk_L2_(Ω)+ N

X

i=1

∂u ∂xi

L2_(Ω)

≥ kukL2_(Ω),

❡ ♦ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ W₀1,2(Ω)✳

❚❡♦r❡♠❛ ✶✳✶✵ ✭❚❡♦r❡♠❛ ❞❡ ❘❡❧❧✐❝❤✲❑♦♥❞r❛❦❤♦✈✮✳ ❙❡❥❛ Ω ⊂ RN ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❡

❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡ C1_{✳ ❊♥tã♦ t❡♠♦s ❛s s❡❣✉✐♥t❡s ✐♠❡rsõ❡s ❝♦♠♣❛❝t❛s✿}

W1,p(Ω)⊂Lq(Ω) ∀q∈[1, p∗), onde 1 p∗ =

1

p−

1

N, se p < N W1,p(Ω)_⊂Lq_(Ω)

∀q_∈[p,+_∞),

W1,p(Ω)_⊂C(Ω),

❡♠ ♣❛rt✐❝✉❧❛r✱ W1,p_(Ω)_⊂_Lp_(Ω) _{❝♦♠ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛ ♣❛r❛ t♦❞♦}_p _{✭❡ t♦❞♦} _N_✮✳

❚❡♦r❡♠❛ ✶✳✶✶✳ ❬❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ P♦✐♥❝❛ré❪ ❙❡❥❛ Ω_⊂ RN _{✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❡} ₁_≤ _p_≤ ∞✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C ✭❞❡♣❡♥❞❡♥t❡ ❞❡ Ω ❡ p✮ t❛❧ q✉❡

ku_kLp_(Ω) ≤Ck∇uk_Lp_(Ω),

♣❛r❛ t♦❞♦ u∈W₀1,p(Ω)✳

❖❜s❡r✈❡ q✉❡ ♦ ❚❡♦r❡♠❛ ✶✳✶✶ ♥ã♦ é ✈á❧✐❞♦ s❡ tr♦❝❛r♠♦s W₀1,2 ♣♦r W1,2_{✱ ♣♦✐s✱ ❛s} ❢✉♥çõ❡s ❝♦♥st❛♥t❡s ♣❡rt❡♥❝❡♥t❡s ❛ W1,2 _{♥ã♦ s❛t✐s❢❛③❡♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ P♦✐♥❝❛ré ✭♣♦✐s} tê♠ ❞❡r✐✈❛❞❛ ♥✉❧❛✮✳

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✶✳✷ ❚❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s

❆ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♥♦ss♦ ♦❜❥❡t✐✈♦✱ ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ❞❡t❛❧✲ ❤❛❞❛ ❡ ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ♥❡st❛ s❡çã♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✼❪ ❡ ❬✶✸❪✳

❙❡❥❛X✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❉❡♥♦t❡♠♦s ♣♦r_L(X)♦ ❡s♣❛ç♦ ❞♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❝♦♥tí♥✉♦s ❞❡ X ❡♠ X ❝♦♠ ❛ ♥♦r♠❛ ✉s✉❛❧✱ ♦✉ s❡❥❛✱ ♣❛r❛ T _{∈ L}(X)✱

kT kL(X)= sup x∈X, x6=0

kT x_kX kx_kX

❉❡✜♥✐çã♦ ✶✳✶✷✳ ❯♠ s❡♠✐❣r✉♣♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s é ✉♠❛ ❢❛♠í❧✐❛ {T(t);t _≥

0} ⊂ L(X) ❞❡ ♦♣❡r❛❞♦r❡s ❛ ✉♠ ♣❛râ♠❡tr♦ q✉❡ t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✭✐✮✳ T(0) =I✳

✭✐✐✮✳ T(t+s) =T(t)T(s), ∀ t, s ≥0✳ ✭✐✐✐✮✳ T : [0,_∞)_−→X é ❝♦♥tí♥✉❛✳

✭✐✐✐✬✮✳ ❙❡kT(t)−Ik →0q✉❛♥❞♦t →0+_{✱ ♦ s❡♠✐❣r✉♣♦ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦✳} ✭✐✐✐✑✮✳ ❙❡ kT(t)x₋x_{k →} 0 q✉❛♥❞♦ t _→ 0+_{✱ ♦ s❡♠✐❣r✉♣♦ é ❢♦rt❡♠❡♥t❡ ❝♦♥tí♥✉♦} ♦✉ C0−semigrupo✳

❖❜s❡r✈❛çã♦ ✶✳✶✸✳ ◆♦t❡ q✉❡ ♥❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r ♦ ✐t❡♠ ✭✐✐✐✬✮ ✐♠♣❧✐❝❛ ♥♦ ✐t❡♠ ✭✐✐✐✑✮✳ ❚❡♦r❡♠❛ ✶✳✶✹✳ ❙❡❥❛{T(t);t_≥0_}✉♠C0−semigrupo✳ ❊♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡sM ≥1 ❡ ω_≥0 t❛✐s q✉❡

kT(t)_{k ≤}M eωt_,

♣❛r❛ t♦❞♦ t_≥0✳

❉❡✜♥✐çã♦ ✶✳✶✺✳ ❙❡ {T(t);t _≥0_} é ✉♠ C0−semigrupo✱ s❡✉ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ é ♦ ♦♣❡r❛❞♦r A:D(A)⊂X→X ❞❛❞♦ ♣♦r

D(A) =

x_∈X; lim t→0+

T(t)x₋x

t existe

Ax= lim t_→0+

T(t)x₋x t

❖ r❡s✉❧t❛❞♦ s❡❣✉✐♥t❡ ❝❛r❛❝t❡r✐③❛ ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ✉♥✐❢♦r♠❡✲ ♠❡♥t❡ ❝♦♥tí♥✉♦✳

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❚❡♦r❡♠❛ ✶✳✶✻✳ ❯♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r A : X → X é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ s❡♠✐✲

❣r✉♣♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A_{∈ L}(X)✳

◆♦ r❡s✉❧t❛❞♦ s❡❣✉✐♥t❡✱ r❡✉♥✐♠♦s ❛❧❣✉♠❛s ♥♦tá✈❡✐s ♣r♦♣r✐❡❞❛❞❡s ❛ r❡s♣❡✐t♦ ❞♦s C0−

semigrupos✳

❚❡♦r❡♠❛ ✶✳✶✼✳ ❙❡❥❛ {T(t);t _≥ 0_} ✉♠ C0 −semigrupo ❡ A s❡✉ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧✳ ❊♥tã♦ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮✳ P❛r❛ t♦❞♦ x_∈X✱ ❛ ❢✉♥çã♦

t7−→T(t)x

é ❝♦♥tí♥✉❛✳

✭✐✐✮✳ P❛r❛ x_∈X✱ lim h→0

1

h

Z t+h

t

T(s)xds=T(t)x✳

✭✐✐✐✮✳ P❛r❛ x∈X✱ t❡♠♦s

Z t

0

T(s)xds∈D(A) ❡

A

Z t 0

T(s)xds

=T(t)x−x.

✭✐✈✮✳ P❛r❛ x ∈ D(A)✱ t❡♠♦s T(t)x ∈ D(A)✱ ❛ ❛♣❧✐❝❛çã♦ t 7→ T(t)x é ❝♦♥t✐♥✉❛♠❡♥t❡

❞✐❢❡r❡♥❝✐á✈❡❧ ❡

d

dtT(t)x=AT(t)x=T(t)Ax.

✭✈✮✳ P❛r❛ x_∈D(A)✱

T(t)x−T(s)x=

Z t

s

T(s)Axds=

Z t

s

AT(s)xds.

❈♦r♦❧ár✐♦ ✶✳✶✽✳ ❙❡❥❛ {T(t);t ≥ 0} ✉♠ C0 −semigrupo ❡ A s❡✉ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧✳ ❊♥tã♦ D(A) = X ❡ A é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❢❡❝❤❛❞♦✳

❖ ❚❡♦r❡♠❛ ❞❡ ❍✐❧❧❡✲❨♦s✐❞❛ ❝❛r❛❝t❡r✐③❛ ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠C0−semigrupo ❞❡ ❝♦♥tr❛çõ❡s✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✼❪ ❡ ❬✶✸❪✳

❉❡✜♥✐çã♦ ✶✳✶✾✳ ❙❡❥❛ A : D(A) _⊂ X _→ X ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❢❡❝❤❛❞♦✳ ❉❡✜♥✐♠♦s ♦

r❡s♦❧✈❡♥t❡ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r ρ(A)✱ ❝♦♠♦

ρ(A) = _{λ_∈C_, ₍_λ₋_A₎ é _bijetor_}_.

❉❡✜♥✐çã♦ ✶✳✷✵✳ ❖ C0 −semigrupo {T(t);t ≥ 0} é ✉♠ C0 −semigrupo ❞❡ ❝♦♥tr❛çõ❡s q✉❛♥❞♦ kT(t)k≤1✱ ♣❛r❛ t♦❞♦ t≥0✳

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❚❡♦r❡♠❛ ✶✳✷✶ ✭❍✐❧❧❡✲❨♦s✐❞❛✮✳ ❙❡❥❛ A : D(A)⊂ X → X ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r✳ ❊♥tã♦ ❛s

s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✭✐✮✳ A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ C0−semigrupo t❛❧ q✉❡

kT(t)_{k ≤}eωt_,

∀t_≥0 (ω >0).

✭✐✐✮✳ A é ❢❡❝❤❛❞♦✱ ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦✱ (ω,∞)⊂ρ(A) ❡

k(λ₋A)−1_{k ≤} 1

λ₋ω, ∀ λ > ω.

❖ ❚❡♦r❡♠❛ ❞❡ ▲✉♠❡r✲P❤✐❧❧✐♣s ❢♦r♥❡❝❡ ♦✉tr❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ❣❡r❛❞♦r❡s ✐♥✜♥✐t❡s✲ ✐♠❛✐s ❞❡ C0 −semigrupos ❞❡ ❝♦♥tr❛çõ❡s✳ ❯♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ t❛❧ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✼❪ ❡ ❬✶✸❪✳

◆♦t❛çã♦ ✶✳✷✷✳ P❛r❛ x∗ ∈X∗ ✭X∗ r❡♣r❡s❡♥t❛ ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞❡ X✮ ❡ x∈X✱ ❞❡♥♦t❛♠♦s

x∗(x) =_hx∗, x_i.

❉❡✜♥✐çã♦ ✶✳✷✸✳ P❛r❛ x_∈X ❞❡✜♥❛ ❛ ❞✉❛❧✐❞❛❞❡ ❞❡ x✱ ❞❡♥♦t❛❞❛ ♣♦r F(x)_⊂X✱ ♣♦r

F(x) = {x∗;x∗ ∈X∗ e Rehx∗, xi=kxk2 ₌_k_x∗_k2_}_.

❖❜s❡r✈❛çã♦ ✶✳✷✹✳ F(x)₆=_∅ ♣❛r❛ t♦❞♦ x_∈X✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤✳

❉❡✜♥✐çã♦ ✶✳✷✺✳ ❯♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r A : D(A) ⊂ X → X é ❞✐ss✐♣❛t✐✈♦ s❡ ♣❛r❛ ❝❛❞❛ x_∈D(A) ❡①✐st❡ x∗ _∈_F₍_x₎ _{t❛❧ q✉❡} _Re_h_x∗_{, Ax}_{i ≤}₀_✳

❚❡♦r❡♠❛ ✶✳✷✻ ✭▲✉♠❡r✲P❤✐❧❧✐♣s✮✳ ❙❡❥❛ A : D(A) ⊂ X → X ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♠ D(A) = X✳ ❱❛❧❡♠✿

✭✐✮ ❙❡ A é ❞✐ss✐♣❛t✐✈♦ ❡ ❡①✐st❡ λ0 > 0 t❛❧ q✉❡ Im(λ0I −A) = X✱ ❡♥tã♦ A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ C0−semigrupo ❞❡ ❝♦♥tr❛çõ❡s ❡♠ X✳

✭✐✐✮ ❙❡ A é ♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ C0 −semigrupo ❞❡ ❝♦♥tr❛çõ❡s ❡♠ X ❡♥tã♦

Im(λI−A) =X ♣❛r❛ t♦❞♦λ >0❡A é ❞✐ss✐♣❛t✐✈♦✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ t♦❞♦x∈D(A) ❡ ♣❛r❛ t♦❞♦ x∗ _∈_F₍_x₎_{✱ t❡♠♦s q✉❡} _Re_h_x∗_{, Ax}_{i ≤}₀_✳

❚❡♦r❡♠❛ ✶✳✷✼✳ ❙❡ A é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❞✐ss✐♣❛t✐✈♦ ❡♠ X ❝♦♠ Im(I−A) = X ❡ X

é r❡✢❡①✐✈♦✱ ❡♥tã♦ D(A) =X✳

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❉❡✜♥✐çã♦ ✶✳✷✽✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠ X✳ ❆ ✐♠❛❣❡♠ ♥✉♠ér✐❝❛ ❞❡ A é ♦

❝♦♥❥✉♥t♦

S(A) = {hx∗, Axi;x∈D(A) com kxk= 1, x∗ ∈X∗ com kx∗k= 1 e hx∗, xi= 1}.

❙❡ ❳ é ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱ ❡♥tã♦ S(A) =_{hAx, x_i;x_∈D(A) ❡ _kx_k= 1_}✳

❚❡♦r❡♠❛ ✶✳✷✾✳ ❙❡❥❛♠ A ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♠D(A) = X✱ ❢❡❝❤❛❞♦✱ ❡S(A) ❛ ✐♠❛❣❡♠ ♥✉♠ér✐❝❛ ❞❡ A✳ ❙❡❥❛ ❛✐♥❞❛Σ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡ ❝♦♥❡①♦ ❞❡C_\_S₍_A₎✳ ❙❡ _{λ /}_∈_S₍_A₎✱

❡♥tã♦ (λI₋A) é ✐♥❥❡t♦r❛✱ t❡♠ ✐♠❛❣❡♠ ❢❡❝❤❛❞❛ ❡ s❛t✐s❢❛③

k(λI−A)xkX ≥d(λ, S(A))kxkX, ∀ x∈D(A).

❆❧é♠ ❞✐ss♦✱ s❡ Σ_∩ρ(A)₆=_∅✱ ❡♥tã♦ Σ_⊂ρ(A) ❡

k(λI−A)−1kL(X) ≤

1

d(λ, S(A)), ∀ λ∈Σ.

✶✳✷✳✶ ❖♣❡r❛❞♦r❡s ❙❡t♦r✐❛✐s ❡ P♦tê♥❝✐❛s ❋r❛❝✐♦♥ár✐❛s

❉❡✜♥✐çã♦ ✶✳✸✵✳ ❙❡❥❛ A : D(A) ⊂ X → X ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r✳ ❉✐③❡♠♦s q✉❡ −A é ✉♠

♦♣❡r❛❞♦r s❡t♦r✐❛❧ s❡ A é ❢❡❝❤❛❞♦✱ ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦ ❡

Σa,ϕ={λ∈C;|arg(λ−a)|< ϕ} ⊂ρ(A),

♣❛r❛ ❛❧❣✉♠ ϕ_∈(π 2, π) ❡

k(λ₋A)−1_k_L(X) ≤

M

|λ₋a_|, ∀ λ∈Σa,ϕ.

❚❡♦r❡♠❛ ✶✳✸✶✳ ❙❡❥❛ A:D(A)⊂X →X ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r t❛❧ q✉❡ −A é s❡t♦r✐❛❧✱ ✐st♦

é✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s a _∈R✱ _{M >} ₀ ❡ _ϕ_∈₍π

2, π) t❛✐s q✉❡

Σa,ϕ={λ ∈C;|arg(λ−a)|< ϕ} ⊂ρ(A)

❡

k(λ₋A)−1_k_L(X) ≤

M

|λ₋a_|, ∀ λ∈Σa,ϕ.

❊♥tã♦ A ❣❡r❛ ✉♠C0−semigrupo {T(t);t ≥0} ⊂ L(X)✱

T(t) = 1 2πi

Z

Γa

eλt₍_λ

−A)−1dλ,

(26)

♦♥❞❡ Γa é ❛ ❢r♦♥t❡✐r❛ ❞❡ Σa,β \ {λ ∈ C;|λ−a| ≤ r} ♣❛r❛ ❛❧❣✉♠ r ♣❡q✉❡♥♦✱ π₂ < β < π✱ ♦r✐❡♥t❛❞❛ ♥♦ s❡♥t✐❞♦ ❞❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❝r❡s❝❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱

kT(t)kL(X) ≤Keat, t ≥0 para algum K >0,

kAT(t)_k_L(X) ≤K1t−1eat, t >0 para algum K1 >0.

❚❡♠♦s ❛✐♥❞❛ q✉❡

d

dtT(t) = AT(t) é limitado, para todo t >0.

❍✐♣ót❡s❡ ✶✳✸✷✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❢❡❝❤❛❞♦ ❡ ❞❡♥s❛♠❡♥t❡ ❞❡✜♥✐❞♦ ♣❛r❛ ♦ q✉❛❧

Σ+_ω =_{λ: 0< ω <_|argλ_{| ≤}π_{} ∪}V _⊂ρ(A)

♦♥❞❡ Vé ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ③❡r♦ ❡

k(λ₋A)−1_{k ≤} M

1_{− |}λ_|, λ ∈Σ

+ ω.

❉❡✜♥✐çã♦ ✶✳✸✸✳ ❙❡ A s❛t✐s❢❛③ ❛ ❤✐♣ót❡s❡ ✶✳✸✷ ❡ α >0✱ ❞❡✜♥✐♠♦s

A−α ₌ 1 2πi

Z

C

λ−α₍

−λ+A)−1dλ, ✭✶✳✶✮

♦♥❞❡ C é ♦ ❝❛♠✐♥❤♦ q✉❡ ♣❡r❝♦rr❡ ♦ r❡s♦❧✈❡♥t❡ ❞❡ A ❞❡ _−∞eiν _❛ _∞_eiν_{, ω < ν < π}_✱ ❡✈✐t❛♥❞♦ ♦ ❡✐①♦ r❡❛❧ ♥❡❣❛t✐✈♦✳

❖❜s❡r✈❡ q✉❡ ♣❛r❛ α = n ∈ N∗✱ ❛ ❉❡✜♥✐çã♦ ✶✳✸✸ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ✉s✉❛❧ ❞❡

A−n _{= (}_A₋1₎n_✳

❉❡✜♥✐çã♦ ✶✳✸✹✳ ❉❡✜♥✐♠♦s ♦ ♦♣❡r❛❞♦r Aα_{✱ ♣❛r❛} _{α >}₀_{✱ ❝♦♠♦ s❡♥❞♦ ❛ ✐♥✈❡rs❛ ❞❡} _A−α_{✱ ❡}

A0 ₌_I_{✱ ❡ ♦ ❞❡♥♦♠✐♥❛♠♦s ❞❡ ♦♣❡r❛❞♦r ♣♦tê♥❝✐❛ ❢r❛❝✐♦♥ár✐❛ ❛ss♦❝✐❛❞♦ ❛} _A_✳ ❚❡♦r❡♠❛ ✶✳✸✺✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r q✉❡ s❛t✐s❢❛③ ❛ ❤✐♣ót❡s❡ ✶✳✸✷✱ ❝♦♠ ω < π

2✳ ❊♥tã♦ ✶✳ Aα _{é ♦♣❡r❛❞♦r ❢❡❝❤❛❞♦ ❝♦♠ ❞♦♠í♥✐♦} _D₍_Aα_{) =}_Im₍_A−α₎_✳

✷✳ ❙❡ α _≥β >0 ❡♥tã♦ D(Aα₎_⊂_D₍_Aβ₎_✳ ✸✳ D(Aα_{) =} _X ♣❛r❛ t♦❞♦ _α_≥₀✳

✹✳ ❙❡ α, β ∈R✱ ❡♥tã♦_Aα+β_x₌_Aα_Aβ_x✱ ♣❛r❛ t♦❞♦ _x_∈_D₍_Aγ₎✱ ♦♥❞❡ _γ _{= max}_{_{α, β, α}₊

β_}✳

(27)

❚❡♦r❡♠❛ ✶✳✸✻✳ ❙❡❥❛−A♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠C0−semigrupo T(t)✳ ❙❡0∈ρ(A)✱ ❡♥tã♦

✶✳ T(t) :X →D(Aα₎_{✱ ♣❛r❛ t♦❞♦} _α_≥₀_✳

✷✳ P❛r❛ ❝❛❞❛ x_∈D(A)✱ t❡♠♦s T(t)Aα_x₌_Aα_T₍_t₎_x.

✸✳ P❛r❛ Re(σ)> δ >0 ❡ ❝❛❞❛ t >0✱ Aα_T₍_t₎ _{é ✉♠ ♦♣❡r❛❞♦r ❧✐♠✐t❛❞♦ ❡}

kAαT(t)k ≤Kαt−αe−δt.

✹✳ ❙❡ 0< α≤1 ❡ x∈D(A−α₎_{✱ ❡♥tã♦}

kT(t)x−xk ≤CαtαkAαxk.

✶✳✷✳✷ P❡rt✉r❜❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❧✐♠✐t❛❞♦s

❖ ✐♠♣♦rt❛♥t❡ ♥❡st❡ ♠♦♠❡♥t♦ é ❞❡t❡r♠✐♥❛r ❡st✐♠❛t✐✈❛s ♣❛r❛ ♦♣❡r❛❞♦r❡s ❝♦♥❤❡❝✐❞♦s q✉❡ s♦❢r❡♠ ❛ ❛çã♦ ❞❡ ♦✉tr♦s ♦♣❡r❛❞♦r❡s ♣❛rt✐❝✉❧❛r❡s✳

◆♦t❛çã♦ ✶✳✸✼✳ ❉❡♥♦t❛r❡♠♦s✱ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡✱ ♣♦r {eAt_;_t _≥₀_} _{♦ s❡♠✐❣r✉♣♦ ❣❡r❛❞♦ ♣♦r} ✉♠ ♦♣❡r❛❞♦r A :D(A)⊂X →X✳

❚❡♦r❡♠❛ ✶✳✸✽✳ ❙❡❥❛ {eAt_;_t _≥₀_} _✉♠ _C

0−semigrupo ❝♦♠ ❣❡r❛❞♦r A:D(A)⊂X →X✳ ❙❡ B ∈ L(X)✱ ❡♥tã♦ A+B : D(A) ⊂ X → X é ♦ ❣❡r❛❞♦r ❞❡ ✉♠ C0 −semigrupo {e(A+B)t_;_t_≥₀_}_{✳ ❙❡}

eAt

≤M eωt✱ ♣❛r❛ t♦❞♦t ≥0✱ ❡♥tã♦

e(A+B)t

≤M e(ω+MkBk)t ♣❛r❛

t♦❞♦ t ≥0✳

❚❡♦r❡♠❛ ✶✳✸✾✳ ❙❡❥❛♠A :D(A)_⊂X _→X t❛❧ q✉❡ ₋Aé s❡t♦r✐❛❧ ❡B :D(B)_⊂X _→X✱

❝♦♠ D(A)⊂D(B)✱ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r t❛❧ q✉❡

kBxk ≤εkAxk+kkxk, para todo x∈D(A),

♣❛r❛ ❛❧❣✉♠ ε > 0 ❡ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ kkxk✳ ❊♥tã♦✱ ❡①✐st❡ δ >0 t❛❧ q✉❡✱ s❡ 0≤ε ≤δ✱ ♦

♦♣❡r❛❞♦r −(A+B) é s❡t♦r✐❛❧✱D(A+B) =D(A) ❡ _{e(A+B)t_;_t_≥₀_} _{é ✉♠} _C

0−s❡♠✐❣r✉♣♦✳ ❈♦r♦❧ár✐♦ ✶✳✹✵✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r s❡t♦r✐❛❧ ❡ B : D(B) ⊂ X → X ✉♠ ♦♣❡r❛❞♦r

❢❡❝❤❛❞♦✱ D(Aα₎_⊂_D₍_B₎_{✱ ♣❛r❛ ❛❧❣✉♠❛} ₀_{< α <}₁_{✳ ❊♥tã♦} ₍_A₊_B₎ _{é s❡t♦r✐❛❧✳}

(28)

✶✳✷✳✸ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❛❜str❛t♦

❈♦♥s✐❞❡r❛♠♦s ♥❡st❛ s❡çã♦ ❡q✉❛çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ❞❛ ❢♦r♠❛

  

ut+Au=f(t, u), t > t0,

u(t0) =u0,

✭✶✳✷✮

♦♥❞❡ A : D(A) ⊂ X → X é ✉♠ ♦♣❡r❛❞♦r s❡t♦r✐❛❧ ❡ Reσ(A) >0 ✭♣❛rt❡ r❡❛❧ ❞♦ ❡s♣❡❝tr♦ ♣♦s✐t✐✈❛✮✱ t❛❧ q✉❡ s✉❛s ♣♦tê♥❝✐❛s ❢r❛❝✐♦♥ár✐❛s Aα _{❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s✱ ❡ ♦s ❡s♣❛ç♦s} _Xα ₌

D(Aα₎ _{❝♦♠ ❛ ♥♦r♠❛ ❞♦ ❣rá✜❝♦} _k_x_k

α = kAαxk ❡stã♦ ❞❡✜♥✐❞♦s ♣❛r❛ α ≥ 0✳ ❆ss✉♠✐♠♦s ❛✐♥❞❛ q✉❡ f :U _→X✱ ❡♠ q✉❡ U _⊂R_×_Xα _{é ❛❜❡rt♦✳}

❯♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✷✮ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ u : [t0, t1) → X✱ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ (t0, t1)✱ ❝♦♠ u(t0) =u0✱ f(·, u(·)) : [t0, t1)→ X ❝♦♥tí♥✉❛✱ u(t)∈D(A)✱ ♣❛r❛ t∈ (t0, t1)✱ ❡ q✉❡ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ✭✶✳✷✮✳

❖ t❡♦r❡♠❛ s❡❣✉✐♥t❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❧♦❝❛✐s ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✳✷✮✳

❚❡♦r❡♠❛ ✶✳✹✶✳ ❙❡❥❛♠ A ✉♠ ♦♣❡r❛❞♦r s❡t♦r✐❛❧✱ 0≤α < 1✱ U ⊂R_×_Xα_{✱ ❡} _f _: _U _→_X_✳ ❙✉♣♦♥❤❛♠♦s q✉❡ f s❡❥❛ ❍♦❧❞❡r ❝♦♥tí♥✉❛ ♥❛ ✈❛r✐á✈❡❧ t ❡ ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❝♦♥tí♥✉❛ ♥❛

✈❛r✐á✈❡❧ x❡♠ U✱ ♦✉ s❡❥❛✱ s✉♣♦♥❤❛♠♦s q✉❡ ♣❛r❛(t1, u1)∈U ❡①✐st❛ ✉♠❛ ✈✐③✐♥❤❛♥ç❛V ⊂U ❞❡ (t1, u1) t❛❧ q✉❡✱ ♣❛r❛ (t, u),(s, v)∈V✱

kf(t, u)₋f(s, v)_kX ≤L(|t−s|θ+ku−vkα), ✭✶✳✸✮

s❡♥❞♦ θ ❡L ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s✳ ❊♥tã♦✱ q✉❛❧q✉❡r q✉❡ s❡❥❛(t0, t1)∈U✱ ❡①✐st❡ ✉♠ ✐♥st❛♥t❡

τ =τ(t0, t1)>0 ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ❛ ❡q✉❛çã♦ ✭✶✳✷✮ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ u ❞❡✜♥✐❞❛ ❡♠ (t0, t0 +τ)✳

❆❝❡r❝❛ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❣❧♦❜❛✐s ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✳✷✮✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

❚❡♦r❡♠❛ ✶✳✹✷✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛s ❤✐♣ót❡s❡s s♦❜r❡A ❡ f ❡♥✉♥❝✐❛❞❛s ♥♦ ❚❡♦r❡♠❛ ✭✶✳✹✶✮

❡st❡❥❛♠ s❛t✐s❢❡✐t❛s✱ ❡ ❛ss✉♠❛♠♦s t❛♠❜é♠ q✉❡✱ ♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ B _⊂ U✱ ❛

✐♠❛❣❡♠ f(B)s❡❥❛ ❧✐♠✐t❛❞❛ ❡♠ X✳ ❙❡ ué ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✷✮ ❡♠ (t0, t1) ❡t1 é ♠❛①✐♠❛❧ ❡♥tã♦✱ ♦✉t1 =∞ ♦✉ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛tn

n→∞

(29)

✶✳✸ ❆tr❛t♦r❡s ♣❛r❛ s✐st❡♠❛s ♣❛r❛❜ó❧✐❝♦s ❝♦♠ ❡str✉t✉r❛

❣r❛❞✐❡♥t❡

◆❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s ❡ ❡st✉❞❛r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❛tr❛t♦r❡s✳

❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ −A : D(A) ⊂ X → X ✉♠ ♦♣❡r❛❞♦r s❡t♦r✐❛❧✱ t❛❧

q✉❡ A−1 _{é ❝♦♠♣❛❝t♦ ❡} _Re₍_ρ₍_A₎₎_⊂₍_−∞_,₋_δ_]_{✱ ♣❛r❛ ❛❧❣✉♠} _{δ >}₀_{✳ ❊♥tã♦ t❡♠♦s}

keAt

kL(X) ≤M e−δt, t >0.

❉❡♥♦t❡♠♦sXα _:=_D₍₍₋_A₎α₎ _{❝♦♠ ❛ ♥♦r♠❛}_k_x_k

α :=kxkXα =k(−A)αxk X✳

❖❜s❡r✈❛çã♦ ✶✳✹✸✳ ❙❡ A t❡♠ r❡s♦❧✈❡♥t❡ ❝♦♠♣❛❝t♦ ❡♥tã♦ Xα _{♣♦❞❡ s❡r ✐♠❡rs♦ ❝♦♠♣❛❝t❛✲} ♠❡♥t❡ ❡♠ Xβ_{✱ ❝♦♠} _{α > β} _≥₀_✳

◆❡st❡ ❝❛s♦✱ t❡♠♦s

eAt

L(Xα_,Xβ₎≤M t α−β

e−δt, β > α≥0, t >0.

P❛r❛ 0< α <1✜①❛❞♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ♣❛r❛❜ó❧✐❝♦

  

u′ =Au+f(u(t))

u(0) =u0 ∈Xα,

✭✶✳✹✮

♦♥❞❡ f : Xα _→_X _{é ❣❧♦❜❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✱ ❣❧♦❜❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r✲} ❡♥❝✐á✈❡❧✳

❈♦♠♦ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♠ ❡ss❛s ❤✐♣ót❡s❡s✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✹✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦✱ ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ t _≥0✱ ❞❛❞❛ ♣♦r

u(t) :=u(t,0, u0) =eAtu0+

Z t

0

eA(t−s)f(u(s))ds

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❚♦❞❛ s♦❧✉çã♦ ❞❡ ✭✶✳✹✮ ♣❡r♠❛♥❡❝❡ ❧✐♠✐t❛❞♦ ♣❛r❛u0 ∈Xα✳ ❉❡ ❢❛t♦✱

ku(t,0, u0)kXα =keAtu₀+

Z t

0

eA(t−s)f(u(s))dskXα

≤ keAtu0kXα +

Z t

0 k

eA(t−s)_k_L(X,Xα₎kf(u(s))k_Xds

≤M e−δt

ku0kXα +

Z t

0

M(t₋s)−α_e−δ(t−s)_Kds

≤M e−δt

ku0kXα +M K

Z t

0

(t₋s)−α_e−δ(t−s)_ds

≤M e−δtku0kXα +

M K δ

Z δt

0

z

δ

−α

e−zdz

≤M e−δt

ku0kXα +M Kδα−1

Z ∞

0

z−α_e−z_dz

≤M e−δt

ku0kXα +M Kδα−1Γ(1−α)≤L,

♦♥❞❡ kf(u(s))_kX ≤K✳

❙❡❥❛T(_·)x:R_→_Xα ❞❡✜♥✐❞♦ ♣♦r

T(t)x=u(t,0, x) =eAt_x₊

Z t

0

eA(t−s)f(T(s)x)ds,

♣❛r❛ t♦❞♦ x _∈ Xα_{✳ ❉❡✜♥✐❞❛ ❞❡st❛ ❢♦r♠❛✱ ❛ ❢❛♠í❧✐❛} _{_T₍_t_); _t _≥ ₀_} _{s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s} ♣r♦♣r✐❡❞❛❞❡s✿

• T(0)x=x✱ ♣❛r❛ t♦❞♦ x∈Xα_✳

• T(t+̺)x=T(t)T(̺)x ♣❛r❛ t♦❞♦ t, ̺≥0✳ • R+_×_Xα _∋₍_{t, x}₎_7−→_T₍_t₎_x _{é ❝♦♥tí♥✉❛✳}

❆ss✐♠✱ ❞✐③❡♠♦s q✉❡ {T(t);t ≥0}é ❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐❣r✉♣♦s ♥ã♦ ❧✐♥❡❛r❡s ❛ss♦❝✐✲ ❛❞♦s ❛♦ ♣r♦❜❧❡♠❛ ✭✶✳✹✮✳

❉❡✜♥✐çã♦ ✶✳✹✹✳ ❙❡❥❛ x∈Xα_{✳ ❉❡✜♥✐♠♦s}

✭❛✮ γ+₍_x_{) =}_{_T₍_t₎_x_;_t_≥₀_} _{❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡} _x_✳

✭❜✮ Hx := {ϕ : (−∞,0] → Xα, ❝♦♥tí♥✉❛ ; ϕ(0) = x, ❡ T(t)ϕ(s) = ϕ(t+s), −∞ ≤

s_{≤ −}t _≤0_}✱ ♣❛r❛ ϕ_∈Hx✱ ❞❡✜♥✐♠♦s γ−

ϕ(x) =∪t≥0ϕ(−t) ❝♦♠♦ ❛ ór❜✐t❛ ♥❡❣❛t✐✈❛ ❞❡ x✳

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✭❝✮ γ(x) = γ+₍_x₎_∪_γ−

ϕ(x) ❛ ór❜✐t❛ ❝♦♠♣❧❡t❛ ❞❡ x✳

❖❜s❡r✈❛çã♦ ✶✳✹✺✳ ❈♦♠♦ Im(T(t)) ♣♦❞❡ ♥ã♦ s❡r t♦❞♦ ♦ Xα_{✱ ❞✐③❡r q✉❡ ❡①✐st❡ ✉♠❛ ór❜✐t❛} ♥❡❣❛t✐✈❛ ♦✉ ❝♦♠♣❧❡t❛ ❞❡x♣♦❞❡ ✐♠♣♦r ❝❡rt❛s r❡str✐çõ❡s ❛x✳ ❆❞✐❛♥t❡ ✈❡r❡♠♦s ✉♠❛ ❝♦♥❞✐çã♦

♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ❡①✐st❛ ✉♠❛ ór❜✐t❛ ❝♦♠♣❧❡t❛ ❞❡ x✳

❉❡✜♥✐çã♦ ✶✳✹✻✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ S ⊂ Xα _{é ✐♥✈❛r✐❛♥t❡ s♦❜} _{_T₍_t_);_t _≥ ₀_} s❡ T(t)S=S✱ ♣❛r❛ t♦❞♦ t _≥0✳ ❉✐③❡♠♦s q✉❡ S é ♣♦s✐t✐✈❛♠❡♥t❡ ✐♥✈❛r✐❛♥t❡ s❡ T(t)S _⊂S✳

▲❡♠❛ ✶✳✹✼✳ S _⊂Xα _{é ✐♥✈❛r✐❛♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ s❡ ♣❛r❛ ❝❛❞❛}_x

∈S✱ ❡①✐st❡ ✉♠❛ ór❜✐t❛

❝♦♠♣❧❡t❛ ❞❡ x✱ q✉❡ ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞❛ ❡♠ ❙✳

❉❡✜♥✐çã♦ ✶✳✹✽✳ ❙❡❥❛ B _⊂Xα_{✳ ❉❡✜♥✐♠♦s✿}

✭❛✮ γ+(B) =∪x_∈Bγ+(x) ❛ ór❜✐t❛ ♣♦s✐t✐✈❛ ❞❡ B✳ ✭❜✮ γ−(B) =_∪x∈B(∪ϕ∈Hxγ

−

ϕ(x)) ❛ ór❜✐t❛ ♥❡❣❛t✐✈❛ ❞❡ B✳ ✭❝✮ γ(B) = _∪x∈Bγ(x) ❛ ór❜✐t❛ ❝♦♠♣❧❡t❛ ❞❡ B✳

✭❝✮ ω(B) =∩s_≥0∪t_≥sT(t)B ♦ ❝♦♥❥✉♥t♦ ω−limite ❞❡ B✳ ✭❞✮ α(B) =∩s≥0∪t≥sγ−(B) ♦ ❝♦♥❥✉♥t♦ α−limite ❞❡ B✳

▲❡♠❛ ✶✳✹✾✳ ❙❡❥❛♠ v _∈ Xα _❡ _B _⊂ _Xα_{✳ ❊♥tã♦✱} _v _∈ _ω₍_B₎ _{s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡♠} s❡q✉ê♥❝✐❛s tn → ∞ ❡ (vn)⊂B t❛❧ q✉❡ T(tn)vn →v✳

▲❡♠❛ ✶✳✺✵✳ ❙❡❥❛♠ v _∈ Xα _❡ _B _⊂ _Xα_{✳ ❊♥tã♦✱} _v _∈ _α₍_B₎ _{s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡♠} s❡q✉ê♥❝✐❛s tn → ∞ ❡ (vn) ⊂ B t❛✐s q✉❡✱ ♣❛r❛ ❝❛❞❛ n ∈ N✱ ❡①✐st❡ ✉♠❛ ór❜✐t❛ ♥❡❣❛t✐✈❛

φn: (−∞,0]→Xα ❞❡ vn ❡ φn(−tn)→v✳

▲❡♠❛ ✶✳✺✶✳ ❙❡❥❛ B _⊂ Xα _{✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦✳ ❊♥tã♦} _γ+₍_B₎ _{é ❧✐♠✐t❛❞♦ ❡} _T₍_t₎_γ+₍_B₎ é ❝♦♠♣❛❝t♦✱ ♣❛r❛ t♦❞♦ t >0✳

❉❡✜♥✐çã♦ ✶✳✺✷✳ ❯♠ s✉❜❝♦♥❥✉♥t♦A❛tr❛✐ ✉♠ ❝♦♥❥✉♥t♦Cs♦❜_{T(t);t _≥0_}s❡dist(T(t)C, A)_→ 0 q✉❛♥❞♦ t→ ∞✳

▲❡♠❛ ✶✳✺✸✳ P❛r❛ t♦❞♦ x _∈ Xα_{✱ ♦ ❝♦♥❥✉♥t♦} _ω₍_{_x_}₎ _{é ♥ã♦ ✈❛③✐♦✱ ❝♦♥❡①♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✲} ✈❛r✐❛♥t❡ ❡ ❛tr❛✐ {x_}✳

(32)

▲❡♠❛ ✶✳✺✹✳ ❙✉♣♦♥❤❛ q✉❡ x_∈Xα _{é t❛❧ q✉❡ ❡①✐st❡ ✉♠❛ ór❜✐t❛ ♥❡❣❛t✐✈❛}_φ _{: (}_−∞_,_0]_→_Xα t❛❧ q✉❡ φ((−∞,0]) é ❝♦♠♣❛❝t♦✳ ❉❡✜♥❛♠♦s

αφ(x) = {v ∈Xα; ❡①✐st❡♠ tn → ∞ t❛❧ q✉❡ φ(−tn)→v}. ❊♥tã♦ αφ(x)6=∅✱ é ❝♦♥✈❡①♦✱ ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡✳

▲❡♠❛ ✶✳✺✺✳ ❙❡ B _⊂ Xα _{é ❧✐♠✐t❛❞♦✱ ❡♥tã♦} _γ₍_B₎ _{é ❧✐♠✐t❛❞♦✳ ❆❧é♠ ❞✐ss♦✱ ❡①✐st❡♠ ✉♠} ✐♥st❛♥t❡ ̺B ❡ ✉♠❛ ❝♦♥st❛♥t❡ N ✭✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ B✮ t❛✐s q✉❡

sup t≥̺B

sup z∈T(t)Bk

z_kα ≤N

❡ t❛♠❜é♠

sup

B_⊂Xα_z_∈sup_ω(B)kzkα ≤N.

▲❡♠❛ ✶✳✺✻✳ ❙❡ B _⊂ Xα _{é ❧✐♠✐t❛❞♦✱ ❡♥tã♦} _ω₍_B₎ _{é ♥ã♦ ✈❛③✐♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡✱ ❡} ❛tr❛✐ B s♦❜ {T(t);t≥0}✳ ❆❧é♠ ❞✐ss♦✱ s❡ B é ❝♦♥❡①♦✱ ❡♥tã♦ ω(B) é ❝♦♥❡①♦✳

❉❡✜♥✐çã♦ ✶✳✺✼✳ ❖ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ A⊂Xα _{é ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ ♣❛r❛} _{_T₍_t_);_t _≥₀_} s❡ A é ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡ ❛tr❛✐ s✉❜❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s ❞❡ Xα _s♦❜ _{_T₍_t_);_t_≥₀_}_✳ ❚❡♦r❡♠❛ ✶✳✺✽✳ ❙❡ N é ❝♦♠♦ ♥♦ ▲❡♠❛ ✶✳✺✺✱ ❡

BN :={u∈Xα; kukα ≤N},

❡♥tã♦ ω(BN) é ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ ♣❛r❛ {T(t);t≥0}✳

✶✳✸✳✶ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ❛tr❛t♦r❡s ♣❛r❛ s✐st❡♠❛s ❣r❛❞✐❡♥t❡s

◆♦ss♦ ♦❜❥❡t✐✈♦ é ❡♥t❡♥❞❡r ❛ ❡str✉t✉r❛ ❞♦s ❛tr❛t♦r❡s ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✹✮✳ ❈♦♠❡ç❛r❡✲ ♠♦s ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❞♦s ❛tr❛t♦r❡s q✉❡ sã♦ ♠❛✐s s✐♠♣❧❡s✱ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ s♦❧✉çõ❡s ❞❡ ❡q✉✐❧í❜r✐♦✳

❉❡✜♥✐çã♦ ✶✳✺✾✳ ❯♠❛ s♦❧✉çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✭✶✳✹✮ é ✉♠❛ s♦❧✉çã♦ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦✱ ♦✉ s❡❥❛✱ é ✉♠❛ s♦❧✉çã♦ ❝♦♥st❛♥t❡ ♥❛ ✈❛r✐á✈❡❧ t✳ ❉❡♥♦t❛♠♦s ♣♦r _E ♦ ❝♦♥❥✉♥t♦

❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✳

◆♦t❡ q✉❡✱ s❡u(t, x) ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦ ❡♥tã♦ du_dt = 0✱ ❧♦❣♦

Au+f(u) = 0. ✭✶✳✺✮

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❉❡✜♥✐çã♦ ✶✳✻✵✳ ❯♠❛ s♦❧✉çã♦ u∗ _{❞❡ ✭✶✳✺✮ é ❤✐♣❡r❜ó❧✐❝❛ s❡} _σ₍_A₊_f′₍_u∗₎₎_{✭♦ ❡s♣❡❝tr♦ ❞❡}

A+f′(u∗)✮ ♥ã♦ ✐♥t❡r❝❡♣t❛ ♦ ❡✐①♦ ✐♠❛❣✐♥ár✐♦✳

❙❡_A˜_:=_A₊_f′₍_u∗₎_{✱ ❡♥tã♦}₋_A˜_{❡stá ❞❡✜♥✐❞♦ ❡♠}_D₍_A₎_{✱ ♣♦✐s s❡}₀_{< α <}₁_❡♥tã♦_D₍_A₎_⊂

D(Aα_{) =} _Xα_{✳ ❈♦♠♦} _f′₍_u∗₎ _{∈ L}₍_Xα_{, X}₎ _{s❡❣✉❡ ❞♦ ❈♦r♦❧ár✐♦ ✭✶✳✹✵✮✱ q✉❡} ₋_A₋_f′₍_u∗₎ _é

s❡t♦r✐❛❧✳ ❊ ❛✐♥❞❛ −A˜t❡♠ r❡s♦❧✈❡♥t❡ ❝♦♠♣❛❝t♦✳

❉❡ ❢❛t♦✱ t♦♠❡B =f′(u∗)✱A é s❡t♦r✐❛❧ ❝♦♠ r❡s♦❧✈❡♥t❡ ❝♦♠♣❛❝t♦ ❡ B é ❢❡❝❤❛❞♦✳

(A+B) = AA−1(A+B) =A(I +A−1B)

⇒(A+B)−1 = (I+A−1B)−1A−1

❈♦♠♦ A t❡♠ r❡s♦❧✈❡♥t❡ ❝♦♠♣❛❝t♦ ❡ 0_∈ ρ(A)✱ s❡❣✉❡ q✉❡ A−1 _{é ❝♦♠♣❛❝t♦✳ ❆ss✐♠✱ ❜❛st❛} ♠♦str❛r q✉❡ (I +A−1_B₎−1 _{é ❧✐♠✐t❛❞♦✱ ♣♦✐s ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❧✐♠✐t❛❞♦ ❝♦♠ ❝♦♠♣❛❝t♦ é} ❝♦♠♣❛❝t♦✳

❖❜s❡r✈❡♠♦s q✉❡I+A−1_B _{é ✐♥✈❡rtí✈❡❧✱ ❥á q✉❡}₍_A₊_B₎_{é ✐♥✈❡rtí✈❡❧✱ ♣♦✐s}₀_∈_ρ₍_A₊_B₎_✱ ❧❡♠❜r❛♥❞♦ q✉❡ u∗ _{é ❤✐♣❡r❜ó❧✐❝♦✳}

P♦rt❛♥t♦✱ (I+A−1_B₎−1 _{é ❧✐♥❡❛r✱ ❡ ✐♥✈❡rtí✈❡❧✱ ❧♦❣♦ ❧✐♠✐t❛❞♦✳} ❆ss✐♠✱ (A+B)−1 _{é ❝♦♠♣❛❝t♦✳}

❖ ❡s♣❡❝tr♦ ❞❡ _A˜ _{é ❝♦♠♣♦st♦ ♣♦r ❛✉t♦✈❛❧♦r❡s ✐s♦❧❛❞♦s ❝♦♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ✜♥✐t❛✱ ✈❡r} ❬✶✷❪✳ ❉❡♥♦t❡♠♦s ♣♦r σ+_{( ˜}_A₎ _{♦s ❡❧❡♠❡♥t♦s ❞♦ ❡s♣❡❝tr♦ ❞❡} _A _✭_σ_{( ˜}_A₎_{✮ q✉❡ t❡♠ ❛ ♣❛rt❡ r❡❛❧} ♣♦s✐t✐✈❛✳ ❖❜s❡r✈❡ q✉❡ ♦ s❡t♦r ❛ss♦❝✐❛❞♦ ❛ _A˜_{✱ ❞❛❞♦ ♣♦r} _Σ_˜_a,ϕ ₌_{_λ_∈ C_;_|_arg₍_λ₋_˜_a₎_|_{< ϕ}_}

❡stá ❝♦♥t✐❞♦ ❡♠ ρ( ˜A)✱ ♣♦✐s A˜ é s❡t♦r✐❛❧✱ ❡ ♣♦rt❛♥t♦ ♦s ❛✉t♦✈❛❧♦r❡s ❡♠ σ+_{( ˜}_A₎ _{❡stã♦✱ ❞❡} ❝❡rt❛ ❢♦r♠❛✱ ❧✐♠✐t❛❞♦s ❡♥tr❡ à ♣❛rt❡ ❛ ❞✐r❡✐t❛ ❞♦ ❡✐①♦ ✐♠❛❣✐♥ár✐♦ ❡ Σ˜a,ϕ✳

❊♠ ❬✽❪ ♣♦❞❡ s❡r ✈✐st♦ q✉❡ s❡−A˜é s❡t♦r✐❛❧✱ ❡♥tã♦ ❡①✐st❡τ _∈Rt❛❧ q✉❡_σ_{( ˜}_A₎é ❞✐s❥✉♥t♦

❞❛ r❡t❛ {λ ∈ C_;_Reλ ₌ _τ_} ❡ ♦ ❝♦♥❥✉♥t♦ _σ₍_eAt˜₎ ♥ã♦ ✐♥t❡r❝❡♣t❛ ♦ ❝♦♥❥✉♥t♦ _{_u _∈ C_; _|_u_| ₌

eτ t_}_{✱ ❡ ❛❧é♠ ❞✐ss♦ ❡①✐st❡♠}_M _≥₁ _❡_{δ >}₀ _{t❛✐s q✉❡}

keAt˜ (I−Q)xkL(X) ≤M e(τ−δ)t, ♣❛r❛ t♦❞♦t ≥0, ❡

keAt˜_Qx

kL(X) ≤M e(τ+δ)t, ♣❛r❛ t♦❞♦ t≤0,

♦♥❞❡ Qé ❛ ♣r♦❥❡çã♦ ❞❡X ♥♦ ❡s♣❛ç♦ ❣❡r❛❞♦ ♣❡❧❛s ❛✉t♦❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❛♦s ❛✉t♦✈❛❧♦r❡s

♣♦s✐t✐✈♦s ❞❡ _A˜_{✳ P♦rt❛♥t♦}

k −A˜φeAt˜ (I ₋Q)x_k_L(X,Xθ₎=k(−A˜)φ−θ(−A˜)θe ˜ At

(I₋Q)x_k_L(X)≤Mφ₋θtφ−θM e(τ−θ)t, ♣❛r❛ θ > φ, t_≥0.