Global nonexistence of solutions with positive initial energy for a class of wave equations

In this study, we consider a class of wave equations with strong damping and source terms associated with initial and Dirichlet boundary conditions. We establish a blow up result for certain solutions with nonpositive initial energy as well as positive initial energy. This further improves the results by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833) and Messaudi and Houari (Math. Meth. Appl. Sci. 2004; 27 : 1687–1696). Copyright © 2008 John Wiley & Sons, Ltd.     

一类拟线性椭圆方程整体解的不存在性

这篇文章主要利用常微分技术讨论了一个二阶拟线性椭圆方程Lpu≡div（｜Du｜^p-2Du）=f（x,u）,x∈R^N的整体解的不存在性.我们只考虑2≤p〈N的情況并且在假设f（x,u）关于第二个变量u没有单调性的情况下得到整体解的不存在性结果.     

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 existence and nonexistence for degenerate parabolic systems

The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system

in the cylinder , where is bounded and are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by the first Dirichlet eigenvalue for the Laplacian on . We prove that there exists a global solution iff .

     

一个耦合系统自由边界问题的整体古典解

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｄｉｓｃｕｓｓｔｈｅｇｌｏｂａｌｃｌａｓｓｉｃａｌｓｏｌｕｔｉｏｎｏｆａｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｑｕａｓｉｓｔａｔｉｏｎａｒｙｐｒｏｂｌｅｍ .Ｔｈｅｐｒｏｂｌｅｍｃｏｍｅｓｆｒｏｍｔｈｅｄｉｓｃｕｓｓｉｏｎｏｆａｇｒｏｗｔｈｍｏｄｅｌｏｆｓｅｌｆｍａｉｎｔａｉｎｉｎｇｐｒｏｔｏｃｅｌｌ(ｓｅｅ [1— 3])ｉｎｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｃａｓｅ .Ｔｈｅｐｒｏｔｏｃｅｌｌｃａｎｂｅｖｉｓｕａｌｉｚｅｄａｓｈａｖｉｎｇａｐｏｒｏｕｓｓｔ…     

Global existence and nonexistence for some degenerate and quasilinear parabolic systems

The author discusses the degenerate and quasilinear parabolic system

     

Global existence and nonexistence of generalized coupled reaction‐diffusion systems

In this paper, we consider the initial boundary value problem for a class of reaction‐diffusion systems with generalized coupled source terms. The assumption on the coupled source terms refers to the single equations and includes many kinds of polynomial growth cases. Under this assumption, the reaction‐diffusion systems have a variational structure, which is the foundation of constructing the potential wells to classify the initial data. In subcritical energy level and critical energy level, which are divided from potential well theory, the global existence solution, blow‐up in finite time solution, and asymptotic behavior of solution are obtained, respectively. Furthermore, we show the sufficient conditions of global well posedness with supercritical energy level by combining with comparison principle and semigroup theory.     

Global existence and nonexistence results for a class of semilinear hyperbolic systems

In this paper, we investigate the Cauchy problem for a class of the system of semilinear hyperbolic equations with damping. With the use of the L_p → L_q type estimation for the corresponding linear problem and the method of comparison of functional, the existence and nonexistence criteria of global solutions are found. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and nonexistence of global positive solutions for a weakly coupled P-Laplacian system

In this paper, we deal with a weakly coupled evolution P-Laplacian system with inhomogeneous terms. We obtain a critical criterion concerning existence and nonexistence of its global positive solutions. Such a criterion is different from that of the weakly coupled evolution P-Laplacian system with homogeneous terms. Further, we demonstrate existence and nonexistence of its global positive solutions.     

Non-existence of positive solution to a quasilinear elliptic system and blow-up estimates for a reaction-diffusion system

IntroductionIn these years, the reaction-diffusion systems of Fujita typea5 ttell as the related elliptic systemt'ith fl g RN, Tnl. n1 3 0, i = 1, 2. \vere studied by' a nu111ber of authors. The probiemsconcerning system (l) inc1ude globa1 existence and g1obal existen(.e numbers. b1ow-up. bloxv-uprates, and blow-up sets. uniqueness or nonuniqueness. et('. FOr s}'stem (2) there are problemssuch as existence or non-existence. uniqueness or nonllniqueI1ess. and so ol1. 1Ieanwhile. itseems that…     

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Existence and Nonexistence of Global Smooth Solutions for Damped p-system with "really Large" Initial Data

In this paper, we extend the results in [4] to general p(v), and remove the condition (A) for existence, but keep it for nonexistence.     

Non-existence of positive radial solutions for a class of quasilinear elliptic system

In this paper, our main purpose is to establish some nonexistence results of positive radial solutions to the quasilinear ordinary differential equation system. The main results of the present paper are new and extend the previously known results.     

Global attractors for the coupled suspension bridge system with temperature

This paper deals with the long‐term properties of the thermoelastic nonlinear string‐beam system related to the well‐known Lazer–McKenna suspension bridge model (0.1) In particular, no mechanical dissipation occurs in the equations, because the loss of energy is entirely due to thermal effects. The existence of regular global attractors for the associated solution semigroup is proved (without resorting to a bootstrap argument) for time‐independent supplies f,g,h and any . Copyright © 2015 John Wiley & Sons, Ltd.     

General decay for a quasilinear system of viscoelastic equations with nonlinear damping

In this paper, we consider a system of coupled quasilinear viscoelastic equations with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.     

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions. Received December 21, 2000; accepted August 30, 2001.     

A finite‐time blow‐up result for a class of solutions with positive initial energy for coupled system of heat equations with memories

In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite‐time blow‐up result for a large class of solutions with positive initial energy.     

Global exact boundary controllability for 1‐D quasilinear wave equations

Based on the local exact boundary controllability for 1‐D quasilinear wave equations, the global exact boundary controllability for 1‐D quasilinear wave equations in a neighborbood of any connected set of constant equilibria is obtained by an extension method. Similar results are also given for a kind of general 1‐D quasilinear hyperbolic equations. Copyright © 2010 John Wiley & Sons, Ltd.