Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 120, pp. 1-19.

Title: Nonlinear boundary dissipation for a coupled system
       of Klein-Gordon equations

Authors: Aldo T. Louredo (Univ. Estadual da Paraiba, Campina Grande, Brazil)
         M. Milla Miranda (Univ. Federal do Rio de Janeiro, Brazil)

Abstract:
 This article concerns the existence of
 solutions and the decay of the energy of the mixed problem
 for the coupled system of Klein-Gordon equations
 $$\displaylines{
 u'' - \Delta u  + \alpha v^{ 2}u=0 \quad\hbox{in }\Omega
 \times (0, \infty), \cr
 v'' - \Delta v  +  \alpha u^{2}v=0 \quad\hbox{in }\Omega 
 \times (0, \infty),
 }$$
 with the nonlinear boundary conditions,
 $$\displaylines{
  \frac{\partial u}{\partial \nu} + h_1(.,u')=0 \quad\hbox{on }
 \Gamma_1 \times (0, \infty), \cr
  \frac{\partial v}{\partial \nu} +  h_2(.,v')=0 \quad\hbox{on }
 \Gamma_1 \times (0, \infty),
 }$$
  and boundary conditions $u=v=0$ on
 $(\Gamma \setminus \Gamma_1) \times (0,\infty)$, where $\Omega$
 is a bounded open set of $\mathbb{R}^n~(n \leq 3)$, $\alpha >0$
 a real number, $\Gamma_1$ a subset of the boundary $\Gamma$ of
 $\Omega$ and $h_i$ a  real function defined on
 $\Gamma_1 \times (0, \infty)$.

 Assuming that each $h_i$ is strongly monotone in the second
 variable, the existence of global solutions of the mixed problem
 is obtained. For that it is used the Galerkin method, the Strauss'
 approximations of real functions and trace theorems for non-smooth
 functions. The exponential decay of the energy for a particular 
 stabilizer is derived by  application of a Lyapunov functional.

Submitted August 3, 2009. Published August 24, 2010.
Math Subject Classifications: 35L70, 35L20, 35L05.
Key Words: Galerkin method; special basis; boundary stabilization.