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.