Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 08, pp. 1-21.
Title: Existence and boundary stabilization of a nonlinear hyperbolic equation
with time-dependent coefficients
Authors: M. M. Cavalcanti (Univ. Estadual de Maringa, PR, Brasil)
V. N. Domingos Cavalcanti (Univ. Estadual de Maringa, PR, Brasil)
J. A. Soriano (Univ. Estadual de Maringa, PR, Brasil)
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.