Stability and numerical study for solutions of some viscoelastic problems with variable exponents’ nonlinearities

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 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 the image processing. In this project, we study the effect of the non-standard nonlinearities of variable exponents on the stability of some viscoelastic equations and systems. For this purpose, we consider several viscoelastic problems with variable-exponent nonlinearities and discuss the long-time behavior and provide some numerical experiments to illustrate the stability results. These problems require the use of non-standard Lebesgue and Sobolev space. Our results, if established, will extend some known stability results in the constant-variable case to the variable-exponent case