Exponential Decay for the Solution of Semilinear Viscoelastic Wave Equations with Localized Damping




Cavalcanti, Marcelo M.
Domingos Cavalcanti, V. N.
Soriano, Juan A.

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Southwest Texas State University, Department of Mathematics


In this paper we obtain an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation utt - Δu + ƒ(x, t, u) + ∫t0 g(t - T) Δu(T) dT + α(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.



Semilinear wave equation, Memory, Localized damping


Cavalcanti, M. M., Domingos Cavalcanti, V. N., & Soriano, J. A. (2002). Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. <i>Electronic Journal of Differential Equations, 2002</i>(44), pp. 1-14.


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