Considera¸c˜ oes Finais

Q Q Q Q Q Q Q Q Q Q Q Q •             •           • Q Q Q Q Q Q Q Q Q Q QQ x₂ x_{1 U1D}ˆ ^C1 D C_{2 D}˜_U2

4.1.1 Considera¸c˜oes Finais

Os exemplos apresentados acima mostram que, para certas superf´ıcies, quase se alcan¸ca o resultado “sharp”devido a Bardos, Lebeau e Rauch [SICON/99] apresentado

na introdu¸c˜ao desta disserta¸c˜ao, mas v´alido apenas para a equa¸c˜ao linear. No entanto, note que temos uma n˜ao-linearidade localizada na regi˜ao de dissipa¸c˜ao. Al´em disso, podemos estender nossos resultados para a equa¸c˜ao da onda semi-linear, tendo em mente que precisamos de uma propriedade de continua¸c˜ao ´unica baseada nas estimativas de Carleman, o que segue do resultado devido a Triggiani and Yao [43].

Em resumo, a hip´otese da regi˜ao geom´etrica de controle em termos de raios da ´otica geom´etrica tem uma rela¸c˜ao pr´oxima quando comparada com a existˆencia de um bom campo vetorial q e, por conseguinte, de um bom multiplicador q · ∇u. Por uma lado, os resultados em termos da an´alise microlocal s˜ao mais gerais do que os apresentados aqui. Mas nossos resultados tamb´em consideram o caso n˜ao linear e nos fornecem exemplos expl´ıcitos de regi˜oes que podem ficar livres de efeitos dissipativos, o que pode ser uma tarefa dif´ıcil se usarmos a hip´otese da regi˜ao geom´etrica de controle sobre superf´ıcies bem gerais. Em nossa opini˜ao, h´a pleno espa¸co para estudos futuros a respeito da rela¸c˜ao entre estes dois diferentes tipos de hip´oteses.

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