On the decay of solutions of a viscoelastic wave equation with variable sources

In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: (Formula presented.) where m(.) and q(.) are two functions satisfying specific conditions. This type of problems appears in fluid dynamics, the electrorheological fluids (smart fluids), which show changing (often dramatically) in the viscosity when an electrical field is applied. The Lebesgue and Sobolev spaces with variable exponents are efficient tools to analyze such problems. In this work, we prove a global existence result using the well-depth method and establish explicit and general decay results under a very general assumption on the relaxation function. Our results extend and generalize many results in the literature.