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Overdamping and energy decay for abstract wave equations with strong damping. (English) Zbl 1302.35264

Summary: In [Q. Appl. Math. 71, No. 1, 183–199 (2013; Zbl 1277.34085)], the first and the second author and G. Perla Menzala find sharp exponential rates for the energy decay of nontrivial solutions to the abstract telegraph equation \[ u_{tt}+2au_{t}+S^{2}u=0, \] where \(\mathcal S\) 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 \[ u_{tt}+2F(S)u_{t}+S^{2}u=0, \] where the damping term involves the action of the positive self-adjoint operator F(S). The main assumption on the continuous function \(F:(0,+\infty) \to (0,+\infty)\) is that \(g(x)=F(x)-x\) changes sign only once, being positive close to zero. We obtain sharp estimates of the form \[ E(t) \leqslant Ce^{2 \alpha t}, \] where \(\alpha>0\) 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.

MSC:

35L90 Abstract hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1277.34085
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