Annales Academić Scientiarum Fennicć
Mathematica
Volumen 34, 2009, 291-302

GENERAL DECAY OF SOLUTIONS TO A VISCOELASTIC WAVE EQUATION WITH NONLINEAR LOCALIZED DAMPING

Wenjun Liu

Nanjing University of Information Science and Technology, College of Mathematics and Physics
Nanjing 210044, P.R. China; wjliu 'at' nuist.edu.cn

Abstract. In this paper we consider the nonlinear viscoelastic equation

u_tt - \Delta u + \int_0^t g(t - \tau) \Delta u(\tau) d\tau + a(x)|u_t|^mu_t + b|u|^\gamma u = 0

in a bounded domain. We prove that, for certain class of relaxation functions and certain initial data, the decay rate of the solution energy is similar to that of the relaxation function regardless of the presence or the absence of the frictional damping. This result improves earlier ones in Berrimi and Messaoudi [1] in which only the exponential decay rate is obtained.

2000 Mathematics Subject Classification: Primary 35B35, 35L20, 35L70.

Key words: General decay, viscoelastic equation, exponential decay, polynomial decay, relaxation function, nonlinear localized damping.

Reference to this article: W. Liu: General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Ann. Acad. Sci. Fenn. Math. 34 (2009), 291-302.

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