Exponential Decay for the Solution of Semilinear Viscoelastic Wave Equations with Localized Damping
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In this paper we obtain an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation u tt - Δu + ƒ(x,t,u) + ∫ t 0 g(t - T) Δu(T) dT + a(x) ut = 0 in Ω x (0,∞). Here the damping term a(x)ut may be null for some part of the domain Ω. By assuming that the kernel g in the memory term decays exponentially, the damping effect allows us to avoid compactness arguments and and to reduce number of the energy estimates considered in the prior literature. We construct a suitable Liapunov functional and make use of the perturbed energy method.
CitationCavalcanti, M. M., Domingos Cavalcanti, V. N., & Soriano, J. A. (2002). Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electronic Journal of Differential Equations, 2002(44), pp. 1-14.
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