Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation

In this paper, we consider the nonlinear viscoelastic equation

with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g and for p>m, we prove that there are solutions with positive initial energy that blow up in finite time.     

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.     

Exponential decay of non‐linear wave equation with a viscoelastic boundary condition

We study in this paper the global existence and exponential decay of solutions of the non‐linear unidimensional wave equation with a viscoelastic boundary condition. We prove that the dissipation induced by the memory effect is strong enough to secure global estimates, which allow us to show existence of global smooth solution for small initial data. We also prove that the solution decays exponentially provided the resolvent kernel of the relaxation function, k decays exponentially. When k decays polynomially, the solution decays polynomially and with the same rate. Copyright © 2000 John Wiley & Sons, Ltd.     

General decay for a differential inclusion of Kirchhoff type with a memory condition at the boundary

In this article, we consider a differential inclusion of Kirchhoff type with a memory condition at the boundary. We prove the asymptotic behavior of the corresponding solutions. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.     

General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping

In this paper we consider the following Timoshenko system: with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ρ₁, ρ₂, k₁, k₂ and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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 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.     

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.     

A blow-up result in a Cauchy viscoelastic problem

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.     

Uniform decay in a Timoshenko-type system with past history

Fernández Sare and Rivera [H.D. Fernández Sare, J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502] considered the following Timoshenko-type system

ρ₁φ_tt−K(φ_x+ψ)_x=0,