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In this work we study a wave equation with a nonlinear time dependent frictional damping of variable exponent type. The existence and uniqueness results are established using Fadeo-Galerkin approximation method. We also exploit the Komornik lemma to prove the uniform stability result for the energy associated to the solution of the problem under consideration.

We consider a Timoshenko system coupled with heat equations modelled by Cattaneo's law. The coupling is through the transverse displacement. Both ends of the beam are dynamic. One end of the beam i...

In this paper, we consider a Cauchy problem of a coupled linearly-damped wave equations with nonlinear sources. In the whole space, we establish the local existence and show that there are solutions with negative initial energy that blow up in a finite time. Moreover, under some conditions on the initial data, we estimate a lower bound of that time.

In this paper we consider wave equation with logarithmic nonlinearity and distributive internal delay. We establish the local existence result using the semigroup theory whereas the global existence by the well-depth method. Finally, under suitable assumptions on the weight of the delay and that of frictional damping, we prove that the system is exponentially stable.

In this paper, we study a beam fixed to a base in a translational motion at one end and to a tip mass at its free end. The beam is subject to some vibrations and it is modelled as a Timoshenko stru...

The following abstract problem: $$\begin{aligned} u_{tt}(t)+A u(t)-\int _{0}^{t}g(t-s)A u(s)ds+\eta (t)h(u_t)=\nabla F (u(t)) \end{aligned}$$ of second order evolution equation with time-dependent dissipation is considered. We prove, under a very general and wide class of kernel, a decay result. We establish optimal explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our result improves and generalizes many other results in the literature.

In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.

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 ini...

This work is concerned with a nonlinear system of two coupled viscoelastic equations. Under some restrictions on the coupled functions, we establish, for a wider class of relaxation functions, a generalized stability result for this system. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl Anal 87(3): 247–263, 2008) and Liu (Non-linear Anal 71, 2257–2267, 2009).

In this paper, we consider a wave equation with past history and a nonlinear source term. We minimize the conditions imposed on the relaxation function g by assuming that g satisfies g′(t) ≤ −ξ(t)G(g(t)), where the two functions ξ and G satisfy some conditions. Using this minimal condition on the relaxation function g, we establish the local and global existence results and general decay rate results. This assumption recovers the exponential and polynomial rates when G(s) = sp and p covers the full admissible range 1,2. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature. In fact, our results generalize, extend, and improve many earlier results in the literature.

In this paper, we consider a linear Cauchy viscoelastic problem with an external source term. We find the critical weaker conditions on the source \({f(x,t)}\) needed to show that, for any compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial.

In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with delay. Under suitable conditions on the initial data and the relaxation function, in the whole space, we prove a finite-time blow-up result.

A viscoelastic Cauchy problem subjected to a nonlinear source term is investigated. The memory term in the system involves a kernel which is regular, as is usually the case, but the system is not dissipative and is considered in the whole space. We prove global existence and nonexistence results. In the case of global existence, we show that solutions go to zero in a polynomial manner as time goes to infinity under some conditions on the source.

In this paper, we consider a system of nonlinear viscoelastic wave equations. We shall show that the global nonexistence results for this system found in the literature can be extended to a bigger region.

A viscoelastic Cauchy problem subjected to a nonlinear source term is investigated. The memory term in the system involves a kernel which is not regular as is usually the case. It is proved that solutions go to zero in a polynomial manner as time goes to infinity.

In this paper, we consider a nonlinear wave equation with delay. We show that under suitable conditions on the initial data, the weights of the damping, the delay term and the nonlinear source, the energy of solutions blows up in a finite time.

In this paper we consider a Cauchy problem for a nonlinear viscoelastic equation with strong damping and dispersive terms. Under certain conditions on the initial data and the relaxation function, we prove a finite-time blow-up result.

In this paper we consider a system of nonlinear viscoelastic wave equations. Under arbitrary positive initial energy and standard conditions on the relaxation functions, we prove a finite-time blow-up result.

In this paper we consider a viscoelastic plate equation with distributed delay and source term. Under suitable conditions on the delay and source term, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a wider class of relaxation functions and improves earlier results in the literature.

In this paper we consider a second-order abstract evolution system with delay. We show that, under suitable conditions on the initial data and the involved functionals, the energy of solutions blows up in finite-time. We also give some concrete examples to illustrate our main result.

In this paper we consider a linear Cauchy viscoelastic problem with an external source term. We show that, for compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial. The finite-speed propagation is used to compensate for the lack of Poincare’s inequality in \({\mathbb{R}^{n}}\).

In this paper we consider a one-dimensional linear thermoelastic system of Timoshenko type, where the heat conduction is given by Green and Naghdi theories. We prove a general decay result, from which the exponential and polynomial decays are only special cases.

In this paper we consider a linear Cauchy viscoelastic problem. We show that, for compactly supported initial data and for an exponentially decaying relaxation function, the decay of the first energy of solution is polynomial. The finite-speed propagation is used to compensate for the lack of Poincare's inequality in R^n.

In this work we consider a Cauchy problem for a nonlinear viscoelastic equation. Under suitable conditions on the initial data and the relaxation function, we prove a finite-time blow-up result.

In this paper, we consider a one-dimensional linear Bresse system with infinite memories acting in the three equations of the system. We establish well-posedness and asymptotic stability results for the system under some conditions imposed into the relaxation functions regardless to the speeds of wave propagations. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with distributive delay, where the heat conduction is given by Green and Naghdi theories. We establish the stability of the system for the case of equal and nonequal speeds of wave propagation.

In this paper we consider a linear Cauchy viscoelastic problem. We show that, for compactly supported initial data and for not necessarily decreasing (oscillating) relaxation function, the decay of the first energy of solutions is polynomial. The proof relies on some appropriately chosen functionals. The combination of which satisfies a certain integral inequality.

In this paper we consider a Cauchy problem for a viscoelastic wave equation. Under suitable conditions on the initial data and the relaxation function, we prove a polynomial decay result of solutions. Conditions used, here, on the relaxation function g and its derivative g ′ are different from the usual ones.

In this article we consider the decay of solutions to a linear Cauchy viscoelastic problem with density. This study includes the exponential and polynomial rates as particular cases. To compensate for the lack of Poincare's inequality in the whole space, we consider the solutions in spaces weighted by the density.

In this article we consider a Cauchy problem of a nonlinear viscoelastic equation of order four. Under suitable conditions on the initial data and the relaxation function, we prove polynomial and logarithmic decay of solutions.