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In this work, a type III thermo‐porous elastic system is considered. First, we use the semigroup theory to demonstrate that the system is well‐posed. Second, we show that the system is exponentially stable under a natural constraint on its parameters. Finally, we provide various numerical examples to illustrate numerically our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a piezoelectric beam model with nonlinear source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system and prove that the energy decays at a rate dictated by the weaker damping. Our results substantially improve some earlier related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the general decay rate of the solutions for a class of fractional Laplace viscoelastic equations in the whole space ℝn,n≥1$$ {\mathbb{R}}^n,n\ge 1 $$, given by utt−Δu+u−∫0tg(t−s)(−Δ)αu(s)ds=0,$$ {u}_{tt}-\Delta u+u-{\int}_0^tg\left(t-s\right){\left(-\Delta \right)}^{\alpha }u(s) ds=0, $$with 0<α<1$$ 0<\alpha <1 $$ and g$$ g $$ satisfying g′(t)≤−η(t)g(t)$$ {g}^{\prime }(t)\le -\eta (t)g(t) $$. We prove the existence of a solution formula then use the energy method in the Fourier space to establish a general decay result which depends on α$$ \alpha $$ and η$$ \eta $$. We also present three illustrative examples by the end. [ABSTRACT FROM AUTHOR]

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]

In this work, we consider the following nonlinear wave equation with variable exponents: utt−div(|∇u|r(.)−2∇u)−Δut+|ut|m(.)−2ut=0,inΩ×(0,T),where Ω is a bounded domain, T>0, and m(.) and r(.) are continuous functions. We will establish several decay results depending on the range of the variable exponents m and r. [ABSTRACT FROM AUTHOR]

In this paper, we study an abstract class of weakly dissipative second‐order systems with finite memory. We establish a new general decay rate for the solution of the system under some appropriate conditions on the memory kernel (relaxation function). Our result improves and generalizes many existing results in the literature. We also give some examples to illustrate our abstract result. [ABSTRACT FROM AUTHOR]

We consider, in this paper, the following nonlinear equation with variable exponents: [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]

In this work, we study the well-posedness and the asymptotic stability of a one-dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χ τ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Timoshenko system in the presence of an infinite memory, where the relaxation function satisfies a relation of the form [ABSTRACT FROM AUTHOR]