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.     

Global nonexistence for an integro‐differential equation

The initial boundary value problem for an integro‐differential equation with nonlinear damping and source terms in a bounded domain is considered. By modifying the method in a work by Autuori et al. in 2010, we establish the nonexistence result of global solutions with the initial energy controlled by a critical value. This improves earlier results in the literatures. Copyright © 2011 John Wiley & Sons, Ltd.     

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)     

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235-1270], especially, the blow-up of weak solutions in the case of non-negative energy.     

具材料阻尼的非线性双曲型方程整体解的不存在性

考虑了一类具材料阻尼的非线性双曲型方程初边值问题整体解的不存在性．分别采用能量方法、Jensen不等式和凹性方法证明了该问题整体解的不存在性定理．作为主要结果的应用,给出了3个例子．     

具正初始能量的耦合波动方程解的整体不存在

考虑带有非线性阻尼和源项的耦合黏弹性波动方程的初边值问题.假设系统具有正初始能量,在适当的条件下证明了解的整体不存在性定理.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.     

一类具有任意大正初始能量的非线性波动方程整体解的不存在性

研究了一类有限区域上具有阻尼和源项的非线性波动方程的初边值问题,证明了具有任意大正初始能量的整体解的不存在性．     

非线性Sobolev-Galpern方程解的blow up

本文研究非线性Ｓｏｂｏｌｅｖ－Ｃａｌｐｅｒｎ方程的初边值问题整体解的不存性即解的爆破问题,用能量估计方法并借助于Ｊｅｎｓｅｎ不等式证明了非线性Ｓｏｂｏｌｉｖ－Ｇａｌｐｅｒｎ方程各种初边值问题在某些假设下不存在整体解。     

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

0IntroductionInthispaper,weconsidertheinitial-boundaryvalueproblemforthefailliliarequationwherefiisaboundeddomaininR"withsmoothboulldaryoff,p22isacollstantalldlp(z,u)l5of'(tl" 'forsomea20andc>0.FOrp(x,ti)=Itll"'u,theauthorsofpaper[4,7llolwereillterestedillllollllegativesolutionandhadobtainedfollowillgresults(alsosee[12]).1)If25a 2相似文献   

Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms

In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.     

Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

The main goal of this work is to study an initial boundary value problem for a quasilinear parabolic equation with logarithmic source term. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions.     

Global nonexistence of positive initial energy solution for coupled quasilinear system

In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.     

A class of fourth order wave equations with dissipative and nonlinear strain terms

We study the initial boundary value problem for a class of fourth order wave equations with dissipative and nonlinear strain terms. By introducing a family of potential wells we not only obtain the invariant sets and vacuum isolating of solutions, but also give some threshold results of global existence and nonexistence of solutions.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

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.     

Porous elastic system with nonlinear damping and sources terms

We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. We also prove the existence of a global attractor.     

关于一类具阻尼项的非线性波方程的Cauchy问题

The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the above-mentioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.     

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.     

ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING:GLOBAL EXISTENCE,DECAY AND BLOWUP OF THE SOLUTION

In this paper we consider the Elastic membrane equation with memory term and nonlinear boundary damping.Under some appropriate assumptions on the relaxation function h and with certain initial data,the global existence of solutions and a general decay for the energy are established using the multiplier technique.Also,we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.