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In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: utt−Δu+∫0tg(t−s)Δu(s)ds+a|ut|m(·)−2ut=|u|q(·)−2u,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. [ABSTRACT FROM AUTHOR]

However, with the advancement of sciences and technology, the study and the understanding of several physical and engineering models have required more sophisticated tools and new mathematical functional spaces. The use of functional analysis, in particular the function spaces such as the classical Lebesgue and Sobolev spaces, has been an essential tool to solve and study the qualitative properties of these solutions. The Lebesgue and Sobolev spaces with variable exponents and some other more general spaces such as Orlicz spaces proved to be the right and efficient means to tackle such problems as well as other models like the image processing. [Extracted from the article]