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

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 $$displaylines{ u'' - Delta u + alpha v^{ 2}u=0 quadhbox{in }Omega imes (0, infty), cr v'' - Delta v + alpha u^{2}v=0 quadhbox{in }Omega imes (0, infty), }$$ with the nonlinear boundary conditions, $$displaylines{ frac{partial u}{partial u} + h_1(.,u')=0 quadhbox{on } Gamma_1 imes (0, infty), cr frac{partial v}{partial u} + h_2(.,v')=0 quadhbox{on } Gamma_1 imes (0, infty), }$$ and boundary conditions $u=v=0$ on $(Gamma setminus Gamma_1) imes (0,infty)$, 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 $Gamma_1 imes (0, infty)$.

Keywords

• Galerkin method
• special basis
• boundary stabilization