Overdamping and energy decay for abstract wave equations with strong damping


In Quart. Appl. Math. 71 (2013), 183-199, the authors find sharp exponential rates for the energy decay of nontrivial solutions to the abstract telegraph equation utt+2aut+S2u=0, where S is a strictly positive self-adjoint operator in a (complex) Hilbert space and a is a positive constant. The aim of this paper is a further extension of these results by considering equations of the form utt+2F(S)u t+S2u=0, where the damping term involves the action of the positive self-adjoint operator F(S). The main assumption on the continuous function F:(0,+∞)→(0,+∞) is that g(x)=F(x)-x changes sign only once, being positive close to zero. We obtain sharp estimates of the form E(t)≤Ce-2αt, where α>0 depends on the relative position of the bottom of the spectrum of S and the point where g vanishes, as well as on the specific behavior of F on the spectrum of S. The general result is then applied to some particular classes of functions F. We also provide a number of applications. 2014-IOS Press and the authors.

Publication Title

Asymptotic Analysis