Electron. J. Diff. Equ., Vol. 2010(2010), No. 120, pp. 1-19.

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

Aldo T. Louredo, M. Milla Miranda

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.

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Aldo T. Lourêdo
Universidade Estadual da Paraíba, DME
CEP 58109095 - Campina Grande - PB, Brazil
email: aldotl@bol.com.br
M. Milla Miranda
Universidade Federal do Rio de Janeiro - IM
CEP 21945-970 - Rio de Janeiro - RJ, Brazil
email: milla@im.ufrj.br

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