Blow‐up of solution of wave equation with internal and boundary source term and non‐porous viscoelastic acoustic boundary conditions

In this paper we study the blow‐up of solution of a mixed problem associated to a nonlinear wave equation with dissipative and source term in a bounded domain of . On a boundary portion of the domain we consider a non‐porous viscoelastic acoustic boundary conditions to a non‐locally reacting boundary.     

Global existence and uniform decay for a nonlinear viscoelastic equation with damping

In this paper we investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

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)     

Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation

In this paper, we prove the existence and the uniqueness of global solution for the Cauchy problem for the generalized Boussinesq equation. Under some assumptions, we also show that the _L∞$L_{\infty }$ norm of small solution of the Cauchy problem for the generalized Boussinesq equation decays to zero as t$t$ tends to the infinity.     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Global existence,asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.     

Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term

We consider the nonlinear model of the wave equation subject to the following nonlinear boundary conditions We show existence of solutions by means of Faedo-Galerkin method and the uniform decay is obtained by using the multiplier technique. Received: 15 June 2000 / Accepted: 4 December 2000 / Published online: 29 April 2002     

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

Energy decay of variable‐coefficient wave equation with nonlinear acoustic boundary conditions and source term

In this paper, we consider a variable‐coefficient wave equation with nonlinear acoustic boundary conditions and source term. Using the Riemannian geometry method, we prove the general energy decay of the system corresponds to the ordinary differential equation (ODE), which certainly is stable under some suitable assumptions.     

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

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , 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.     

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.     

Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions

This article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term. Both dissipation and the source are fully nonlinear   and the growth rate of the source map may include critical exponents (for Sobolev’s embedding H¹→L²$H^1\to L^2$). Besides continuity and monotonicity, no growth or regularity assumptions are imposed on the damping. We analyze the system in the presence of Neumann-type boundary conditions including the mixed cases: Dirichlet–Neumann–Robin.     

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].     

Time-periodic solutions of a nonlinear wave equation

The existence of a time-periodic solution of an n-dimensional nonlinear wave equation is established with n=2 and 3.     

Blow-up of solutions for a nonlinear beam equation with fractional feedback

A nonlinear beam equation describing the transversal vibrations of a beam with boundary feedback is considered. The boundary feedback involves a fractional derivative. We discuss the asymptotic behavior of solutions. In fact, we prove that solutions blow up in finite time under certain assumptions on the nonlinearity.     

Global existence,blow up and asymptotic behaviour of solutions for nonlinear Klein–Gordon equation with dissipative term

We study the Cauchy problem of nonlinear Klein–Gordon equation with dissipative term. By introducing a family of potential wells, we derive the invariant sets and prove the global existence, finite time blow up as well as the asymptotic behaviour of solutions. In particular, we show a sharp condition for global existence and finite time blow up of solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

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.