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

Louredo, Aldo Trajano ; Milla Miranda, M.

Electronic journal of differential equations, Tome 2010 (2010)

Résumé

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.

Détail

Classification : 35L70, 35L20, 35L05
Keywords: Galerkin method, special basis, boundary stabilization