General decay result for nonlinear viscoelastic equations

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the $L^1(0,\infty )$ relaxation function g namely $g^{\prime }(t)\le ?\xi (t)H(g(t))$, where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when $H(s)=s^p$ and p covers the full admissible range $[1,2)$. We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature.     

General decay of solutions in one-dimensional porous-elastic system with memory

In this work we consider a one-dimensional porous-elastic system with memory effects. It is well-known that porous-elastic system with a single dissipation mechanism lacks exponential decay. In contrary, we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system, depending on the kernel of the memory term and the wave speeds of the system. In fact, we prove a general decay result, for which exponential and polynomial decay results are special cases. Our result is new and improves previous results in the literature.     

General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type

In this paper we consider linear porous-thermoelasticity systems, in a bounded domain, where the memory-type damping is acting on a part of the boundary. We establish a general decay result, for which the usual exponential and polynomial decay rates are just special cases. Our work allows certain relaxation functions which are not necessarily of exponential or polynomial decay and, therefore, generalizes and improves on earlier results from the literature.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback

In this paper, we consider a degenerate viscoelastic Petrovsky-type plate equation \[ K(\mbox{\boldmath $x$})u_{tt}+\Delta^2u-\int_0^tg(t-s)\Delta^2u(s)ds+f(u)=0 \] with boundary feedback. Under the weaker assumption on the relaxation function, the general energy decay is proved by priori estimates and analysis of Lyapunov-like functional. The exponential decay result and polynomial decay result in some literature are special cases of this paper.     

Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations

In this paper, we consider a nonlinear system of two coupled viscoelastic equations which describes the interaction between two different fields arising in viscoelasticity. We prove the well-posedness and, for a wider class of relaxation functions, establish a generalized stability result for this system.     

Exponential energy decay of solutions of viscoelastic wave equations

In this paper we consider the nonlinear viscoelastic equation

     

General decay for a class of viscoelastic problem with not necessarily decreasing kernel

In this paper, a problem which arises in a class of viscoelasticity is considered. We obtain the decay rate of the energy, for certain class of relaxation functions not necessarily exponentially or polynomially decaying to zero.     

On the uniform decay in viscoelastic problems in

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 Poincaré’s inequality in .     

General decay of energy for a viscoelastic equation with nonlinear damping

In this paper we are concerned with a nonlinear viscoelastic equation with nonlinear damping. The general uniform decay of the energy is obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

A general decay result of a viscoelastic equation with infinite history and nonlinear damping

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and in the presence of an infinite-memory term. We prove an explicit and general decay result using some properties of the convex functions. Our approach allows a wider class of kernels, from which those of exponential decay type are only special cases.     

General energy decay in a Timoshenko-type system of thermoelasticity of type III with a viscoelastic damping

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.     

Exponential decay for a quasilinear viscoelastic equation

We consider the following nonlinear viscoelastic equation together with Dirichlet-boundary conditions, in a bounded domain Ω and ρ > 0. We prove an exponential decay result for a class of relaxation functions. Our result is established without imposing the usual relation between g and its derivative (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal decay rates for the viscoelastic wave equation

In this paper, we consider a viscoelastic equation with minimal conditions on the relaxation function g, namely, , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of gat infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=s^p and p covers the full admissible range [1,2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.     

General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity

This work deals with the study of a new class of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity. A decay result of the energy of solutions for the problem without imposing the usual relation between a certain relaxation function and its derivative is established. This result generalizes earlier ones to an arbitrary rate of decay, which is not necessarily of exponential or polynomial decay.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions

In this paper, we consider the weak viscoelastic equation $$u_{tt} - \Delta u + \alpha(t) \int\limits_{0}^{t} g(t-s)\Delta u(s)\, {\rm d}s=0$$ with a homogeneous Dirichlet condition on a portion of the boundary and acoustic boundary conditions on the rest of the boundary. We establish a general decay result, which depends on the behavior of both α and g, by using the perturbed energy functional technique. This is an extension and improvement of the previous result from Park and Park (Nonlinear Anal 74(3):993–998, 2011) (i.e., the similar problem with ${\alpha(t) \equiv 1}$ ) to the time-dependent viscoelastic case.