Existência de solução e estabilidade na fronteira da equação da onda semilinear.

Abstract

We study the existence and asymptotic behavior of the weak solution to the problem u00 􀀀� (t) u + h(u) = f; u = 0 sobre 􀀀�0; @u @ + u0 = 0 em 􀀀�1; u(0) = u0; u0(0) = u1 em (2) where Q is a cylindrical domain, T > 0 a real number, subject to certain boundary conditions 􀀀� = 􀀀�0 [ 􀀀�1, 􀀀�0 \ 􀀀�1 = ; with med(􀀀�0); med(􀀀�1) > 0 and h continues function satisfying the Strauss's conditions sh(s) 0, 8s 2 R. The existence of strong solution is made using the Faedo-Galerkin's method with a special basis to V \ H2( ) as done in [16] and the result of compactness as done in [12]. The existence of weak solution uses the theorem of Strauss as done in [24] and results and general trace as done in [20]. The asymptotic behavior is done using the Liapunov functional, with multiplicative techniques as done in Kormonik-Zuazua [9].