Nonlinear Viscoelastic Wave Equation Research Articles

Global existence and blow‐up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source

AbstractIn this paper we investigate the global existence and finite time blow‐up of solutions to the nonlinear viscoelastic equation associated with initial and Dirichlet boundary conditions. Here ∂j denote the sub‐differential of j. Under suitable assumptions on g(·), j(·) and the parameters in the equation, we obtain the global existence of generalized solutions, weak solutions for the equation. The finite time blow‐up of weak solutions for the equation is also established provided the initial energy is negative and the exponent p is greater than the critical value k + m. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation

In this paper, we consider the nonlinear viscoelastic wave equation u t t − Δ u + ∫ 0 t h ( t − τ ) div [ a ( x ) ∇ u ( τ ) ] d τ + | u | γ u = 0 with boundary dissipation. Under suitable conditions on the initial data and the relaxation function, we prove the existence and uniqueness of its global solution by means of the Galerkin method and show the uniform decay rate of the energy.

On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution

In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function g and with certain initial data, the global existence of solutions and a general decay for the energy have been established.

Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of nonlinear viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We will prove that the energy associated to the system is unbounded. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. The key ingredient in the proof is a method used in Vitillaro (Arch Ration Mech Anal 149:155–182, 1999) and developed in Said-Houari (Diff Integr Equ 23(1–2):79–92, 2010) for a system of wave equations, with necessary modification imposed by the nature of our problem.

Blow-up of solutions of a nonlinear viscoelastic wave equation

In this paper, we consider the nonlinear viscoelastic equation u t t − Δ u + ∫ 0 t g ( t − τ ) Δ u ( τ ) d τ − △ u t = ∣ u ∣ p − 2 u , in Ω × [ 0 , T ] , with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g , we prove that there are solutions with positive initial energy that blow up in finite time.

Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.

Blow up of Solutions of a Nonlinear Viscoelastic Wave Equation

We consider a nonlinear viscoelastic wave equation with nonlinear source term. Under suitable conditions on g, it is proved that any weak solution with negative initial energy blows up in finite time if p>2.

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations u t t − Δ u + ∫ 0 t g 1 ( t − τ ) Δ u ( τ ) d τ + | u t | m − 1 u t = f 1 ( u , v ) , v t t − Δ v + ∫ 0 t g 2 ( t − τ ) Δ v ( τ ) d τ + | v t | r − 1 v t = f 2 ( u , v ) in Ω × ( 0 , T ) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in R n , n = 1 , 2 , 3 . Under suitable assumptions on the functions g i ( ⋅ ) , f i ( ⋅ , ⋅ ) ( i = 1 , 2 ) , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

AbstractIn this paper the nonlinear viscoelastic wave equation in canonical form equation image with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd.

Blow up and global existence in a nonlinear viscoelastic wave equation

AbstractIn this paper the nonlinear viscoelastic wave equation associated with initial and Dirichlet boundary conditions is considered. Under suitable conditions on g, it is proved that any weak solution with negative initial energy blows up in finite time if p > m. Also the case of a stronger damping is considered and it is showed that solutions exist globally for any initial data, in the appropriate space, provided that m ≥ p.