Blow-up of solutions of wave equation with a nonlinear boundary condition and interior focusing source of variable order of growth

In this paper, being investigated an initial-boundary value problem for a one-dimensional wave equation with a nonlinear source of variable order and nonlinear dissipation at the boundary. The existence of a local solution of the problem under consideration is proved. Then the question of the absence of global solutions is investigated. Depending on the relationship between the order of growth of the nonlinear source and the nonlinear boundary dissipation, different results are obtained on the blow-up of weak solutions in a finite time interval.     

Global existence and nonexistence for a nonlinear wave equation with damping and source terms

In this paper we consider a nonlinear wave equation with damping and source term on the whole space. For linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. The criteria to guarantee blowup of solutions with positive initial energy are established both for linear and nonlinear damping cases. Global existence and large time behavior also are discussed in this work. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blow-up for wave equation with weak boundary damping and source terms

In this paper, we consider the wave equation with nonlinear boundary damping and source terms. This work is devoted to prove a finite time blow-up result under suitable condition on the initial data and positive initial energy. The main goal of the present paper is to generalize our previous result in Ha (2012) treating the boundary damping term in a more general setting.     

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].     

Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term

The wave equation with a source term is considered

     

Singularity formation for the inhomogeneous nonlinear wave equation with damping

It is well known that the damping term will give more smooth effect to obtain global solutions. In this paper, we consider the effect of damping term on the solutions to system of inhomogeneous wave equation with damping term. We can obtain the singularity that will be formed in finite time for some large initial data. Copyright © 2016 John Wiley & Sons, Ltd.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Global solutions for a nonlinear wave equation

In this work the existence of a global solution for the mixed problem associated to the nonlinear equation

is proved in a Hilbert space framework by using Galerkin method.     

Decay rate for a class of viscoelasticity equation with nonlinear damping on the boundary

In this paper we study the equation of viscoelasticity

utt−uxxt−Fx(ux)=f(x,t)     

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.     

A note on the instability of a focusing nonlinear damped wave equation

We investigate the initial value problem for a nonlinear damped wave equation in two space dimensions. We prove local well‐posedness and instability by blow‐up of the standing waves. Copyright © 2014 John Wiley & Sons, Ltd.     

On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions

We study the global existence of solutions of the nonlinear degenerate wave equation (ρ?0)

     

Strong solutions for semilinear wave equations with damping and source terms

This article deals with local existence of strong solutions for semilinear wave equations with power-like interior damping and source terms. A long-standing restriction on the range of exponents for the two nonlinearities governs the literature on wellposedness of weak solutions of finite energy. We show that this restriction may be eliminated for the existence of higher regularity solutions by employing natural methods that use the physics of the problem. This approach applies to the Cauchy problem posed on the entire ? ⁿ as well as for initial boundary problems with homogeneous Dirichlet boundary conditions.     

Blow-up of solutions for a viscoelastic equation with nonlinear damping

The initial boundary value problem for a viscoelastic equation with nonlinear damping in a bounded domain is considered. By modifying the method, which is put forward by Li, Tasi and Vitillaro, we sententiously proved that, under certain conditions, any solution blows up in finite time. The estimates of the life-span of solutions are also given. We generalize some earlier results concerning this equation.      

Blow-up for semilinear wave equation with boundary damping and source terms

In this paper, we consider the semilinear wave equation with boundary damping and source terms. This work is devoted to prove a finite time blow-up result under suitable condition on the initial data and positive initial energy.     

Local energy decay for the wave equation with a nonlinear time-dependent damping

This article addresses a wave equation on a exterior domain in ? ^d (d odd) with nonlinear time-dependent dissipation. Under a microlocal geometric condition we prove that the decay rates of the local energy functional are obtained by solving a nonlinear non-autonomous differential equation