Well-posedness and Exponential Decay of Energy for the Solution of a Wave Equation with Nonlinear Source and Localized Damping Termes

Abstract

We consider the wave equation with a locally damping and a nonlinear source term in a bounded domain. ytt − Δy + a(x)g(yt) = |y|p−2y, where p > 2. The damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary. We show, for certain initial data and suitable conditions on g,a and p that this solution is global we use the Faedo-Galerkin method. Also we established the exponential decay of the energy when the nonlinear damping grows linearly by introducing a suitable Lyapunov functional.