Overdamping and energy decay for abstract wave equations with strong damping

Abstract

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

u_tt+2au_t+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

u_tt+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.