Stabilization of a locally damped incompressible wave equation

We show that the solutions of an incompressible vector wave equation with a locally distributed nonlinear damping decay in an algebraic rate to zero, that is, denoting by E(t) the total energy associated to the system, there exist positive constants C (which depends on E(0)) and γ satisfying, for t⩾0: E(t)⩽C(1+t)−γ.