❆▲❉❖ ❍❊◆❘■◗❯❊ ❉❊ ❙❖❯❩❆ ▼❊❉❊■❘❖❙
❈❖▼P❖❘❚❆▼❊◆❚❖ ❆❙❙■◆❚Ó❚■❈❖ ❉❊ ❙■❙❚❊▼❆❙ ❉❊ ❇❘❊❙❙❊ ❈❖▼ ❉■❙❙■P❆➬➹❖ ❋❘■❈❈■❖◆❆▲ ❊ ❉■❙❙■P❆➬➹❖ ◆❆
❋❘❖◆❚❊■❘❆
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❱■➬❖❙❆
Ficha catalográfica preparada pela Biblioteca Central da Universidade Federal de Viçosa - Câmpus Viçosa
T
Medeiros, Aldo Henrique de Souza,
1992-M488c
2016
Comportamento assintótico de sistemas de Bresse com
dissipação friccional e dissipação na fronteira / Aldo Henrique
de Souza Medeiros. – Viçosa, MG, 2016.
viii, 97f. : il. (algumas color.) ; 29 cm.
Orientador: Margareth da Silva Alves.
Dissertação (mestrado) - Universidade Federal de Viçosa.
Referências bibliográficas: f. 95-97.
1. Matemática aplicada. 2. Equações diferenciais parciais.
3. Análise matemática. 4. Teoria assintótica. I. Universidade
Federal de Viçosa. Departamento de Matemática. Programa de
Pós-graduação em Matemática. II. Título.
❆▲❉❖ ❍❊◆❘■◗❯❊ ❉❊ ❙❖❯❩❆ ▼❊❉❊■❘❖❙
❈❖▼P❖❘❚❆▼❊◆❚❖ ❆❙❙■◆❚Ó❚■❈❖ ❉❊ ❙■❙❚❊▼❆❙ ❉❊ ❇❘❊❙❙❊ ❈❖▼ ❉■❙❙■P❆➬➹❖ ❋❘■❈❈■❖◆❆▲ ❊ ❉■❙❙■P❆➬➹❖ ◆❆
❋❘❖◆❚❊■❘❆
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳
❆P❘❖❱❆❉❆✿ ✶✾ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳
❆♥❞❡rs♦♥ ❆❧❜✉q✉❡rq✉❡ ❞❡ ❆r❛✉❥♦
✭❈♦♦r✐❡♥t❛❞♦r✮ ❆♥❞ré ❏✉♥q✉❡✐r❛ ❞❛ ❙✐❧✈❛ ❈♦rrê❛✭❚✐t✉❧❛r ■♥t❡r♥♦✮
❍✉② ❍♦❛♥❣ ◆❣✉②❡♥ ✭❚✐t✉❧❛r ❊①t❡r♥♦✮
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❆❞ã♦ ❡ ❱❡r❛✳
❆ ❢é ❡ ❛ r❛③ã♦ ❝♦♥st✐t✉❡♠ ❝♦♠♦ q✉❡ ❛s ❞✉❛s ❛s❛s ♣❡❧❛s q✉❛✐s ♦ ❡s♣ír✐t♦ ❤✉♠❛♥♦ s❡ ❡❧❡✈❛ à ❝♦♥t❡♠♣❧❛çã♦ ❞❛ ✈❡r❞❛❞❡✳
❏♦ã♦ P❛✉❧♦ ■■
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r s♦✉ ♠✉✐tíss✐♠♦ ❣r❛t♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❆❞ã♦ ❡ ❱❡r❛✱ ♣❡❧♦ ❡①❡♠♣❧♦✱ ❝❛r✐♥❤♦ ❡ ♠♦t✐✈❛çã♦✱ ✈♦❝ês sã♦ s❡♠ ❞ú✈✐❞❛ ♦s ♠❡✉s ♣r✐♠❡✐r♦s ❡ ❡t❡r♥♦s ♣r♦❢❡ss♦r❡s✱ ❣r❛ç❛s ❛ ✈♦❝ês ❝♦♥s❡❣✉✐ ❝❤❡❣❛r ♦♥❞❡ ❡st♦✉✱ ❡ s❡✐ q✉❡ ♣♦ss♦ ❝♦♥t❛r ❝♦♠ ✈♦❝ês ♣❛r❛ ❝♦♥s❡❣✉✐r ✐r ♠❛✐s ❛❧é♠✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ▼❛r❣❛r❡t❤✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❛♣r❡♥❞✐③❛❞♦ ✈❛❧✐♦s♦✱ ♣❡❧❛s s✉❛s ❝♦rr❡çõ❡s ❡ ✐♥❝❡♥t✐✈♦✳ ❊♥✜♠ ♣❡❧❛ ♣❡ss♦❛ ♠❛r❛✈✐❧❤♦s❛ q✉❡ é✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s ❡ ♠❡✉s ❛✈ós✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❢♦rç❛ ❡ ❛❧❡❣r✐❛✳ ❆❣r❛❞❡ç♦ ❞❡ ✉♠❛ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ q✉❡ s❡♠♣r❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦✱ ❞❛♥❞♦ ❛♣♦✐♦✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ❡ ❞❡ ❡st✉❞♦s✳ ❱♦❝ês ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ✈ã♦ ❝♦♥t✐♥✉❛r ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❝♦♠ ❝❡rt❡③❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉▼❆✲❯❋❱✱ ♣♦r ❝♦❧❛❜♦r❛r❡♠ ❝♦♠ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ♣❡❧♦s ❡✜❝✐❡♥t❡s s❡r✈✐ç♦s ♣r❡st❛❞♦s✳
❆❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✈✐✐
❆❜str❛❝t ✈✐✐✐
■♥tr♦❞✉çã♦ ✐①
✶ Pr❡❧✐♠✐♥❛r❡s ✶
✶✳✶ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❊s♣❛ç♦s ❢✉♥❝✐♦♥❛✐s ❡ ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❙❡♠✐❣r✉♣♦s ❞❡ ❝❧❛ss❡ C0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✶✳✹ ❊s♣❛ç♦s Lp(0, T;X) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✺ ❉✐str✐❜✉✐çõ❡s ✈❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✻ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❙✐st❡♠❛ ❞❡ ❇r❡ss❡ ✷✵
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✶ ❈♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞♦ t✐♣♦ ❉✐r✐❝❤❧❡t✲❉✐r✐❝❤❧❡t✲❉✐r✐❝❤❧❡t ✷✷ ✷✳✷✳✷ ❈♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞♦ t✐♣♦ ❉✐r✐❝❤❧❡t✲◆❡✉♠❛♥♥✲◆❡✉♠❛♥♥ ✸✷
✸ ❊st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❞♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ✸✻ ✸✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❋❛❧t❛ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹ ❊st❛❜✐❧✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❞♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ✻✸ ✹✳✶ ❊st❛❜✐❧✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✺ ❊st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ♣❛r❛ ✉♠ s✐st❡♠❛ ❇r❡ss❡ ❝♦♠ ❝♦♥tr♦❧❡
♥❛ ❢r♦♥t❡✐r❛ ✼✵
✺✳✶ ❊♥❡r❣✐❛ ❛ss♦❝✐❛❞❛ ❛♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✺✳✷ ❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✺✳✸ ❊st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✽✽
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✽✾
❘❡s✉♠♦
▼❊❉❊■❘❖❙✱❆✳ ❍✳ ❙✳✱ ▼✳❙❝✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳ ❈♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ s✐st❡♠❛s ❞❡ ❇r❡ss❡ ❝♦♠ ❞✐ss✐♣❛çã♦ ❢r✐❝❝✐♦♥❛❧ ❡ ❞✐ss✐♣❛çã♦ ♥❛ ❢r♦♥t❡✐r❛✳ ❖r✐❡♥t❛❞♦r❛✿ ▼❛r❣❛r❡t❤ ❞❛ ❙✐❧✈❛ ❆❧✈❡s✳ ❈♦♦r✐❡♥t❛❞♦r✿ ❆♥❞❡rs♦♥ ▲✉✐③ ❞❡ ❆❧❜✉q✉❡rq✉❡ ❆r❛✉❥♦✳
◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ s✐st❡♠❛s ❞✐ss✐♣❛t✐✈♦s ❝♦♠ ❛♣❧✐❝❛çõ❡s à ♠♦❞❡❧❛❣❡♠ ❞❡ ♠❛t❡r✐❛✐s ❡❧ást✐❝♦s✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❡st✉❞❛✲ s❡ ❛ ❡①✐stê♥❝✐❛✱ ✉♥✐❝✐❞❛❞❡ ❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ ❞♦✐s s✐st❡♠❛s t✐♣♦✲ ❇r❡ss❡✱ ✉♠ ❝♦♠ ❞✐ss✐♣❛çã♦ ❞❛❞❛ ♣❡❧♦ ❛tr✐t♦ ❡ ♦✉tr♦ ❝♦♠ ❡❢❡✐t♦s ❞✐ss✐♣❛t✐✈♦s ♥❛ ❢r♦♥t❡✐r❛✳ ❖ ♦❜❥❡t✐✈♦ é ❡st❛❜❡❧❡❝❡r ❝♦♥❞✐çõ❡s q✉❡ ❛ss❡❣✉r❡♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❛ ♣♦❧✐♥♦♠✐❛❧ ❞♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦✳ P❛r❛ ✐ss♦✱ ✉s❛r❡♠♦s ❛ ❛❜♦r❞❛❣❡♠ ❞❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❞❡ ❝❧❛ss❡ C0✱
♣r♦♣r✐❡❞❛❞❡s ❞♦ ❝♦♥❥✉♥t♦ r❡s♦❧✈❡♥t❡ ❡ ❞♦ ♦♣❡r❛❞♦r r❡s♦❧✈❡♥t❡ ❞♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠ C0✲s❡♠✐❣r✉♣♦ ❡ té❝♥✐❝❛s ♠✉❧t✐♣❧✐❝❛t✐✈❛s✳
❆❜str❛❝t
▼❊❉❊■❘❖❙ ❆✳ ❍✳ ❙✳✱ ▼✳❙❝✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❋❡❜r✉❛r②✱ ✷✵✶✻✳ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ❇r❡ss❡ s②st❡♠s ✇✐t❤ ❢r✐❝t✐♦♥❛❧ ❞✐ss✐♣❛t✐♦♥ ❛♥❞ ❞✐ss✐♣❛t✐♦♥ ♦♥ t❤❡ ❜♦r❞❡r✳ ❆❞✐✈✐s❡r✿ ▼❛r❣❛r❡t❤ ❞❛ ❙✐❧✈❛ ❆❧✈❡s✳ ❈♦✲❆❞✈✐s❡r✿ ❆♥❞❡rs♦♥ ▲✉✐③ ❞❡ ❆❧❜✉q✉❡rq✉❡ ❆r❛✉❥♦✳
■♥ t❤✐s ♣❛♣❡r ✇❡ st✉❞② t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ❞✐ss✐♣❛t✐✈❡ s②st❡♠s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ♠♦❞❡❧✐♥❣ ♦❢ ❡❧❛st✐❝ ♠❛t❡r✐❛❧s✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✐t ✐s st✉❞✐❡❞ ❡①✐st❡♥❝❡✱ ✉♥✐q✉❡♥❡ss ❛♥❞ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t✇♦ t②♣❡✲❇r❡ss❡ s②st❡♠s✱ ✇✐t❤ ❞✐ss✐♣❛t✐♦♥ ❣✐✈❡♥ ❜② ❢r✐❝t✐♦♥ ❛♥❞ ♦♥❡ ✇✐t❤ ❞✐ss✐♣❛t✐✈❡ ❡✛❡❝ts ❛t t❤❡ ❜♦r❞❡r✳ ❚❤❡ ❣♦❛❧ ✐s ❡st❛❜❧✐s❤ ❝♦♥❞✐t✐♦♥s t❤❛t ❡♥s✉r❡ t❤❡ ❡①♣♦♥❡♥t✐❛❧ st❛❜✐❧✐t② ❛♥❞ ♣♦❧②♥♦♠✐❛❧ t❤❡ s❡♠✐❣r♦✉♣ ❛ss♦❝✐❛t❡❞✳ ❋♦r t❤✐s✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ s❡♠✐❣r♦✉♣ t❤❡♦r② ♦♣❡r❛t♦rs ❧✐♥❡❛r ❝❧❛ss C0 ✱ r❡s♦❧✈❡♥t s❡t ♣r♦♣❡rt② ❛♥❞ r❡s♦❧✈✐♥❣ ♦♣❡r❛t♦r
✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ ❛ C0 ✲s❡♠✐❣r✉♣♦ ❛♥❞ t❡❝❤♥✐❝❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❡✳
■♥tr♦❞✉çã♦
❖ ❡st✉❞♦ ❞♦s ♠♦❞❡❧♦s ♣❛r❛ ✈✐❣❛s ❝♦♠ ❝❛r❣❛s ❛t✉❛♥t❡s ✐♥t❡r♥❛ ♦✉ ❡①t❡r♥❛♠❡♥t❡ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛❧t❛ ❡♥❣❡♥❤❛r✐❛✱ ❥á q✉❡ ❛ ✈✐❣❛ é ✉♠ ♠♦❞❡❧♦ ❞❡ ❡str✉t✉r❛ ✢❡①í✈❡❧ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ❡♠ ♣r♦❥❡t♦s ❞❡ ❡str✉t✉r❛ ❡ ♠❡❝â♥✐❝♦s✱ t❛✐s ❝♦♠♦ ♣r♦❥❡t♦s ❞❡ ♣♦♥t❡s✱ ❡❞✐❢í❝✐♦s✱ ❛✈✐õ❡s✱ ♣❧❛t❛❢♦r♠❛s ❞❡ ♣❡tró❧❡♦✱ ❞❡♥tr❡ ♦✉tr♦s ❬✶❪✳ ◆❛s ú❧t✐♠❛s ❞é❝❛❞❛s✱ ✐♠♣♦rt❛♥t❡s ♠❡❝❛♥✐s♠♦s ❞✐ss✐♣❛t✐✈♦s ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❡st❛❜✐❧✐③❛r ♠♦❞❡r♥❛s ❡str✉t✉r❛s ❡♠ ❡♥❣❡♥❤❛r✐❛ q✉❛♥❞♦ s✉❜♠❡t✐❞❛s ❛ ♦s❝✐❧❛çõ❡s ♥ã♦ ❞❡s❡❥á✈❡✐s✳ ❉❡ ♠♦❞♦ q✉❡ ❝♦♥❤❡❝❡r ❡ ❡♥t❡♥❞❡r ♦s ❡❢❡✐t♦s ❞❡ ❛❧❣✉♥s ♠❡❝❛♥✐s♠♦s ❞✐ss✐♣❛t✐✈♦s s❡ t♦r♥❛ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❝♦♥tr♦❧❛r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ❣r❛♥❞❡s ❡str✉t✉r❛s ❝✉❥❛s ♦s❝✐❧❛çõ❡s sã♦ ♠♦❞❡❧❛❞❛s ♣♦r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s q✉❡ ❡✈♦❧✉✐ ❝♦♠ ♦ t❡♠♣♦✳
❖ ❡st✉❞♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ s✐st❡♠❛s ❞✐ss✐♣❛t✐✈♦s é ✉♠ r❛♠♦ ❢ért✐❧ ♣❛r❛ ❛ ♣❡sq✉✐s❛ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s✳ P❛r❛ s❡ ♦❜t❡r ❡ss❡ ❝♦♠♣♦rt❛♠❡♥t♦✱ ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s ❛♥❛❧ít✐❝♦s tê♠ s✐❞♦ ✉t✐❧✐③❛❞♦s ♣♦r ✈ár✐♦s ❛✉t♦r❡s✱ s❡♠♣r❡ ❛❞❡q✉❛❞♦s ❛♦s ♣r♦❜❧❡♠❛s ❡♠ q✉❡stã♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✉s❛r❡♠♦s ♦ ♠ét♦❞♦ q✉❡ ❡①♣❧♦r❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ❞✐ss✐♣❛t✐✈♦ ✭✈❡r ❬✷✻❪✱ ❬✷✼❪✮✳
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ✐♥✈❡st✐❣❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❛s s♦❧✉çõ❡s ❞❡ s✐st❡♠❛s ❞❡ ❇r❡ss❡✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r♦❜❧❡♠❛ ❞♦ ❛r❝♦ ❝✐r❝✉❧❛r ✭♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ✈❡❥❛ ▲❛❣♥❡s❡ ❡ ♦✉tr♦s ❬✶✺❪✮✳ ❈♦♥s✐❞❡r❛♠♦s ✉♠ ❛r❝♦ ❝✐r❝✉❧❛r ❞❡ r❛✐♦ R ❡ ❝♦♠♣r✐♠❡♥t♦ L ❡♠ s✉❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ❝♦♥st✐t✉í❞♦ ❞❡ ♠❛t❡r✐❛❧
❧✐♥❡❛r✱ ✐s♦tró♣✐❝♦ ❡ ❧✐♥❡❛r♠❡♥t❡ ❡❧ást✐❝♦❀ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ✈✐❣❛ é ♠♦❞❡❧❛❞♦ ♣❡❧❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦
ρ1ϕtt−Sx−lN = F1 ❡♠ (0, L)×(0,∞), ρ2ψtt−Mx+S = F2 ❡♠ (0, L)×(0,∞), ρ1wtt−Nx+lS = F3 ❡♠ (0, L)×(0,∞),
✭✶✮
♦♥❞❡ t ❞❡♥♦t❛ ❛ ✈❛r✐á✈❡❧ t❡♠♣♦r❛❧ ❡ x ❛ ✈❛r✐á✈❡❧ ❡s♣❛❝✐❛❧✳ ❆s ❢✉♥çõ❡s ω =
ω(x, t), ϕ = ϕ(x, t) ❡ ψ = ψ(x, t) sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ❞❡s❧♦❝❛♠❡♥t♦s
❧♦♥❣✐t✉❞✐♥❛✐s✱ ✈❡rt✐❝❛✐s ❡ ❞♦ â♥❣✉❧♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✳ ❆q✉✐✱
N =k0(ωx−lϕ), S =k(ϕx+ψ+lω) ❛♥❞ M =bψ ✭✷✮ sã♦ ❛s r❡❧❛çõ❡s t❡♥sã♦✲❞❡❢♦r♠❛çã♦ ♣❛r❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡❧ást✐❝♦✳ ❆❧é♠ ❞✐ss♦✱
ρ1 = ρA, ρ2 = ρI✱ k = k′GA✱ k0 = E A✱ b = E I✱ l = R−1✱ ρ é ❛ ❞❡♥s✐❞❛❞❡ ❞♦
♠❛t❡r✐❛❧✱ E é ♦ ♠ó❞✉❧♦ ❞❡ ❡❧❛st✐❝✐❞❛❞❡✱ G é ♦ ♠♦❞✉❧♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✱ k′ é ♦
①
❢❛t♦r ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✱A é ❛ ár❡❛ ❞❛ s❡çã♦ tr❛♥s✈❡rs❛❧✱ I é ♦ ♠♦♠❡♥t♦ ❞❡ ✐♥ér❝✐❛
❞❛ s❡çã♦ tr❛♥✈❡rs❛❧ ❡ R é ♦ r❛✐♦ ❞❡ ❝✉r✈❛t✉r❛ ✭✜❣✉r❛ ✶✮✳ ❋✐♥❛❧♠❡♥t❡✱ F1✱F2 ❡F3
sã♦ ❢♦rç❛s ❡①t❡r♥❛s✳
❋✐❣✉r❛ ✶✿ P❡❞r♦ ❘♦❜❡rt♦ ❞❡ ❧✐♠❛✱ ❙✐st❡♠❛ ❞❡ ❇r❡ss❡ t❡r♠♦❡❧ást✐❝♦ ♥ã♦ ❧✐♥❡❛r✿ ❊①✐stê♥❝✐❛ ❣❧♦❜❛❧ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧✱ P●▼❆❈ ✷✵✶✺✳
❙✉❜st✐t✉✐♥❞♦ ✭✷✮ ❡♠ ✭✶✮ ❡♥❝♦♥tr❛♠♦s ♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ❝❧áss✐❝♦
ρ1ϕtt−k(ϕx+ψ+lw)x−lk0(wx−lϕ) = F1 ❡♠ (0, L)×(0,∞),
ρ2ψtt−bψxx+k(ϕx+ψ+lω) = F2 ❡♠ (0, L)×(0,∞),
ρ1wtt−k0(ωx−lϕ)x+lk(ϕx+ψ+lω) = F3 ❡♠ (0, L)×(0,∞). ✭✸✮
❈♦♥s✐❞❡r❛♠♦s ❡st❡ ♠♦❞❡❧♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛s
ϕ(0, t) =ϕ(L, t) =ψ(0, t) =ψ(L, t) =ω(0, t) = ω(L, t) = 0, t∈(0,∞), ✭✹✮
♦✉
ϕ(0, t) = ϕ(L, t) = ψx(0, t) =ψx(L, t) =ωx(0, t) =ωx(L, t) = 0, t∈(0,∞), ✭✺✮ ❡ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s
ϕ(·,0) =ϕ0, ϕt(·,0) =ϕ1, ψ(·,0) =ψ0, ψt(·,0) =ψ1, ω(·,0) =ω0, ωt(·,0) =ω1
✭✻✮ ❙❡❥❛ (ϕ, ψ, ω) ✉♠❛ s♦❧✉çã♦ r❡❣✉❧❛r ❞♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ✭✸✮✲✭✻✮✱ ♣❛r❛
F1 =F2 =F3 = 0✳ ❆ ❡♥❡r❣✐❛ t♦t❛❧ ❛ss♦❝✐❛❞❛ é ❞❛❞❛ ♣♦r
E(t) = 1 2
Z L
0
ρ1|ϕt|2+ρ2|ψt|2+ρ1|ωt|2
dx
+
Z L
0
k|ϕx+ψ+lω|2+b|ψx|2 +k0|ωx−lϕ|2
dx.
❈♦♠ ❛❧❣✉♥s ❝á❧❝✉❧♦s s✐♠♣❧❡s ✈❡♠♦s q✉❡
d
①✐
■st♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ❡♥❡r❣✐❛ s❡ ♠❛♥té♠ ❝♦♥st❛♥t❡ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ✭♦ s✐st❡♠❛ ✭✶✮ ✲ ✭✻✮ é ❝♦♥s❡r✈❛t✐✈♦✮ ❡✱ ♣♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ♥ã♦ ❞❡❝❛✐✳ ❊♥tr❡t❛♥t♦✱ ❛❝r❡s❝❡♥t❛♥❞♦ t❡r♠♦s ❞✐ss✐♣❛t✐✈♦s ❛♦ s✐st❡♠❛✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❛❧❣✉♠ t✐♣♦ ❞❡ ❞❡❝❛✐♠❡♥t♦ ❞❛s s♦❧✉çõ❡s✳ ❯♠❛ q✉❡stã♦ ✐♠♣♦rt❛♥t❡ ♥♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ é ❡♥❝♦♥tr❛r ✉♠❛ ❞✐ss✐♣❛çã♦ ♠í♥✐♠❛✱ ❛tr❛✈és ❞❛ q✉❛❧ ❛s s✉❛s s♦❧✉çõ❡s ❞❡❝❛❡♠ ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ③❡r♦ ❝♦♠ ♦ t❡♠♣♦✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ♣❡sq✉✐s❛❞♦r❡s ❞❛ ár❡❛ tê♠ s❡ ❞❡❞✐❝❛❞♦ ❛ ✐♥✈❡st✐❣❛r ❡ss❡s ♣r♦❜❧❡♠❛s✳
❉✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ✭❞❛♠♣✐♥❣✮ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♥♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ❡ ✈ár✐♦s r❡s✉❧t❛❞♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❡ ♣♦❧✐♥♦♠✐❛❧ ❢♦r❛♠ ♦❜t✐❞♦s✳ ◆❛ ♠❛✐♦r✐❛ ❞❡ss❡s tr❛❜❛❧❤♦s✱ q✉❛♥❞♦ s♦♠❡♥t❡ ✉♠❛ ❞✐ss✐♣❛çã♦ é ❡❢❡t✐✈❛ s♦❜r❡ ♦ s✐st❡♠❛✱ ❛ ❝♦♥❝❧✉sã♦ é q✉❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ é ✈á❧✐❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s ✈❡❧♦❝✐❞❛❞❡s ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s sã♦ ❛s ♠❡s♠❛s✱ ✐st♦ é✱
k ρ1
= b
ρ2 ❡
k=k0. ✭✼✮
❆ s❡❣✉✐r✱ r❡❝♦r❞❛♠♦s ❛❧❣✉♥s ❞❡ss❡s tr❛❜❛❧❤♦s✳
❊♠ ❋❛t♦r✐ ❡ ❘✐✈❡r❛ ❬✶✻❪✱ ♦s ❛✉t♦r❡s ❛♥❛❧✐s❛r❛♠ ♦ s✐st❡♠❛ t❡r♠♦❡❧ást✐❝♦ ❞❡ ❇r❡ss❡ ❝♦♠ F1 =F2 =F3 = 0 ❡ ❧❡✐s ❝♦♥st✐t✉t✐✈❛s ❞❛❞❛s ♣♦r
S =κ(ϕx+ψ+lw), M =bψx−δθ, N =κ0(wx−lϕ). ✭✽✮ ❊♠ ❛❞✐çã♦✱ ❡❧❡s ✉s❛r❛♠ ❛ ▲❡✐ ❞❡ ❋♦✉r✐❡r ♣❛r❛ ♦ ✢✉①♦ ❞♦ ❝❛❧♦r
q=κθx (q ✢✉①♦✱ θ ❞✐❢❡r❡♥ç❛ ❞❡ t❡♠♣❡r❛t✉r❛) ❡ ❛ ❡q✉❛çã♦ ❞♦ ❜❛❧❛♥ç♦ ❞❛ ❡♥❡r❣✐❛ é ❞❛❞❛ ♣♦r
cθt−kθxx+mψxt= 0. ✭✾✮
❙✉❜st✐t✉✐♥❞♦ ✭✽✮ ❡♠ ✭✶✮ t❡♠♦s ♦ s✐st❡♠❛
ρ1ϕtt−k(ϕx+ψ+lw)x−lk0(wx−lϕ) = 0 ❡♠ (0, L)×(0,∞),
ρ2ψtt−bψxx+k(ϕx+ψ+lω) +γψt = 0 ❡♠ (0, L)×(0,∞),
ρ1wtt−k0(ωx−lϕ)x+lk(ϕx+ψ +lω) = 0 ❡♠ (0, L)×(0,∞),
cθt−kθxx+mψxt = 0 ❡♠ (0, L)×(0,∞), ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✭✹✮ ❡
θ(0,·) =θ(0,·) = 0, t ≥0
♦✉ ✭✺✮ ❡
θx(0,·) = θx(0,·) = 0, t ≥0.
①✐✐
♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛ ❡ ❞❛ r❡❣✉❧❛r✐❞❛❞❡ ❞♦s ❞❛❞♦s ✐♥✐❝✐❛✐s✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡s ✐♥tr♦❞✉③✐r❛♠ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ✉♠ s❡♠✐❣r✉♣♦ ❞✐ss✐♣❛t✐✈♦ ❞❡❝❛✐❛ ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡✳ ❊st❡ r❡s✉❧t❛❞♦ ♣❡r♠✐t✐✉✲❧❤❡s ♠♦str❛r ❛❧❣✉♠❛ ♦t✐♠❛❧✐❞❛❞❡ ♣❛r❛ ❛ t❛①❛ ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧✳ ❊st❡ tr❛❜❛❧❤♦ ♠❡❧❤♦r❛ ♦s r❡s✉❧t❛❞♦s ❞❡ ▲✐✉ ❡ ❘❛♦ ❬✶✾❪✳
❖ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ✭✶✮ ❝♦♠ ❞✐ss✐♣❛çã♦ ❞❛❞❛ ♣❡❧♦ ❛tr✐t♦ ❛❣✐♥❞♦ s♦❜r❡ ♦ â♥❣✉❧♦ ❞❡ ❝✐s❛❧❤❛♠❡♥t♦✱ ✐st♦ é✱ q✉❛♥❞♦
S =κ(ϕx+ψ+lw), M =bψx, N =κ0(wx−lϕ),
F1 =F3 = 0 ❡ F2 =−γψt, ✭✶✵✮
γ > 0✱ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ❡♠ ❆❧❛❜❛✉✲❇♦✉ss♦✉✐r❛ ❬✶✵❪✳ ❙✉❜st✐t✉✐♥❞♦ ✭✶✵✮ ❡♠ ✭✶✮
❡♥❝♦♥tr❛♠♦s ♦ s✐st❡♠❛ ❇r❡ss❡ ❝♦♥s✐❞❡r❛❞♦
ρ1ϕtt−k(ϕx+ψ+lw)x−lk0(wx−lϕ) = 0,
ρ2ψtt−bψxx+k(ϕx+ψ+lω) +γψt = 0,
ρ1wtt−k0(ωx−lϕ)x+lk(ϕx+ψ+lω) = 0,
✭✶✶✮
(x, t) ∈ (0, L)×(0,∞)✱ s✉❥❡✐t♦ às ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✭✹✮ ♦✉ ✭✺✮ ❡ ❝♦♥❞✐çõ❡s
✐♥✐❝✐❛✐s ✭✻✮✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s ♠♦str❛r❛♠ q✉❡ ❡ss❡ ♠❡❝❛♥✐s♠♦ ❞✐ss✐♣❛t✐✈♦ é s✉✜❝✐❡♥t❡♠❡♥t❡ ❢♦rt❡ ♣❛r❛ ❡st❛❜✐❧✐③❛r ♦ s✐st❡♠❛ ❞❡s❞❡ q✉❡ ❛ ❝♦♥❞✐çã♦ ✭✼✮ s❡❥❛ ❛ss❡❣✉r❛❞❛✳ ◗✉❛♥❞♦ ✭✼✮ ♥ã♦ é ✈á❧✐❞❛✱ ❡❧❡s ♠♦str❛r❛♠ ❛ ❢❛❧t❛ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❞♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✭✺✮ ❞♦ t✐♣♦ ❉✐r✐❝❧❡t✲◆❡✉♠❛♥♥✲◆❡✉♠❛♥♥❀ s✉♣♦♥❞♦ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✭✹✮ ♦✉ ✭✺✮ ❡♥❝♦♥tr❛r❛♠ ❛ t❛①❛ ❞❡ ❞❡❝❛✐♠❡♥t♦ t−6+ǫ ♣❛r❛ ǫ ♣❡q✉❡♥♦ ❡ q✉❡ ❛ t❛①❛ ♣♦❞❡ s❡r ♠❡❧❤♦r❛❞❛ t♦♠❛♥❞♦ ❞❛❞♦s ✐♥✐❝✐❛✐s ♠❛✐s r❡❣✉❧❛r❡s✳ ❖ tr❛❜❛❧❤♦ ❞❡ ❋❛t♦r✐ ❬✾❪ ♠❡❧❤♦r❛ ♦ r❡s✉❧t❛❞♦ ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧ ❡♥❝♦♥tr❛❞♦ ❡♠ ❆❧❛❜❛✉✲❇♦✉ss♦✉✐r❛ ❬✶✵❪✳ ❊♠ ❬✾❪✱ ♦s ❛✉t♦r❡s ♣r♦✈❛r❛♠ q✉❡ ♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ❞❡❝❛✐ ❝♦♠ t❛①❛t−1/2 ♦✉t−1/4✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s✳ ❊♠ ✉♠ ❝❛s♦
♣❛rt✐❝✉❧❛r✱ ❡❧❡s ♠♦str❛r❛♠ q✉❡ ❛ t❛①❛ ❞❡ ❞❡❝❛✐♠❡♥t♦ ♦❜t✐❞❛ é ót✐♠❛✳
◆♦✉♥ ❡ ❲❡❤❜❡ ❬✷✸❪ ❡st❡♥❞❡r❛♠ ♦ tr❛❜❛❧❤♦ ❞❡ ❆❧❛❜❛✉✲❇♦✉ss♦✉✐r❛ ❬✶✵❪ ❡ ❝♦♥s✐❞❡r❛r❛♠ ♦ ✐♠♣♦rt❛♥t❡ ❝❛s♦ ❡♠ q✉❡ ❛ ❞✐ss✐♣❛çã♦ é ❧♦❝❛❧♠❡♥t❡ ❞✐str✐❜✉í❞❛✱ ♦✉ s❡❥❛✱ γ ∈ L∞(0, L)✱ γ(x) ≤ 0 q✳t✳♣✱ ❡ ❡❧❡s ❛ss✉♠✐r❛♠ q✉❡ ❡①✐st❡♠ a, b t❛✐s
q✉❡ 0 ≤ a < b ≤ L ❡ γ ≥ γ0 > 0 ❡♠ ]a, b[✳ ❖s ❛✉t♦r❡s ❡♥❝♦♥tr❛r❛♠ ♣❛r❛
❛ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ r❡s✉❧t❛❞♦s ❛♥á❧♦❣♦s ❛♦s ❞❡ ❬✶✵❪✱ ♠❛s ♠❡❧❤♦r❛r❛♠ ❛s t❛①❛s ❞❡ ❞❡❝❛✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧✳ ❊♠ ❬✷✽❪✱ ❙♦r✐❛♥♦ ❡ ♦✉tr♦s ♣r♦✈❛♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠❡s♠♦ s✐st❡♠❛ ♥♦ ❝❛s♦ ❡♠ q✉❡ γ ♣♦❞❡ ♠✉❞❛r ❞❡ s✐♥❛❧ ✭♦ s✐st❡♠❛ ♣♦❞❡ s❡r
♥ã♦✲❞✐ss✐♣❛t✐✈♦✮ ❡ γ = 1
L
RL
0 γ(s)ds > 0✱ ❞❡s❞❡ q✉❡ ❛ ❝♦♥❞✐çã♦ ✭✼✮ s❡❥❛ ✈❡r✐✜❝❛❞❛
❡ ✐♠♣♦♥❞♦ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s s♦❜r❡γ ❡ γ✳
❙❛♥t♦s ❡ ♦✉tr♦s ❬✶✼❪ ❝♦♥s✐❞❡r❛r❛♠ ♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ❝♦♠ ❤✐stór✐❛
ρ1ϕtt−k(ϕx+ψ+lw)x−lk0(wx−lϕ) = 0,
ρ2ψtt−bψxx+
Z ∞
0
g(s)ψxx(t−s)ds+k(ϕx+ψ+lω) = 0,
①✐✐✐
♣❛r❛ (x, t)∈(0, L)×(0,∞)✱ ❝♦♠ ♦ ♥ú❝❧❡♦ g s❛t✐s❢❛③❡♥❞♦
g ∈C1(R+)∩L1(R+), g(t)>0, q0, q1 >0 : −q0g(t)≤g′(t)≤ −q1g(t),∀t≥0.
❊❧❡s t❛♠❜é♠ ♦❜t✐✈❡r❛♠ ✭✼✮ ❝♦♠♦ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❞♦ s❡♠✐❣r✉♣♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ❊♠ ❝❛s♦ ❝♦♥trár✐♦✱ ❡❧❡s ♣r♦✈❛r❛♠ q✉❡ ♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ é ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡ ❡stá✈❡❧ ❝♦♠ t❛①❛ ót✐♠❛ ❞❡ ❞❡❝❛✐♠❡♥t♦✳
◆❛❥❞✐ ❡ ❲❡❤❜❡ ❬✷✷❪ ❝♦♥s✐❞❡r❛♠ ♦ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ♥♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❞✐ss✐♣❛çã♦ t❡r♠❛❧ é ❧♦❝❛❧♠❡♥t❡ ❞✐str✐❜✉í❞❛ ♥♦ â♥❣✉❧♦ ❞❡ ✜❧❛♠❡♥t♦✱ ❞❛❞♦ ♣♦r
ρ1ϕtt−k(ϕx+ψ+lw)x−lk0(wx−lϕ) = 0 ❡♠ (0, L)×(0,∞),
ρ2ψtt−bψxx+k(ϕx+ψ+lω) +α(x)θx = 0 ❡♠ (0, L)×(0,∞),
ρ1wtt−k0(ωx−lϕ)x+lk(ϕx+ψ+lω) = 0 ❡♠ (0, L)×(0,∞),
ρ3θt−θxx+T0(αψx)t = 0 ❡♠ (0, L)×(0,∞).
❖s ❛✉t♦r❡s ❣❡♥❡r❛❧✐③❛r❛♠ ♦s r❡s✉❧t❛❞♦s ❞❡ ❋❛t♦r✐ ❡ ❘✐✈❡r❛ ❬✶✻❪✱ ♣♦✐s ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞❛♠♣✐♥❣ α ♥ã♦ é ❝♦♥st❛♥t❡✱ ♠❛s ❡❧❡ é ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❡♠ W2,∞(0, L) ❡
❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛ ♥♦ ✐♥t❡r✈❛❧♦ ]a, b[⊂]0, L[ ✭♦s ❝❛s♦s a = 0 ♦✉ b =L ♥ã♦ sã♦
❡①❝❧✉í❞♦s✮ ❡ t❛♠❜é♠ ♠❡❧❤♦r❛r❛♠ ❛ t❛①❛ ❞❡ ❞❡❝❛✐♠❡♥t♦✳
❘❡❝❡♥t❡♠❡♥t❡✱ ●❛r❜✉❣✐♦ ❬✶✷❪ ❡♠ s✉❛ t❡s❡ ❞❡ ❉♦✉t♦r❛❞♦ ❡st✉❞♦✉ ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❛❧✐t❛t✐✈❛s ❡ ❛ss✐♥tót✐❝❛s ♣❛r❛ s✐st❡♠❛s t❡r♠♦❡❧ást✐❝♦s ❞❡ ❇r❡ss❡✱ ♦♥❞❡ ♦ ✢✉①♦ ❞❡ ❝❛❧♦r s❡❣✉❡ ❛ ❧❡✐ ❞❡ ❋♦✉r✐❡r✱ ❛ ❧❡✐ ❞❡ ❈❛tt❛♥❡♦ ❡ ❛ t❡r♠♦❡❧❛st✐❝✐❞❛❞❡ ❞♦ t✐♣♦ ■■■✳ ◆♦ ❈❛♣ít✉❧♦ ✷ ❞❡ s❡✉ tr❛❜❛❧❤♦✱ ❆ ♠♦❞❡❧❛❣❡♠ ❞❛ ✈✐❣❛ ❞❡ ❇r❡ss❡✱ ❡❧❡ ❛♣r❡s❡♥t❛ ❛ ❞❡❞✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ♠♦✈✐♠❡♥t♦ ✭✶✮✳ ◆ã♦ ❛♣r❡s❡♥t❛♠♦s ♥❡st❛ ❞✐ss❡rt❛çã♦ ♦ ❡st✉❞♦ ❞❡ss❛ ♠♦❞❡❧❛❣❡♠✱ ♠❛s ♥♦s r❡❢❡r✐♠♦s ❛♦ tr❛❜❛❧❤♦ ❬✶✷❪ ♣❛r❛ ♦s ✐♥t❡r❡ss❛❞♦s✳
P❛r❛ ♠♦❞❡❧♦s ❞❡ s✐st❡♠❛ ❞❡ ❇r❡ss❡ ❝♦♠ ❞✐ss✐♣❛çõ❡s ♥❛ ❢r♦♥t❡✐r❛ só ❝♦♥❤❡❝❡♠♦s ♦ tr❛❜❛❧❤♦ ❞❡ ❆❧✈❡s ❡t ❛❧❧ ❬✷✵❪ ✱ ♥♦ q✉❛❧ ♦s ❛✉t♦r❡s ♠♦str❛r❛♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❞♦ s✐st❡♠❛
ρ1ϕtt−k(ϕx+ψ+lw)x−lk0(wx−lϕ) +kθx = 0,
ρ2ψtt−bψxx+k(ϕx+ψ+lω) = 0,
ρ1wtt−k0(ωx−lϕ)x+lk(ϕx+ψ+lω) = 0,
✭✶✷✮
(x, t)∈(0, L)×(0,∞)✱ ❝♦♠ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭✻✮ ❡ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛
k(ϕx+ψ+lω)(0, t) = γ1 ϕt(0, t), t∈(0,∞),
bψx(0, t) =γ2 ψt(0, t), t ∈(0,∞),
k0(ωx−lϕ)(0, t) =γ3 ωt(0, t), t∈(0,∞),
✭✶✸✮
♦♥❞❡ γi >0✱ ♣❛r❛i= 1,2❡ 3✱ ❡
ϕ(L, t) = ψ(L, t) =ω(L, t) = 0, t ∈(0,∞). ✭✶✹✮
①✐✈
❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❡st✉❞❛r ❛ ❜♦❛ ❝♦❧♦❝❛çã♦ ❡ ♦❜t❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss✐♥tót✐❝❛s ✭❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❡st❛❜✐❧✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✮ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❡st✉❞❛❞♦s ❡♠ ❆❧❛❜❛✉✲❇♦✉ss♦✉✐r❛ ❬✶✵❪✱ ❋❛t♦r✐ ❬✾❪ ❡ ❆❧✈❡s ❡t ❛❧❧ ❬✷✵❪✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❡♠ ❝✐♥❝♦ ❝❛♣ít✉❧♦s✳ ◆♦ ❈❛♣ít✉❧♦ ✶ ❛♣r❡s❡♥t❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❢❡rr❛♠❡♥t❛s ❞❛ t❡♦r✐❛ ❞❛ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧✱ ❞❡ ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♥❛ r❡t❛✱ ❞❡ s❡♠✐❣r✉♣♦s ❞❡ ❝❧❛ss❡ C0 ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧ ❡
❡st❛❜✐❧✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✳ ❊st❡s r❡s✉❧t❛❞♦s s❡rã♦ ✉t✐❧✐③❛❞♦s ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦✳ ◆♦ ❈❛♣ít✉❧♦ ✷ ♣r♦✈❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ s✐st❡♠❛ ✭✶✶✮ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞♦ t✐♣♦ ✭✹✮ ♦✉ ❞♦ t✐♣♦ ✭✺✮ ❡ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭✻✮✳ ◆♦ss❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ é ❛ t❡♦r✐❛ ❞❡ s❡♠✐❣r✉♣♦s ❧✐♥❡❛r❡s ❞❡ ❝❧❛ss❡ C0✱
❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ❚❡♦r❡♠❛ ✶✳✺✻ ✭▲✉♠♠❡r✲P❤✐❧❧✐♣s✮ ❡ ♦ ❚❡♦r❡♠❛ ✶✳✺✽✳
◆♦ ❈❛♣ít✉❧♦ ✸ ♠♦str❛♠♦s q✉❡ ♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ✭✶✶✮ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✭✹✮ ♦✉ ✭✺✮ é ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❡stá✈❡❧ q✉❛♥❞♦ ❛ ❤✐♣ót❡s❡ ✭✼✮ é ✈á❧✐❞❛✳ ▼♦str❛♠♦s t❛♠❜é♠ q✉❡ ♣❛r❛ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ✭✺✮ ❞♦ t✐♣♦ ❉✐r✐❝❤❧❡t✲ ◆❡✉♠❛♥♥✲◆❡✉♠❛♥♥ ❛ ❝♦♥❞✐çã♦ ✭✼✮ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ✉s❛❞❛ ♥❡st❡ ❝❛♣ít✉❧♦ é ♦ ❚❡♦r❡♠❛ ✶✳✻✺ ❞❡✈✐❞♦ à Prüss ❬✷✻❪✳
◆♦ ❈❛♣ít✉❧♦ ✹ ♠♦str❛♠♦s q✉❡ ♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ✭✶✶✮ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞♦ t✐♣♦ ❉✐r✐❝❤❧❡t✲◆❡✉♠❛♥♥✲◆❡✉♠❛♥♥ é ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡ ❡stá✈❡❧ q✉❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✼✮ ♥ã♦ é ✈á❧✐❞❛✱ ♦✉ s❡❥❛✱ k
ρ1 6= b
ρ2 ♦✉ k 6=k0✳ ◗✉❛♥❞♦ k
ρ1 6= b
ρ2 ❡ k = k0 ❡♥❝♦♥tr❛♠♦s ❛ t❛①❛ ❞❡ ❞❡❝❛✐♠❡♥t♦ t
−1/2 ❡ q✉❛♥❞♦ k 6= k0
❡♥❝♦♥tr❛♠♦s ❛ t❛①❛ t−1/4✳ ❆ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ✉s❛❞❛ ♥❡st❡ ❝❛♣ít✉❧♦ é ♦
❚❡♦r❡♠❛ ✶✳✻✼ ❞❡✈✐❞♦ à ❇♦r✐❝❤❡✈ ❡ ❚♦♠✐❧♦✈ ❬✹❪✳
◆♦ ❈❛♣ít✉❧♦ ✺ ♠♦str❛♠♦s q✉❡ ♦ s❡♠✐❣r✉♣♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ✭✶✷✮✲✭✶✹✮ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ✭✻✮ é ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❡stá✈❡❧✳ ◆♦✈❛♠❡♥t❡ ❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ é ♦ ❚❡♦r❡♠❛ ✶✳✻✺ ❞❡✈✐❞♦ à Prüss ❬✷✻❪✳
❋✐♥❛❧♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ ❡♠ t♦❞❛ ❛ ❞✐ss❡rt❛çã♦✱ C r❡♣r❡s❡♥t❛ ✉♠❛
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s r❡✈❡r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳
✶✳✶ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧
◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✱ ❝♦♥s✉❧t❛r ❈❛✈❛❧❝❛♥t✐ ❬✽❪✱ ❇r❡③✐s ❬✻❪ ❡ ❖❧✐✈❡✐r❛ ❬✷✹❪✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ X q✉❡ é t❛♠❜é♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦
❝♦♠♣❧❡t♦ ❝♦♠ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ ♥♦r♠❛ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❉❡✜♥✐çã♦ ✶✳✷ ✭❋♦r♠❛ ❙❡sq✉✐❧✐♥❡❛r✮✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦✳ ❯♠❛
❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r ❞❡ V✱ é ✉♠❛ ❛♣❧✐❝❛çã♦ a : V ×V → C✱ (u, v) → a(u, v)✱
q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
✭✐✮ a(u+v, w) =a(u, w) +a(v, w) ♣❛r❛ t♦❞♦ u✱ v✱ w∈V✳
✭✐✐✮ a(λu, w) = λa(u, w) ♣❛r❛ t♦❞♦ u✱ v ∈V ❡ λ∈C✳
✭✐✐✐✮ a(u, v+w) =a(u, v) +a(u, w) ♣❛r❛ t♦❞♦ u✱ v ❡ w∈V✳
✭✐✈✮ a(u, λw) = λa(u, w) ♣❛r❛ t♦❞♦ u✱ w∈V ❡ λ∈C✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❯♠❛ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r s♦❜r❡ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ N✱ a(·,·)✱ é
❞❡♥♦♠✐♥❞❛ ❧✐♠✐t❛❞❛ ♦✉ ❝♦♥tí♥✉❛ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C >0 t❛❧ q✉❡
|a(u, v)| ≤CkukNkvkN, ♣❛r❛ t♦❞♦ u, v ∈ N.
❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠❛ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r s♦❜r❡ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ N✱ a(·,·)✱ é
❞✐t❛ ❝♦❡r❝✐✈❛ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ β >0 t❛❧ q✉❡
|a(v, v)| ≥βkvk2N, ♣❛r❛ t♦❞♦ v ∈ N.
✷ ✶✳✶✳ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲
❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦✳ ❯♠ ❢✉♥❝✐♦♥❛❧ T :V → C
é ❞✐t♦ ❧✐♥❡❛r s❡
✭✐✮ T(u+v) = T(u) +T(v) ♣❛r❛ t♦❞♦u ❡ v ∈V✳
✭✐✐✮ T(λu) =λT(u) ♣❛r❛ t♦❞♦ u∈V ❡ λ∈C✳
❡ é ❞✐t♦ ❛♥t✐❧✐♥❡❛r s❡
✭✐✮ T(u+v) = T(u) +T(v) ♣❛r❛ t♦❞♦u ❡ v ∈V✳
✭✐✐✮ T(λu) =λT(u) ♣❛r❛ t♦❞♦ u∈V ❡ λ∈C✳
❉❡✜♥✐çã♦ ✶✳✻✳ ❯♠ ❢✉♥❝✐♦♥❛❧ T :N →C✱ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ N✱ é ❞✐t♦
❧✐♠✐t❛❞♦ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C >0 t❛❧ q✉❡
|T(u)| ≤CkukN, ♣❛r❛ t♦❞♦ u∈ N.
❚❡♦r❡♠❛ ✶✳✼✳ ❙❡ N é ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❡ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❡♥tã♦
L(N, X) = {f : N → X; ❢ é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❧✐♠✐t❛❞♦} ❝♦♠ ❛ ♥♦r♠❛ ||f||L(N,X)= sup{|f(x)| ; ||x||N = 1} é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❖❧✐✈❡✐r❛ ❬✷✹❪✱ ♣✳ ✷✼✳
❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡ N é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦✱ ❡♥tã♦ ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ L(N,C) s❡rá ❞❡♥♦t❛❞♦ ♣♦r N′ ❡ ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞❡ N✳
❚❡♦r❡♠❛ ✶✳✾ ✭❍❛❤♥✲❇❛♥❛❝❤✮✳ ❙❡❥❛♠ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❡ ✉♠❛
❛♣❧✐❝❛çã♦ p:V →[0,∞) s❛t✐s❢❛③❡♥❞♦
p(u+v) ≤ p(u) +p(v), ∀u, v ∈V, p(αu) = |α|p(u), ∀u∈V, α∈C.
❙❡ f : Z → C é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❞❡✜♥✐❞♦ ♥♦ s✉❜❡s♣❛ç♦ Z ⊂ V ❝♦♠ |f(w)| ≤ p(w)✱ ❡♥tã♦ f ♣♦ss✉✐ ✉♠❛ ❡①t❡♥sã♦ ❧✐♥❡❛r F :V →C ❞♦♠✐♥❛❞❛ ♣♦r p✱
♦✉ s❡❥❛✱
|F(u)| ≤p(u), ∀u∈V.
F é ❝❤❛♠❛❞❛ ❞❡ ❡①t❡♥sã♦ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤ ❞❡ f✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❇♦t❡❧❤♦ ❡t ❛❧✳ ❬✺❪✱ ♣✳ ✺✽✳
❚❡♦r❡♠❛ ✶✳✶✵ ✭▲❛①✲▼✐❧❣r❛♥✮✳ ❙❡❥❛♠H é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡a:H×H →C
✉♠❛ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r ❧✐♠✐t❛❞❛ ❡ ❝♦❡r❝✐✈❛✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ❢✉♥❝✐♦♥❛❧ T :H →
C ❛♥t✐❧✐♥❡❛r ❧✐♠✐t❛❞♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ u∈H t❛❧ q✉❡
a(u, v) =T(v) ♣❛r❛ t♦❞♦v ∈H.
✸ ✶✳✶✳ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲
Pr♦♣♦s✐çã♦ ✶✳✶✶✳ ❙❡❥❛♠ ♥ú♠❡r♦s r❡❛✐s a, b≥0 ❡ p≥1✳ ❊♥tã♦ (a+b)p ≤2p(ap+bp).
❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠á①✐♠♦ ♦❜t❡♠♦s
(a+b)p ≤ (2 max{a, b})p = 2pmax{ap, bp}
≤ 2p(ap+bp).
Pr♦♣♦s✐çã♦ ✶✳✶✷ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✮✳ ❙❡ a, b≥ 0 ❡ p, q > 1 sã♦ t❛✐s q✉❡ 1
p+
1
q = 1 ❡♥tã♦
ab≤ a p
p + bq
q.
❉❡♠♦♥str❛çã♦✳ ❱❡r ❘✳●✳ ❇❛rt❧❡ ❬✸❪✱ ♣✳ ✺✻✳
❯♠❛ ✈❛r✐❛çã♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ q✉❡ s❡rá ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦ é ❞❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦✳
❈♦r♦❧ár✐♦ ✶✳✶✸✳ ❙❡❥❛♠ a, b≥ 0 ❡ p, q > 1 t❛✐s q✉❡ 1
p +
1
q = 1✳ P❛r❛ t♦❞♦ ε > 0
t❡♠✲s❡
ab≤c(ε)ap+εbq.
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s
ab = (qε)1q 1
(qε)1q
ab
= a
(qε)1q
!
(qε)1qb
.
❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ s❡❣✉❡ q✉❡
ab ≤ 1 p
a
(qε)1q
!p
+1
q
(qε)1qb
q
= 1
p(qε)pq
ap +εbq ∀ ε >0.
❚♦♠❛♥❞♦ c(ε) = 1
p(qε)pq
t❡♠♦s✱
✹ ✶✳✶✳ ❆◆➪▲■❙❊ ❋❯◆❈■❖◆❆▲
❚❡♦r❡♠❛ ✶✳✶✹ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✮✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧
❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h·,·iV✳ ❊♥tã♦ ♣❛r❛ t♦❞♦s u✱ v ∈V t❡♠♦s |hu, viV| ≤ kukVkvkV;
❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {u, v} é ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❖❧✐✈❡✐r❛ ❬✷✹❪✱ ♣✳ ✶✷✶✳
❚❡♦r❡♠❛ ✶✳✶✺✳ ❙❡ M é ✉♠ s✉❜❡s♣❛ç♦ ❢❡❝❤❛❞♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✱ ❡♥tã♦ H =M⊕M⊥✱ ✐st♦ é✱ ❝❛❞❛ u∈H ❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ r❡♣r❡s❡♥t❛çã♦ ♥❛ ❢♦r♠❛
u=p+q ❝♦♠ p∈M ❡ q ∈M⊥,
♦♥❞❡ M⊥ ={q ∈H :hp, qiH = 0 ♣❛r❛ t♦❞♦p∈M }✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❇♦t❡❧❤♦ ❡t ❛❧✳ ❬✺❪✱ ♣✳ ✶✶✶✳
❉❡✜♥✐çã♦ ✶✳✶✻ ✭❘❡s♦❧✈❡♥t❡✮✳ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ A : D(A) ⊂
X → X✳ ❉✐③❡♠♦s q✉❡ λ ∈ C ❡stá ♥♦ ❝♦♥❥✉♥t♦ r❡s♦❧✈❡♥t❡ ❞❡ A✱ ♦ q✉❛❧ s❡rá
❞❡♥♦t❛❞♦ ♣♦r ̺(A)✱ s❡ ♦ ♦♣❡r❛❞♦r
R(λ,A) = (λI− A)−1
❡①✐st❡✱ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡♠ X ❡ é ❧✐♠✐t❛❞♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱
̺(A) ={λ∈C; (λI− A)−1❡①✐st❡, D (λI− A)−1é ❞❡♥s♦ ❡♠ ❳ ❡
(λI− A)−1é ❧✐♠✐t❛❞♦}.
◆❡st❡ ❝❛s♦✱ R(λ,A) ❞❡♥♦♠✐♥❛✲s❡ ♦ ♦♣❡r❛❞♦r r❡s♦❧✈❡♥t❡ ❞❡ A✳ ❉❡✜♥✐çã♦ ✶✳✶✼ ✭❊s♣❡❝tr♦✮✳ ❖ ❡s♣❡❝tr♦ ❞❡ A é ♦ ❝♦♥❥✉♥t♦
σ(A) =C\̺(A)
❢♦r♠❛❞♦ ♣♦r três s✉❜❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s
✭✐✮ ❖ ❡s♣❡❝tr♦ ♣♦♥t✉❛❧ ❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❞❡ s❡✉s ❛✉t♦✈❛❧♦r❡s✱ ❞❡♥♦t❛❞♦ ♣♦r
σp(A)❀
✭✐✐✮ ❖ ❡s♣❡❝tr♦ ❝♦♥tí♥✉♦ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r σc(A)✱ é ♦ ❝♦♥❥✉♥t♦ ❞♦s λ ∈ C t❛✐s q✉❡ λI − A é ✉♠ ♦♣❡r❛❞♦r ✐♥❥❡t✐✈♦✱ t❡♠ ✐♠❛❣❡♠ ❞❡♥s❛ ❡♠ X✱ ♠❛s
(λI− A)−1 :R(λI − A)→X é ♥ã♦ ❧✐♠✐t❛❞♦❀
✭✐✐✐✮ ❖ ❡s♣❡❝tr♦ r❡s✐❞✉❛❧ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r σr(A)✱ é ♦ ❝♦♥❥✉♥t♦ ❞♦s λ ∈ C t❛✐s q✉❡ λI − A é ✉♠ ♦♣❡r❛❞♦r ✐♥❥❡t✐✈♦ ♠❛s s✉❛ ✐♠❛❣❡♠ ♥ã♦ é ❞❡♥s❛ ❡♠ X✳
❉❡✜♥✐çã♦ ✶✳✶✽✳ ❯♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r T : D(T) ⊂ N1 → N2 é ❢❡❝❤❛❞♦ s❡ ♣❛r❛
✺ ✶✳✷✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❊❙P❆➬❖❙ ❉❊ ❙❖❇❖▲❊❱
▲❡♠❛ ✶✳✶✾✳ ❙❡❥❛♠ X é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ S :X →X ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r
❝♦♥tí♥✉♦ ❝♦♠ ✐♥✈❡rs♦ ❝♦♥tí♥✉♦✳ ❙❡ B ∈L(X) s❛t✐s❢❛③
kBkL(X) <
1
kS−1kL (X)
,
❡♥tã♦ S+B é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ✐♥✈❡rsí✈❡❧✱ ❝♦♠ ✐♥✈❡rs❛ ❝♦♥tí♥✉❛✳
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ S+B é ❜✐❥❡t✐✈♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ w ∈ X ❡ ❞❡♥♦t❡♠♦s
♣♦r P ♦ ♦♣❡r❛❞♦r
P(x) =S−1(w)−S−1B(x).
◆♦t❡ q✉❡ P é ✉♠❛ ❝♦♥tr❛çã♦✱ ♣♦✐s
kP(x)−P(y)kX = k −S−1B(x) +S−1B(y)kX ≤ kS−1kL(X)kBkL(X)kx−ykX ≤ kx−ykX.
P❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ z ∈ X t❛❧
q✉❡ P(z) = z✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ z ∈X ❞❡ ♠♦❞♦ q✉❡ z =S−1(w)−S−1B(z) ⇔ (S+B) (z) =w.
▲♦❣♦✱ t❡♠♦s q✉❡ S+B é ✉♠ ♦♣❡r❛❞♦r ❜✐❥❡t✐✈♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✐♥✈❡rsí✈❡❧✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ S+B é ✉♠ ♦♣❡r❛❞♦r ❝♦♥tí♥✉♦✱ s❡❣✉❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦
●rá✜❝♦ ❋❡❝❤❛❞♦✱ q✉❡ (S+B)−1 t❛♠❜é♠ é ✉♠ ♦♣❡r❛❞♦r ❝♦♥tí♥✉♦✳
❚❡♦r❡♠❛ ✶✳✷✵✳ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❢❡❝❤❛❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H t❛❧ q✉❡ ♦ ♦♣❡r❛❞♦r r❡s♦❧✈❡♥t❡ (λ0I − A)−1 ❡①✐st❡ ❡ é ❝♦♠♣❛❝t♦ ♣❛r❛ ❛❧❣✉♠ λ0✳
❊♥tã♦ ♦ ❡s♣❡❝tr♦ σ(A) = C\ρ(A) é ❝♦♥st✐t✉í❞♦ ❛♣❡♥❛s ❞❡ ❛✉t♦✈❛❧♦r❡s ❞❡ A ❝♦♠
♠✉❧t✐♣❧✐❝✐❞❛❞❡ ✜♥✐t❛✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❑❛t♦ ❚✱ ❬✸✶❪✱ ♣✳ ✶✽✼✳
✶✳✷ ❊s♣❛ç♦s ❢✉♥❝✐♦♥❛✐s ❡ ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈
◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡s❝r❡✈❡r ❛s ♥♦t❛çõ❡s ❡ ❞❡✜♥✐çõ❡s ❞❡ ❡s♣❛ç♦s ❢✉♥❝✐♦♥❛✐s q✉❡ s❡rã♦ ✉s❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❝♦♥s✉❧t❛r ❇ré③✐s ❬✻❪✳
❉❡✜♥✐r❡♠♦s ❛ s❡❣✉✐r ♦s ❡s♣❛ç♦s ❢✉♥❝✐♦♥❛✐s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ◆❡st❛s ❞❡✜♥✐çõ❡s✱ Ω⊂Rn é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳
❉❡✜♥✐çã♦ ✶✳✷✶✳ ❙❡❥❛u:Ω→R❝♦♥tí♥✉❛✳ ❖ s✉♣♦rt❡ ❞❡u✱ q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r
✻ ✶✳✷✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❊❙P❆➬❖❙ ❉❊ ❙❖❇❖▲❊❱
❢♦r ✉♠ ❝♦♠♣❛❝t♦ ❞♦ Ω ❡♥tã♦ ❞✐③❡♠♦s q✉❡ u♣♦ss✉✐ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳ ❉❡♥♦t❛♠♦s
♣♦r C0(Ω) ❛♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ Ω ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳
❉❡✜♥✐çã♦ ✶✳✷✷✳ Cm(Ω) é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s
❞❡ ♦r❞❡♠ ≤ m ❝♦♥tí♥✉❛s ❡♠ Ω ✭m ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦ ♦✉ m = ∞✮✳ ❉❡♥♦t❛r❡♠♦s ♣♦r C0(Ω) = C(Ω).
❉❡✜♥✐çã♦ ✶✳✷✸✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ϕ : Ω → R q✉❡ ♣♦ss✉❡♠ t♦❞❛s ❛s
❞❡r✐✈❛❞❛s ❛té ❛ ♦r❞❡♠m ❝♦♥tí♥✉❛s ❡♠Ω ❡ q✉❡ tê♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ s❡♥❞♦ q✉❡
❡ss❡ s✉♣♦rt❡ ❞❡♣❡♥❞❡ ❞❡ ϕ✱ é ❞❡♥♦t❛❞♦ ♣♦r C0m(Ω) ✭♦✉ C0∞ s❡ m=∞✮✳
❉❡✜♥✐çã♦ ✶✳✷✹✳ ❯♠❛ s✉❝❡ssã♦ (ϕν)ν∈N ❞❡ ❢✉♥çõ❡s ❞❡ C0∞(Ω) ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦
q✉❛♥❞♦ ❡①✐st❡ K ⊂Ω ❝♦♠♣❛❝t♦ t❛❧ q✉❡✿
∗ suppϕν ⊂K, ∀ ν∈N;
∗ P❛r❛ ❝❛❞❛ α∈Nn
Dαϕν →0 ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ K, ♦♥❞❡ Dα ❞❡♥♦t❛ ♦ ♦♣❡r❛❞♦r ❞❡r✐✈❛çã♦ ❞❡ ♦r❞❡♠ α ❞❡✜♥✐❞♦ ♣♦r
∂|α| ∂xα1
1 ∂x
α2
2 ...∂xαnn
,
❝♦♠ α= (α1, α2, ..., αn)∈Nn ❡ |α|=α1+α2+...+αn✳
❉❡✜♥✐çã♦ ✶✳✷✺✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧C0∞(Ω)❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡✜♥✐❞❛
❛❝✐♠❛ é r❡♣r❡s❡♥t❛❞♦ ♣♦r D(Ω) ❡ ❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s ❡♠ Ω.
❉❡✜♥✐çã♦ ✶✳✷✻✳ ❙❡❥❛ 1≤ p≤+∞. ❉❡♥♦t❛♠♦s ♣♦r Lp(Ω) ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤
❞❛s ✭❝❧❛ss❡s ❞❡✮ ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ❡♠ Ω ❝♦♠ ✈❛❧♦r❡s ❡♠ R, t❛✐s q✉❡ |u|p é ✐♥t❡❣rá✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ▲❡❜❡s❣✉❡ ❡♠ Ω ❝♦♠ ♥♦r♠❛
||u||Lp =
Z
Ω|
u(x)|pdx
1 p
♣❛r❛ 1≤p < +∞.
P❛r❛ p=∞✱ ❞❡♥♦t❛♠♦s ♣♦rL∞(Ω)♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❛s ✭❝❧❛ss❡s ❞❡✮ ❢✉♥çõ❡s
♠❡♥s✉rá✈❡s ❞❡ u ❞❡✜♥✐❞❛s s♦❜r❡ Ω q✉❡ sã♦ ❡ss❡❝✐❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛s ❝♦♠ ❛ ♥♦r♠❛
❞❛❞❛ ♣♦r
||u||L∞ = sup ess
x∈Ω |
u(x)|= inf{C ∈R;|u(x)| ≤C q✳t✳♣✳ ❡♠ Ω}.
❉❡✜♥✐çã♦ ✶✳✷✼✳ ❙❡❥❛♠ 1 ≤ p < ∞✳ ❉✐r❡♠♦s q✉❡ f : Ω → R é ❧♦❝❛❧♠❡♥t❡
✐♥t❡❣rá✈❡❧ ❡♠ Lp(Ω)✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r f ∈ Lp
loc(Ω)✱ s❡ f ❢♦r ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡ ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ⊂Ω t✐✈❡r♠♦s
Z
K|
✼ ✶✳✷✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❊❙P❆➬❖❙ ❉❊ ❙❖❇❖▲❊❱
❚❡♦r❡♠❛ ✶✳✷✽ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✮✳ ❙❡❥❛♠ f1 ∈ Lp1(Ω), f2 ∈ Lp2(Ω),· · · , f
n ∈ Lpn(Ω)✱ n ∈ N✱ ❝♦♠ p1,· · · , pn > 1 ❡
1
p1
+· · ·+ 1
pn
= 1✳
❊♥tã♦ f1·...·fn∈L1(Ω) ❡
Z
Ω|
f1·...·fn|dx≤ ||f1||Lp1 ·...· ||fn||Lpn.
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇ré③✐s ❬✻❪✱ ♣✳ ✾✷✳
❚❡♦r❡♠❛ ✶✳✷✾✳ ❙❡❥❛♠ I = (a, b)✱ −∞ ≤a < b≤ ∞ ❡ s❡❥❛ u∈L1loc(I) t❛❧ q✉❡
Z
I
uϕxdx= 0 ∀ϕ ∈C01(I).
❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡ u=C ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ I✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✵✺✳
❉❡✜♥✐çã♦ ✶✳✸✵✳ ❙❡❥❛♠ I = (a, b)✱ −∞ ≤a < b≤ ∞✱ ❡ p ∈R ❝♦♠ 1≤p≤ ∞✳
❖ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ W1,p(I) é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦
W1,p(I) =
u∈Lp(I); ∃ux∈Lp(I) ❝♦♠
Z b a
uϕxdx=−
Z b a
uxϕdx ∀ϕ∈C01(I)
❖ ❡s♣❛ç♦ W1,p(I) é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ♥♦r♠❛ kukW1,p = (kuk
p
Lp+kuxk p Lp)
1 p .
◗✉❛♥❞♦ p = 2✱ ❞❡♥♦t❛♠♦s H1(I) = W1,2(I)✳ ❖ ❡s♣❛ç♦ H1(I) é ✉♠ ❡s♣❛ç♦ ❞❡
❍✐❧❜❡rt ❡q✉✐♣❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦
hu, viH1 =hu, viL2 +hux, vxiL2 =
Z b a
(uv+uxvx) dx.
❚❡♦r❡♠❛ ✶✳✸✶✳ ❙❡❥❛u∈W1,p(I)❝♦♠ 1≤p≤ ∞ ❡I ⊂R❧✐♠✐t❛❞♦ ♦✉ ✐❧✐♠✐t❛❞♦✳
❊♥tã♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ eu∈C(I) t❛❧ q✉❡
u=ue q.t.p. ❡♠(0, L) ❡
e
u(y)−ue(x) =
Z y x
ux(t)dt ∀x, y ∈I.
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✵✹✳
Pr♦♣♦s✐çã♦ ✶✳✸✷✳ ❖ ❡s♣❛ç♦ W1,p(I)é r❡✢❡①✐✈♦ ♣❛r❛ 1< p <∞ ❡ s❡♣❛rá✈❡❧ ♣❛r❛
1≤p <∞✳
✽ ✶✳✷✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❊❙P❆➬❖❙ ❉❊ ❙❖❇❖▲❊❱
❉❡✜♥✐çã♦ ✶✳✸✸✳ ❉❛❞♦ ✉♠ ✐♥t❡✐r♦m ≥2❡ ✉♠ ♥ú♠❡r♦ r❡❛❧1≤p≤ ∞❞❡✜♥✐♠♦s✱
♣♦r r❡❝♦rrê♥❝✐❛✱ ♦ ❡s♣❛ç♦
Wm,p(I) = u∈Wm−1,p(I);D1u∈Wm−1,p(I) ,
❝♦♠ ❛ ♥♦t❛çã♦ D1u=ux✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛
kukWm,p =kukLp + m
X
i=1
kDiukLp.
❊ t❛♠❜é♠ ❞❡✜♥✐♠♦s
Hm(I) =Wm,2(I),
❡q✉✐♣❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r
hu, viH2 =hu, viL2 + m
X
i=1
hDiu, DiviL2 =
Z b a
uvdx+
m
X
i=1
Z b a
DiuDivdx.
❆ s❡❣✉✐r ❡stã♦ ❛❧❣✉♥s r❡s✉❧t❛❞♦s✱ ❞❡♥tr❡ ❡❧❡s ♦s ❞❡ ✐♠❡rsõ❡s✱ q✉❡ s❡rã♦ ✉s❛❞♦s ♥♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ ♥ã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❞❡♠♦♥str❛çõ❡s✱ ♠❛s s❡rã♦ ✐♥❞✐❝❛❞❛s ❛s r❡s♣❡❝t✐✈❛s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳
❚❡♦r❡♠❛ ✶✳✸✹✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C ✭q✉❡ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❡
|I| ≤ ∞✮ t❛❧ q✉❡
kukL∞ ≤CkukW1,p, ∀u∈W1,p(I), ∀1≤p≤ ∞.
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ W1,p(I) ֒→ L∞(I) ❝♦♠ ❛ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ♣❛r❛ t♦❞♦
1≤p≤ ∞✳
❆❧é♠ ❞✐ss♦✱ s❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ ❡♥tã♦
❆ ✐♠❡rsã♦ W1,p(I)֒→C(I) é ❝♦♠♣❛❝t❛ ♣❛r❛ t♦❞♦1< p ≤ ∞✳ ❆ ✐♠❡rsã♦ W1,1(I)֒→Lq(I) é ❝♦♠♣❛❝t❛ ♣❛r❛ t♦❞♦ 1≤q <∞✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✶✷✳
❈♦r♦❧ár✐♦ ✶✳✸✺✳ ❙✉♣♦♥❤❛ q✉❡ I s❡❥❛ ✉♠ ✐♥t❡r✈❛❧♦ ✐❧✐♠✐t❛❞♦ ❡ u∈W1,p(I)✱ ❝♦♠
1≤p <∞✳ ❊♥tã♦
lim
x∈I
|x|7→∞
u(x) = 0.
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✶✹✳
✾ ✶✳✷✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❊❙P❆➬❖❙ ❉❊ ❙❖❇❖▲❊❱
❡
(uv)x = uxv + u vx.
❆❞❡♠❛✐s✱ ✈❛❧❡ ❛ ❢♦r♠✉❧❛ ❞❡ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s
Z z y
uxvdx=u(z)v(z)−u(y)v(y)−
Z z y
uvxdx, ∀x, y ∈I.
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✶✺✳
❈♦r♦❧ár✐♦ ✶✳✸✼✳ ❙❡❥❛ G ∈ C1(R) t❛❧ q✉❡ G(0) = 0✱ ❡ s❡❥❛ u ∈ W1,p(I) ❝♦♠
1≤p≤ ∞✳ ❊♥tã♦
G◦u∈W1,p(I) ❡ (G◦u)x = (Gx◦u)ux. ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✶✺✳
❉❡✜♥✐çã♦ ✶✳✸✽✳ ❉❛❞♦ 1≤p < ∞✱ ❞❡♥♦t❛♠♦s ♣♦r W01,p(I) ♦ ❢❡❝❤♦ ❞❡ C01(I) ❡♠
W1,p(I)✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛ ❞❡ W1,p(I)✳
❖ ❡s♣❛ç♦ H01(I) =W01,2(I) é ❡q✉✐♣❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❡ H1(I)✳
❚❡♦r❡♠❛ ✶✳✸✾✳ ❙❡❥❛ u∈W1,p(I)✳ ❊♥tã♦ u∈W01,p(I) s❡✱ s♦♠❡♥t❡ s❡✱ u= 0 ❡♠
∂I✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✶✼✳
❚❡♦r❡♠❛ ✶✳✹✵ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ P♦✐♥❝❛ré✮✳ ❙✉♣♦♥❤❛♠♦s I ✉♠ ✐♥t❡r✈❛❧♦
❧✐♠✐t❛❞♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ Cp ≥ 0✱ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✐♥t❡r✈❛❧♦ I✱ t❛❧ q✉❡
kukW1,p ≤CpkuxkLp ∀u∈W
1,p
0 (I).
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❡♠ W01,p(I)✱ kuxkLp é ✉♠❛ ♥♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ❞❡
W1,p(I)✳
❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ❬✻❪✱ ♣✳ ✷✶✽✳
❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❞♦ RN✳ P❛r❛ m ∈ N ❡ 1 ≤ p ≤ ∞✱ Wm,p(Ω) é ❞❡✜♥✐❞♦
❝♦♠♦ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s u∈Lp(Ω)❝✉❥❛s ❞❡r✐✈❛❞❛s ❞✐str✐❜✉❝✐♦♥❛✐s ❛té ❛ ♦r❞❡♠
m t❛♠❜é♠ ❡stã♦ ❡♠ Lp(Ω)✳ ➱ ❜❡♠ ❝♦♥❤❡❝✐❞♦✱ ✈❡r ❍✳ ❇r❡③✐s ❬✻❪✱ q✉❡ Wm,p(Ω) é
✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ♥♦r♠❛
kukWm,p(Ω) =
X
|α|≤m
kDαukpLp(Ω)
1/p
,
♦♥❞❡ α = (α1,· · ·, αn) ∈ Nn✱ |α| = α1 +· · ·+αn✱ Dαu = ∂
α1 +···+αn ∂xα1
1 ···∂x αn n
u. ◗✉❛♥❞♦ p= 2✱ ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛♠♦s Wm,p(Ω) ♣♦rHm(Ω) ❡ ❡st❡ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt
✶✵ ✶✳✸✳ ❙❊▼■●❘❯P❖❙ ❉❊ ❈▲❆❙❙❊ C0
Pr♦♣♦s✐çã♦ ✶✳✹✶✳ ❙❡❥❛ Ω ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❞♦ RN ❝♦♠ ❢r♦♥t❡✐r❛ ∂Ω ❞❡
❝❧❛ss❡ C1✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡ Ω ❡ Nt❛❧ q✉❡
♣❛r❛ t♦❞♦ u∈H1(Ω) t❡♠✲s❡
kukL2(Ω) ≤C
k∇ukL2(Ω)+
Z
Ω u dx
❉❡♠♦♥str❛çã♦✳ ❱❡r ▲✐✉ ❡ ❩❤❡♥❣ ❬✸✸❪✱ ♣✳ ✶✶✳
✶✳✸ ❙❡♠✐❣r✉♣♦s ❞❡ ❝❧❛ss❡
C
0◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡s❝r❡✈❡r ❛s ♥♦t❛çõ❡s✱ ❞❡✜♥✐çõ❡s ❡ ❛❧❣✉♥s t❡♦r❡♠❛s s♦❜r❡ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡C0 q✉❡ s❡rã♦ ✉s❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s
❝♦♥s✉❧t❛r ●♦♠❡s ❬✶✸❪ ♦✉ P❛③② ❬✷✺❪✳
❉❡✜♥✐çã♦ ✶✳✹✷ ✭❙❡♠✐❣r✉♣♦✮✳ ❙❡❥❛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✳
❉❡✜♥✐çã♦ ✶✳✹✸ ✭●❡r❛❞♦r ■♥✜♥✐t❡s✐♠❛❧✮✳ ❈♦♥s✐❞❡r❡
D(A) ={x∈X ; lim
h→0
S(h)−I
h x ❡①✐st❡}.
❖ ♦♣❡r❛❞♦r A ❞❡✜♥✐❞♦ ♣♦r
Ax= lim
h→0
S(h)−I
h x, ∀x∈D(A)
é ❞✐t♦ ❣❡r❛❞♦r ✐♥✜♥✐t❡s✐♠❛❧ ❞♦ s❡♠✐❣r✉♣♦ S✳
Pr♦♣♦s✐çã♦ ✶✳✹✹✳ ❙❡❥❛ {S(t)}t≥0 ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ❝❧❛ss❡ C0 ❝♦♠ ❣❡r❛❞♦r
✐♥✜♥✐t❡s✐♠❛❧ A✳
✐✮ ❙❡ x∈D(A)✱ ❡♥tã♦ S(t)x∈D(A) ∀t≥0 ❡ d
