We study in an abstract setting the indirect stabilization of systems of two wave-likeequations coupled by a localized zero order term. Only one of the two equations isdirectly damped. The main novelty in this paper is that the coupling operator is notassumed to be coercive in the underlying space. We show that the energy of smoothsolutions of these systems decays polynomially at infinity, whereas it is known thatexponential stability does not hold (see [F. Alabau, P. Cannarsa and V. Komornik,J. Evol. Equ. 2 (2002) 127–150]). We give applications ofour result to locally or boundary damped wave or plate systems. In any space dimension, weprove polynomial stability under geometric conditions on both the coupling and the dampingregions. In one space dimension, the result holds for arbitrary non-empty open damping andcoupling regions, and in particular when these two regions have an empty intersection.Hence, indirect polynomial stability holds even though the feedback is active in a regionin which the coupling vanishes and vice versa.
