Indirect linear locally distributed damping of coupled systems

The aim of this paper is to prove indirect internal stabilization results for different coupled systems with linear locally distributed damping (coupled wave equations, wave equations with different speeds of propagation). In our case, a linear local damping term appears only in the first equation whereas no damping term is applied to the second one (this is indirect stabilization, see [11]). Using thepiecewise multiplier method we prove that the full system is stabilized and that the total energy of the solution of this system decays polynomially