Salim A. Messaoudi

• PhD
• Professor at University of Sharjah

In this work, we consider a coupled system of two wave equations with only one thermal effect in a one-dimensional domain. We give the well-posedness using the semigroup theory and show that the solution of the system decays polynomially by constructing an appropriate Lyapunov function.

The main goal of this work is to investigate the following quasilinear wave equation where \(\Omega \) is a bounded domain of \({\mathbb {R}}^{n}\), \(T>0\), m(.), r(.) are variable exponents and \(\alpha (t)\) is the time-varying damping coefficient. We will establish several decay results under specific conditions on variable exponents and the ti...

This work is concerned with a coupled nonlinear mathematical model for a suspension bridge with past history. The vibrations of both the road bed in the vertical plain and main cable from which the road bed is suspended by the tie cables are taken into consideration. Using the semi-group approach, we give a thorough and careful existence and unique...

This paper is concerned with a thermoelastic swelling system with Coleman-Gurtin’s law, when the heat flux q is given by where \(\theta \) is the temperature supposed to be known for negative times. \(\Psi \) is the convolution thermal kernel, a nonnegative bounded convex function on \([0, + \infty )\) belongs to a broad class of relaxation functio...

The asymptotic stability of the vertical vibrations of a suspension bridge and the main cables from which the bridge is suspended by the cables is investigated. The history of the materials of the bridge and the effect of heat governed by Gurtin-Pipkin's thermal law on main cables are taking into consideration. We show that the damping induced by t...

In this work, we consider a one‐dimensional truncated Timoshenko system coupled with a heat equation, where the heat flux is given by the generalized dual‐phase lag model. By using the semigroup theory and some nonclassical differential operators, we establish the well‐posedness of the problem. Then, we use the multiplier method to show that the on...

Suspension bridges are essential in lifeline civil structures that have been constructed in many countries due to their superior effectiveness when it comes to long spans. In this paper, we study a mathematical model for a one-dimensional suspension bridge problem with viscoelastic damping and nonlinear frictional damping. The model takes into cons...

The paper deals with the local and global well-posedness of coupled system of (α, β)Laplacian type of wave equations with initial and Dirichlet-boundary conditions and variable exponents of nonlinearity. Furthermore, explicit decay rates of the solutions are established, under suitable assumptions on the parameters of the problem. This work general...

In this paper, we consider a one-dimensional finite-memory Bresse system with homogeneous Dirichlet-Neumann-Neumann boundary conditions. We prove some general decay results for the energy associated with the system in the case of equal and non-equal speeds of wave propagation under appropriate conditions on the relaxation function. In addition, we...

This paper is concerned with the well-posedness and stability of a one-dimensional thermoelastic truncated Timoshenko system of Type III. In order to establish the well-posedness, we first solve an auxiliary problem and give the proof in details, using the semigroup theory and some non traditional operators. Then, we use this result to solve our or...

This work focusses on the well–posedness and the exponential stability of a simplified porous elastic system. By omitting the second time derivative term of the function φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \...

In this work, we establish the well-posedness and the asymptotic stability of a linear thermoelastic Gurtin–Pipkin–Timoshenko system free of second spectrum. We give a detailed proof of the well-posedness, using the semigroup theory and with the help of some new operators. Then, we prove that the system is exponentially stable irrespective of the p...

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.

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions wit...

In this article, we investigate the following weakly dissipative plate equation: Under some mild conditions on the relaxation function g, we show that the solution energy has general decay estimate. We also give some examples to illustrate our result. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen’s...

In this article, we consider a coupled system of two hyperbolic equations with variable exponents in the damping and source terms, where the dampings are modilated with time-dependent coefficients \(\alpha(t), \beta(t)\). First, we state and prove an existence result of a global weak solution, using Galerkin's method with compactness arguments. The...

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...

In this work we study the well-posedness and the asymptotic stability of a linear thermoelastic Timoshenko system free of second spectrum, where the heat conduction is given by Cattaneo’s law. We give a detailed proof of the well-posedness, using the semigroup theory. Then, we prove that the system is exponentially stable irrespective of the coeffi...

In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space \begin{document}$ \mathbb{R}^n $\end{document}, \begin{document}$ n\geq 1 $\end{document}, given by \begin{document}$ \begin{equation*} u_{tt}+\Delta^2 u+ u+ \int_0^t g(t-s)A u(s)ds = 0, \end{equation*} $\end{docu...

This paper considers a one-dimensional thermoelastic Timoshenko beam system with suspenders, general weak internal damping with time varying coefficient, and time-varying delay terms. Under suitable conditions on the nonlinear terms, we prove a general stability result for the beam model, where exponential and polynomial decay are special cases. We...

In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the vari...

In this paper, we consider a nonlinear beam equation with a strong damping and the p(x)-biharmonic operator. The exponent p(·) of nonlinearity is a given function satisfying some condition to be specified. Applying Faedo–Galerkin’s method, the existence of weak solutions is proved. Using Nakao’s lemma, the asymptotic behavior of weak solutions is e...

In this paper, we consider a plate equation with a strong damping and a logarithmic nonlinearity. Using the Galerkin method, we establish the existence of the solutions of the problem and we prove an exponential decay result, using the multiplier method. This result extends and improves many results in the literature.

This work is concerned with a coupled system of two biharmonic equations with variable exponents in the damping and source terms. Using the energy approach and for certain solution with positive initial data, we prove the blow-up theorem. Then, we establish the global existence as well as energy decay results of solutions, under appropriate conditi...

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...

In this work we investigate a thermoelastic shear beammodel with thermal dissipation. We prove a well posedness result by the use of the Faedo--Galerkin method and the exponential stability by the multiplier method. cOur result improves the stability results obtained for certain thermoelasticTimoshenko type systems in the sense that we do not requi...

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponents nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions wi...

With the advancement of science and technology, many physical and engineering models require more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, electrorheological fluids (smart fluids) have the property that the viscosity changes (often drastically) when exposed to an electrical fiel...

In this work, we consider a swelling porous system where the damping terms are on the boundary. We establish an explicit and general decay result, without imposing restrictive growth assumption near the origin on the damping terms. Our result allows a larger class of damping terms, and the usual exponential and polynomial decay estimates are only s...

In this paper, we study the stability of a Bresse system with memory-type boundary conditions. For a wider class of kernel functions, we establish an optimal explicit energy decay result. Our stability result improves many earlier results in the literature. Finally, we also give four numerical tests to illustrate our theoretical results using the c...

In this paper, we consider the following viscoelastic swelling porous-elastic system $$\begin{aligned} \left\{ \begin{array}{ll} \rho _z z_{tt} -a_1z_{xx}- a_2 u_{xx} + \int \limits _0^t g(t-s) z_{xx}(x,s) \ ds =0, &{}{\text {in }}\, (0,1)\times (0,\infty ),\\ \rho _u u_{tt}- a_3 u_{xx} - a_2 z_{xx} = 0, &{}{\text {in }}\, (0,1)\times (0,\infty ),...

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on...

In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary...

In this paper, we are interested in a viscoelastic Moore–Gibson–Thompson equation with a type-II memory term and a relaxation function satisfying g′(t)≤-η(t)g(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{...

In this work, we consider a swelling porous-elastic system coupled with a heat system of second sound. We establish the existence of solutions then we prove an exponential decay result. Unlike the “purely” porous system, this result is obtained without the equal-speed requirement. We also perform some numerical tests to illustrate our theoretical f...

In this article, we consider a coupled system of two nonlinear hyperbolic equations, where the exponents in the damping and source terms are variables. First, we prove a theorem of existence and uniqueness of weak solution, by using the Faedo Galerkin approximations and the Banach fixed point theorem. Then, using the energy method, we show that cer...

In this paper, we consider the following nonlinear viscoelastic wave equation with variable exponents: $$u_{tt}-\Delta u+\int_{0}^{t} g(t-\tau)\Delta u(x,\tau)\rm{d}\tau+\mu u_{t}=\vert u\vert^{p(x)-2}u,$$ where μ is a nonnegative constant and the exponent of nonlinearity p(·) and g are given functions. Under arbitrary positive initial energy and s...

In this paper, we establish general decay estimates of the solution for the Cauchy problem of a Moore–Gibson–Thompson equation with a viscoelastic term using the energy method in the Fourier space. We first discuss the well-posedness. Then we establish our main result and present two illustrative examples by the end.

In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and gene...

This paper is concerned with the asymptotic behavior of the solution of a laminated Timoshenko beam system with viscoelastic damping. We extend the work known for this system with finite memory to the case of infinite memory. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula, which gives th...

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)...

This work is concerned with a one-dimensional thermoelastic porous system with infinite memory effect. We show that the stability of the system holds for a much larger class of kernels than the ones considered in the literature such as the one in [1] and [2]. More precisely, we consider the kernel g:[0,+∞)→(0,+∞) satisfyingg′(t)≤−γ(t)G(g(t)), where...

In this paper, we consider a one-dimensional nite-memory Bresse system with homogeneous Dirichlet-Neumann-Neumann boundary conditions. We prove some general decay results for the energy associated with the system in the case of equal and non-equal speeds of wave propagation under appropriate conditions on the relaxation function. In addition, we sh...

We study the following wave equation $$u_{tt}-\Delta u+\alpha (t)\left| u_{t}\right| ^{m(\cdot )-2}u_{t}=0$$ u tt - Δ u + α ( t ) u t m ( · ) - 2 u t = 0 with a nonlinear damping having a variable exponent m ( x ) and a time-dependent coefficient $$\alpha (t)$$ α ( t ) . We use the multiplier method to establish energy decay results depending on bo...

In this paper we study and obtain the existence of asymptotically almost periodic solutions to some classes of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type in a separable Hilbert space H. To illustrate our abstract results, the existence of asymptotically almost periodic solutions to the well-known Kirchoff plate equa...

This paper is concerned with the following memory-type Timoshenko system $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$with Dirichlet bo...

In this work, we investigate a one-dimensional porous-elastic system with thermoelasticity of type III. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical res...

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results ex...

This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels gi : [0, +∞) → (0, +∞) satisfying gi⁰(t) ≤ −ξi(t)Hi(gi(t)), ∀ t ≥ 0 and for i = 1, 2, where ξi and Hi are fun...

Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrat...

This work is concerned with a system of wave equations with variable-exponent nonlinearities acting in both equations. We, first, discuss the well-posedness then prove a blow up result for solutions with negative initial energy.

In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying \begin{document}$ g'(t) \leq - \xi(t) H(g(t)) $\end{document}, where \begin{document}$ H $\end{document} is an increasing, convex function and \begin{document}$ \xi $\end{document} is a nonincr...

In this article, we consider a coupled system of two nonlinear hyperbolic equations, where the exponents in the damping and source terms are variables. First, we prove a theorem of existence and uniqueness of weak solution, by using the Faedo Galerkin approximations and the Banach fixed point theorem. Then, using the energy method, we show that cer...

In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: u t t − Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s + a | u t | m ( · ) − 2 u t = | u | q ( · ) − 2 u , where m(.) and q(.) are two functions satisfying specific conditions. This type of problems appears in fluid dynamics, the electrorheological fluids (s...

This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function ki\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...

In this article, we investigate a one-dimensional thermoelastic laminated beam system with viscoelastic dissipation on the effective rotation angle and through heat conduction in the interfacial slip equations. Under general conditions on the relaxation function and the relationship between the coefficients of the wave propagation speed of the firs...

We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results i...

In this paper, we consider a thermoelastic laminated beam with a finite memory acting on the effective rotation angle. We establish an explicit and optimal stability estimate for the solution energy with minimal conditions on the relaxation function, from which the exponential and polynomial stability are just particular cases. This new result impr...

We consider a viscoelastic plate equation with nonlinear source and partially hinged boundary conditions. Our goal is to show analytically that the solution blows up in �finite time. The background of the problem comes from the modeling of the downward displacement of a suspension bridge using a thin rectangular plate. This result shows that in the...

In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical illustrations are presented at the end. The numerical solution is computed using the popular finite element method in...

In this note we study an abstract class of weakly dissipative second‐order systems with finite memory. We establish the polynomial decay of Rivera, Naso and Vegni for the solution of the system under a very weak condition on the relaxation function.

The present article investigates a thermo-viscoelastic laminated beam system, where the heat conduction is given by Maxwell–Cattaneo’s law (popularly known as second sound). We establish explicit and general stability results for the solution energy in the case of equal-speed propagation and nonequal speed. Our results contain the exponential and p...

In this paper, we are concerned with a memory-type Timoshenko system with Dirichlet boundary conditions and a very general class of relaxation functions. We prove the existence and uniqueness of solutions of the system as well as some new decay results which generalize and improve many earlier ones in the literature. We consider the case of equal-s...

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.

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and a relaxation function satisfying g′(t) ≤ −ξ(t)G(g(t)). Using the Galaerkin method, we establish the existence of the solution and prove an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This w...

In this paper, we consider a weakly dissipative viscoelastic equation with a nonlinear damping. A general decay rate is proved for a wide class of relaxation functions. To support our theoretical findings, some numerical results are provided.

In this paper, we study a fourth-order plate problem as a model for a suspension bridge in the presence of a nonlinear frictional damping and a hanger restoring force. We establish the existence of a global weak solution and prove a stability result.

In this paper, we consider a viscoleastic equation with a nonlinear feedback localized on a part of the boundary and a relaxation function satisfying g′(t) ≤−ξ(t)G(g(t)). We establish an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our results are obtained without imposing any res...

Feng and Apalara (2019) [16] investigated the one-dimensional porous-elastic system with finite memory under the assumptions of equal-speed wave propagations and positive definite energy associated with the solution, and established an optimal explicit energy decay rate. In this paper, we consider the case of non-equal wave speeds and positive semi...

In this paper, we consider the following pseudo-parabolic equation with variable exponents: ut-Δu-Δut+∫0tg(t-τ)Δu(x,τ)dτ=|u|p(x)-2u.Under suitable assumptions on the initial datium u0, the relaxation function g and the variable exponents p, we prove that any weak solution, with initial data at arbitrary energy level, blows up in finite time.

Due to an error in the typesetting process, reference [1] is incorrectly published in the original publication of the article.

This work is concerned with a coupled system of nonlinear viscoelastic wave equations that models the interaction of two viscoelastic fields. This system has been extensively studied by many authors for relaxation functions decaying exponentially, polynomially, or with some general decay rate. We prove a new general decay result that improves most...

This paper is concerned with the following memory-type Bresse system ρ1ϕtt − k 1 (ϕx + ψ + lw)x − lk 3 (wx − lϕ) = 0, ρ2ψtt − k 2 ψxx + k 1 (ϕx + ψ + lw) + Z0t g(t − s)ψxx(·, s)ds = 0, ρ1wtt − k 3 (wx − lϕ)x + lk 1 (ϕx + ψ + lw) = 0, with homogeneous Dirichlet-Neumann-Neumann boundary conditions, where (x, t) ∈ (0, L) ×(0, ∞), g is a positive stric...

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 exa...

We consider a quasilinear heat system in the presence of an integral term and establish a general and optimal decay result from which improves and generalizes several stability results in the literature.

In this paper, we consider a viscoelastic plate equation with a logarithmic nonlinearity. Using the Galaerkin method and the multiplier method, we establish the existence of solutions and prove an explicit and general decay rate result. This result extends and improves many results in the literature such as Gorka [19], Hiramatsu et al. [27] and Han...

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle - a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively...

In this paper, we consider a nonlinear wave equation with damping and source terms of variable-exponent types. First, we use the stable-set method to prove a global result. Then, by applying an integral inequality due to Komornik, we obtain the stability result.

This paper is concerned with a problem of a logarithmic nonlinear wave equation with delay. The local existence result has been established using the semigroup theory. In addition, for negative initial energy, a finite-time blow-up result is proved.

Under the necessary compatibility condition and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and stability of solution in [Lm+1(Ω)]d×(W1,m+1m(Ω)∩L02(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...

In this paper, we consider the following nonlinear waveequation with variable exponents: utt−div(|∇u|r(⋅)−2∇u)+|ut|m(⋅)−2ut=0.By using a lemma by Komornik, we prove the decay estimates for the solution under suitable assumptions on the variable exponents m,r and the initial data. We also give two numerical applications to illustrate our theoretical...

We consider a quasilinear heat system in the presence of an integral term and establish a general and optimal decay result from which improves and generalizes several stability results in the literature.

In this paper, we consider a viscoelastic plate equation with a velocity-dependent material density and a logarithmic nonlinearity. Using the Faedo-Galaerkin approximations and the multiplier method, we establish the existence of the solutions of the problem and we prove an explicit and general decay rate result. These results extend and improve ma...

We establish a general decay rate for a viscoelastic problem with a nonlinear boundary feedback and a relaxation function satisfying g ′ (t) ≤ −ξ(t)g p (t), t ≥ 0, 1 ≤ p < 3/2. This work generalizes and improves earlier results in the literature. In particular those of [5], [11] and [17]. © 2018 Juliusz Shauder Centre for Nonlinear Studies Niolaus...

In this paper we study the well-posedness and the asymptotic stability of a one-dimensional thermoelastic Bresse system, where the heat conduction is given by Cattaneo’s law effective in the shear angle displacements. We establish the well-posedness of the system and prove that the system is exponentially stable depending on the parameters of the s...

In this paper, we consider a plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping. Using the Galaerkin method, we establish the existence of solutions of the problem and we prove an explicit and general decay rate result, using the multiplier method and some properties of the convex functions. Our result is...

We consider the following nonlinear parabolic equation: ut-div(|∇u|p(x)-2∇u)=f(x,t) , where f:Ω×(0,T)→R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate th...

This paper is concerned with the following memory-type Timoshenko system ρ1ψtt-K(ψx+ψ)x=0 ρ2ψtt-bψxx+K(ψx+ψ)+∫0tg(t-s)ψxx(s)ds=0, (x,t)∈(0,L)×(0,∞), with Dirichlet boundary conditions, where g is a positive non-increasing function satisfying, for some constant 1≤p < 3/2, g'(t)≤-Ξ(t)gp(t),for all t≥0. We prove some decay results which generalize and...

In this paper, we consider a linear thermoelastic Timoshenko system with variable physical parameters, where the heat conduction is given by Cattaneo’s law and the coupling is via the displacement equation. We discuss the well-posedness and the regularity of solution using the semigroup theory. Moreover, we establish the exponential decay result pr...

The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results re...

In this paper, we investigate the well-posedness as well as optimal decay rate estimates of the energy associated with a Kirchhoff-Carrier problem in ndimensional bounded domain under an internal finite memory. The considered class of memory kernels is very wide and allows us to derive new and optimal decay rate estimates then those ones considered...