Exponential decay for the damped wave equation in unbounded domains

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, . . . ), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity

Stabilization of locally coupled wave-type systems

In this paper, we consider a system of two wave equations on a bounded domain, that are coupled by a localized zero order term. Only one of the two equations is supposed to be damped. We show that the energy of smooth solutions of this system decays polynomially at infinity. This result is proved in an abstract setting for coupled second order evolution equations and is then applied to internal and boundary damping for wave and for plate systems. In one space dimension, this yields polynomial stability for any non-empty open coupling and damping regions, in particular if these two regions have empty intersection