On the Decay of Solutions of a Damped Viscoelastic Wave Equation with Variable-exponent Nonlinearities

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The stabilization of linear and nonlinear wave equations by means of internal or boundary feedbacks has attracted a considerable attention and much effort has been devoted to establish decay rates for the last half century. Various results regarding existence, polynomial, exponential and general decay have been proved. With the advancement of sciences and technology, many physical and engineering models require more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, the electrorheological fluids (smart uids) have the property that the viscosity changes (often dramatically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like filtration processes through a porous media and image processing. In this work, we consider the following nonlinear viscoelastic problem with variable exponents: ... where ... is a bounded domain and T > 0. This can be regarded as a model for the propagation of waves in a viscoelastic material in the presence of a non-standard frictional damping due to the nature of the "smart" material. Our goal is to show that weak solutions decay exponentially or polynomially depending the values of the variable exponent m.

Brief Biography

Dr. Salim Messaoudi is a Professor of Applied Mathematics in the University of Sharjah, UAE. He got his PhD in 1989 from Carnegie-Mellon University, USA. Since then he taught in various Arabia Universities, in Particular KFUPM, and visited other universities in Europe, USA and Brazil.
He published a good number of papers, gave more than 70 talks in conferences and supervised around 30 MSc and PhD students. He won many awards, in particular, the Custodian of the holy mosques Translation Award in 2015.