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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 with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples. [ABSTRACT FROM AUTHOR]

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]

The main goal of this work is to investigate the following weakly damped nonlinear suspensionbridge equation utt(x, y, t) + ∆2u(x, y, t) + α(t)g(ut) = 0, and establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions α and g and obtained without any restriction growth assumption on g at the origin. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen’s inequality and the generalized Young inequality are used to establish the stability results. [ABSTRACT FROM AUTHOR]

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 coefficients. We also illustrate our theoretical results by performing some numerical tests. [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 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 conservative Lax–Wendroff method scheme. [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 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 special cases. We also give some illustrative examples. [ABSTRACT FROM AUTHOR]

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 special cases. We also give some illustrative examples. [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]

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) . By constructing appropriate Lyapunov functionals in the Fourier space, we establish a general decay estimate of the solution under the condition β - γ α - ϱ 2 > 0. We then give the decay rate of the L 2 -norm of the solution. We also give two examples to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

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 findings. [ABSTRACT FROM AUTHOR]

We study the following wave equation 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 α (t) . We use the multiplier method to establish energy decay results depending on both m and α . We also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme. [ABSTRACT FROM AUTHOR]

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 extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type. [ABSTRACT FROM AUTHOR]

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 in the porous elastic system, we make the system free of the damaging consequences of the second spectrum. We consider an elastic dissipation and prove the well–posedness by the use of Hille–Yosida therorem and some arguments of elliptic equations. Next, we use the multiplier method and establish an exponential decay result without any restriction on coefficients of the system. [ABSTRACT FROM AUTHOR]

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 results. [ABSTRACT FROM AUTHOR]

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 space, combined with time-stepping finite differences. [ABSTRACT FROM AUTHOR]

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 certain solutions with positive initial energy blow up in finite time. We also give some numerical applications to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider a wave equation with a nonlinear source term in addition to a frictional damping subjected to constant time delay. We establish the local existence of a unique solution and then show that there are solutions with positive initial energy that blow up in a finite time. [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 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-speeds and the case of non-equal-speeds of propagation. We also give some numerical illustrations and related comparisons. [ABSTRACT FROM AUTHOR]

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. [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]

In this paper, we consider a one-dimensional porous thermoelastic system with second sound and nonlinear feedback. We show the well-posedness, using the semigroup theory, and establish an explicit and general decay rate result, using some properties of convex functions and the multiplier method. Our result is obtained without imposing any restrictive growth assumption on the damping term. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the following nonlinear waveequation with variable exponents: u t t − div (| ∇ u | r (⋅) − 2 ∇ u) + | u t | m (⋅) − 2 u t = 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 results. [ABSTRACT FROM AUTHOR]

The aim of this article is to establish the well-posedness as well as an exponential stability result for the standard linear solid vibrating systems of thermoelasticity type III. Numerical experiments using finite differences are given to confirm our analytical result. [ABSTRACT FROM AUTHOR]

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 related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs. [ABSTRACT FROM AUTHOR]

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 provided that the stability function $$\chi _{r}(x)=0$$ . Otherwise, we show that the solution decays polynomially. [ABSTRACT FROM AUTHOR]

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

In this paper we consider a system of Timoshenko-thermoelasticity with second sound in the presence of an infinite history term. We prove, under suitable conditions on the coefficients and the relaxation function, a general decay result from which the exponential and polynomial decay estimates are only special cases. We also establish a lack of exponential decay result and discuss the optimality for some situation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear wave equation with variable exponents: u t t − Δ u + a u t | u t | m ( ⋅ ) − 2 = b u | u | p ( ⋅ ) − 2 , where a , b are positive constants. By using the Faedo–Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents m and p . We also prove the finite time blow-up of solutions and give a two-dimension numerical example to illustrate the blow up result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a plate equation with a logarithmic nonlinearity in the presence of a frictional damping. We prove, for a suitably chosen initial data, a global existence as well as a decay result. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following wave equation: in whichis a nonnegative constant and the exponents of nonlinearityare given functions. For arbitrary positive initial energy, we prove a finite-time blow-up result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a Timoshenko-type system of thermoelasticity of type III in the presence of a distributed delay. We prove the well-posedness and two exponential stability results in the presence as well as in the absence of an extra frictional damping. In case of absence of the frictional damping, our stability result is obtained under the equal-speed of propagation and a smallness condition on the weight of delay. [ABSTRACT FROM AUTHOR]

In this paper we establish a general decay rate for a quasilinear viscoelastic problem where the relaxation function satisfies g ′ ( t ) ≤ − ξ ( t ) g p ( t ) , t ≥ 0 , 1 ≤ p < 3 2 . This work generalizes and improves earlier results in the literature. [ABSTRACT FROM AUTHOR]

Recently, Ferrero and Gazzola, [Disc. Cont. Dyn. Syst. 35: 5879-5908 (2015)], suggested and investigated a rectangular plate model describing the statics and dynamics of a suspension bridge. The plate is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. This reliable model aims to describe more accurately the motion of suspension bridges compared to all previous known models. In the present paper, we consider a plate equation in the presence of memory and subject to the above-mentioned boundary conditions. We give a rigorous well-posedness result and establish the existence of a global attractor. [ABSTRACT FROM AUTHOR]

In this paper we investigate a porous thermoelastic system where the heat conduction is given by Cattaneo’s law and where the energy associated with the solution is not necessary positive definite ( μ ξ = b 2 ). We introduce a stability number χ and prove an exponential (resp. polynomial) decay result depending on χ . [ABSTRACT FROM AUTHOR]

In this paper, we consider a wave equation with a strong damping and a strong constant (respectively, distributed) delay. We prove the well-posedness and establish an exponential decay result under a suitable assumption on the weight of the damping and the weight of the delay. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a semilinear problem describing the downward displacement of a suspension bridge in the presence of a hanger restoring force h ( u ) , an external force f which includes gravity, and a delay term which accounts for its history. The goal is to establish a well-posedness result and the existence of a finite-dimensional global attractor. [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 article we study the fourth-order viscoelastic plate equation ... in the bounded domain ... with non traditional boundary conditions. We establish the well-posedness and a decay result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi's theory. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and in the presence of infinite memory term. In the domain as well as on a part of the boundary, we use the multiplier method and some properties of the convex functions to prove an explicit and general decay result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay, where the heat flux is given by Cattaneo's law. We prove an exponential decay result under a smallness condition on the delay and a stability number introduced first in Santos et al. (J Diff Eqs 253:2715-2733, ), using a method different from that of Santos et al. (J Diff Eqs 253:2715-2733, ). We also reproduce the polynomial decay of Santos et al. (J Diff Eqs 253:2715-2733, ) using the multiplier method in the case of absence of delay. The polynomial decay issue in the presence of a small delay is an open question. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of two wave equations with nonlinear damping and source terms acting in both equations in the presence of infinite-memory terms and prove an explicit and general decay result. Our approach allows a wide class of kernels, among which those of exponential decay type are only special cases. [ABSTRACT FROM AUTHOR]