Stabilization of a locally damped incompressible wave equation

We show that the solutions of an incompressible vector wave equation with a locally distributed nonlinear damping decay in an algebraic rate to zero, that is, denoting by E(t) the total energy associated to the system, there exist positive constants C (which depends on E(0)) and γ satisfying, for t?0: E(t)?C(1+t)^−γ.     

Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

We consider the Cauchy problem for the damped wave equation

     

Instability results for the damped wave equation in unbounded domains

We extend some previous results for the damped wave equation in bounded domains in to the unbounded case. In particular, we show that if the damping term is of the form αa with bounded a taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive α.In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrödinger operators to the unbounded case, which we believe to be of interest in their own right.     

Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity

where 0,$"> and space dimensions . Assume that the initial data

where \frac{n}{2},$"> weighted Sobolev spaces are

Also we suppose that

0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">

where

Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property

for all 0,$"> where

     

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form αu^?(t)+u^″(t)=βAu(t)+γAu^′(t)+f(t,u(t)), t?0, satisfying αβ<γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.     

ATTRACTOR FOR WEAKLY DAMPED DRIVEN LONG WAVE-SHORT WAVE RESONANCE EQUATION

The existence of solution and universal attractor for weakly damped driven long wave-short wave resonance equation are obtained.     

The damped wave equation with unbounded damping

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.     

Decay property of solutions for damped wave equations with space-time dependent damping term

We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|^ρ−1u. Here, with b₀>0, α,β?0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L² decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N−α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L²-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,α,β) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.     

Pullback attractors for a nonautonomous damped wave equation with infinite delays in weighted space

In this paper, we investigate the existence of solutions for a damped wave equation with infinite delays in the weighted space and , respectively. In addition, we study the existence of pullback attractor for the process associated to the problem by a direct and simple compactness method.     

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin–Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.     

On the strongly damped wave equation with constraint

A weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is defined. The main idea consists in the use of duality techniques in Sobolev–Bochner spaces, aimed at providing a suitable “relaxation” of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite “physical” energy.     

Local energy decay for the damped Klein-Gordon equation in exterior domain

We prove uniform local energy decay for the solution of the dissipative Klein–Gordon equation on an exterior domain under some geometric condition called “exterior geometric control”.     

Random attractor of the stochastic strongly damped wave equation

In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with homogeneous Neumann boundary condition. We investigate the existence of a random attractor for the random dynamical system associated with the equation.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation

The existence and uniqueness of the local generalized solution to the initial boundary value problem for the three-dimensional damped nonlinear hyperbolic equation