Existence & stability of biharmonic systems modelling the motion of several structural elements

With the advancement of sciences and technology, many physical and engineering models require more sophisticated mathematical tools to be investigated 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 appropriate for studying such problems as well as other models like image processing. In this project, we consider some biharmonic wave systems in the presence of non-standard nonlinearities due to the smart nature of the material. These systems describe the interaction between the motions of plates and waves, such as the motion of a suspension bridge and the cables. For this purpose, we discuss several biharmonic wave systems with variable-exponent nonlinearities and use these modern functional spaces to establish the existence of solutions and discuss their stability in the presence and absence of viscoelastic terms. Our results, if obtained, will generalize and improve all the existing results in the Literature and open the door for more research in this direction