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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.

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