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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]

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two "smart" materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state (0 , 0) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings. [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]