Nesta seção, provamos o Teorema 4.3.
Lema 4.12. Nas hipóteses do Teorema4.8, o sistema dinâmico correspondente é gradiente.
Demonstração. Consideremos o funcional de Lyapunov Ψ(S(t)z) = E(t), onde E(t) é a
energia do sistema. Então, d dtΨ(u(t)) =−k∇ut(t)k 2 2+ 1 2 Z ∞ 0 µ′(s)k∆ηt(s)k22ds≤ 0.
4.4 Caracterização do atrator global 59 Portanto, o funcional de Lyapunov é decrescente. Suponhamos agora que, para uma trajetória z(t) = (u(t), ut(t), ηt), temos Ψ(z(t)) = z(t) para todo t≥ 0. Logo,
d dtΨ(z(t)) = 0, ∀ t ≥ 0, ou seja, k∇ut(t)k22+ 1 2 Z ∞ 0 −µ ′(s)k∆ηt(s)k2 2ds = 0, ∀ t ≥ 0.
Como ambos termos do lado esquerdo possuem o mesmo sinal, concluímos que k∇ut(t)k22 = 0 e
Z ∞
0
µ′(s)k∆ηt(s)k22ds = 0, ∀ t ≥ 0.
Assim, ut(t) = 0 e ηt = 0, para todo t. Assim, concluímos que z(t) = (u, 0, 0), o que
mostra que z(t) é uma solução estacionária do problema (4.5)-(4.8). Conforme a Definição 1.29, mostramos que Ψ(t) é um funcional de Lyapunov estrito sobre H. Portanto, o sistema em questão é gradiente.
Prova do Teorema 4.3: Como foi mostrado que o sistema(H, S(t)) é gradiente, segue do Teorema 1.30 que, o único atrator global A de (H, S(t)) é caracterizado por
A = M+(N ),
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