Stabilization for the wave equation with Neumann boundary condition by a locally distributed damping

Département de Mathématiques, E. N. S. Cachan, Antenne de Bretagne, Campus de Ker Lann, 35170 Bruz, France

Abstract

We consider the problem of the wave equation with Neumann boundary condition damped by a locally distributed linear damping a(x)u'. When the damping region ω : = {x, a(x) ≥ a > 0} contains a neighborhood of the boundary of the domain, E. Zuazua proved that the energy decays exponentially to zero. Using a piecewise multiplier method introduced by K. Liu, we prove that the energy decays exponentially to zero under weaker geometrical conditions. We give explicit examples when the domain is a polyhedron, and in the case of a disc. The proof is based on the construction of multipliers adapted to the geometrical conditions.