Indirect internal stabilization of weakly coupled evolution equations

Abstract.

Let two second order evolution equations be coupled via the zero order terms, and suppose that the first one is stabilized by a distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behaviour of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay polynomially at infinity, and that this decay rate is, in some sense, optimal. The stabilization result for abstract evolution equations is also applied to study the asymptotic behaviour of various systems of partial differential equations.