Stabilization and blow up study of some hyperbolic systems in the presence of Viscoelasticity

With the advancement of sciences and technology, many physical and engineering models require more sophisticated mathematical functional spaces to be studied and well understo...ood. For example, in fluid dynamics, the electrorheological fluids (smart fluids) 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 image processing. In this project, we study the effect of the non-standard nonlinearities of variable exponents on the stability of wave equations and systems. For this purpose, we consider several wave and viscoelastic “weakly” damped problems with variable-exponent nonlinearities and discuss the stability in the presence and absence of forcing terms. These problems require the use of non-standard Lebesgue and Sobolev spaces. Our results, if established, will extend some known stability results in the constant-variable case to the variable-exponent case.