Stabilization of the wave equation with Neumann boundary condition and localized nonlinear damping

Abstract

We show that the solutions of the wave equation with potential, Neumann boundary conditions and a locally distributed nonlinear damping, decay to zero, with an algebraic rate, that is, the total energy E(t) satisfies for t >= 0: E(t) <= C(1 + t)(-gamma), where C is a positive constant depending on E(0) and gamma > 0 is a constant. We assume geometrical conditions as in P. Martinez [7]. In the one/two-dimensional cases, we obtain exponential decay rate when the nonlinear dissipation behaves linearly close to the origin. The same result holds in higher dimension if the dissipative localized term behaves linearly.