Electron. J. Diff. Eqns., Vol. 1998(1998), No. 08, pp. 1-21.

Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients

M. M. Cavalcanti, V. N. Domingos Cavalcanti & J. A. Soriano

Abstract:
In this article, we study the hyperbolic problem
$$ K(x,t)u_{tt} - \sum_{j=1}^n\left(a(x,t)u_{x_j}\right) + F(x,t,u,\nabla u) = 0 $$
coupled with boundary conditions
$$u=0,\quad\hbox{on }\Gamma_1\,, \quad {\partial u \over\partial\nu} + \beta(x)u_t =0\quad\hbox{ on }\Gamma_0\,.$$
Here the variable $x$ belongs to a bounded region of ${\Bbb R}^n$, whose boundary is partitioned into two disjoint sets $\Gamma_0,\Gamma_1$.
We prove existence, uniqueness, and uniform stability of strong and weak solutions when the coefficients and the boundary conditions provide a damping effect.

Submitted July 6, 1997. Published March 10, 1998.
Math Subject Classification: 35B40, 35L80.
Key Words: Boundary stabilization, asymptotic behaviour.

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Marcelo M. Cavalcanti
Universidade Estadual de Maringa,
87020-900 Maringa-PR, Brazil
e-mail: marcelo@gauss.dma.uem.br
Valeria N. Domingos Cavalcanti
Universidade Estadual de Maringa,
87020-900 Maringa-PR, Brazil
e-mail: valeria@gauss.dma.uem.br
Juan Amadeo Soraino
Universidade Estadual de Maringa,
87020-900 Maringa-PR, Brazil
e-mail: soriano@gauss.dma.uem.br

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