Total versus single point blow-up in a localized heat system

This paper considers a heat system with localized sources and local couplings subject to null Dirichlet boundary conditions, for which both total and single point blow-up are possible. The aim of the paper is to identify the total and single point blow-up via a complete classification for all the nonlinear parameters in the model. As preliminaries of the paper, simultaneous versus non-simultaneous blow-up of solutions is involved, too. The results are then compared with those for another kind of heat system coupled via localized sources in a previous paper of the authors.     

A Quasilinear Parabolic System with Nonlocal Boundary Conditions and Localized Sources

This paper investigates the properties of solutions to a quasilinear parabolic system with nonlocal boundary conditions and localized sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and blow-up rate estimates are also derived.     

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.     

一类具非局部源抛物型方程组解的一致爆破性质

考虑带有齐次Dirichlet边界条件,具非局部源项的半线性抛物型方程组正解的爆破性质,首先给出了该问题的解在有限时刻爆破的充分条件,以及解的两个分量同时爆破的必要条件,并建立了解的一致爆破模式.     

耦合热方程组解的多重 Blow-up 速率

This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. Based on a complete classification for all the four nonlinear parameters, we establish multiple blow-up rates for the system under various dominations. We also determine uniform blow-up profiles for the three cases where localized source couplings dominate the system.     

UNIFORM BLOW-UP PROFILES FOR HEAT EQUATIONS WITH COUPLING NONLOCAL SOURCES OF ASYMMETRIC MIXED TYPE NONLINEARITIES＊

This article deals with a nonlocal heat system subject to null Dirichlet boundary conditions,where the coupling nonlocal sources consist of mixed type asymmetric nonlinearities.We at first give the cri...     

Blow-Up Behaviors of Solutions to Reaction-Diffusion Equations With Nonlocal Sources and Variable Exponents北大核心CSCD

该文考虑了一类带有变指数非局部项的反应扩散方程的爆破问题。首先,由不动点原理,证明了问题解的局部存在性和唯一性。其次,利用上下解方法,给出在齐次Dirichlet边界条件下,问题的解在有限时间发生爆破的充分条件,即变指数大于零且初始值足够大,并对爆破时间的上下界进行了估计。     

Non-simultaneous blow-up in localized reaction–diffusion equations

This article deals with blow-up solutions in reaction–diffusion equations coupled via localized exponential sources, subject to null Dirichlet conditions. The optimal and complete classification is obtained for simultaneous and non-simultaneous blow-up solutions. Moreover, blow-up rates and blow-up sets are also discussed. It is interesting that, in some exponent regions, blow-up phenomena depend sensitively on the choosing of initial data, and the localized nonlinearities play important roles in the blow-up properties of solutions.     

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

This paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball.     

一类具有Dirichlet边界条件的反应扩散方程的爆破解

针对一类具有Dirichlet边界条件的非线性反应扩散方程的爆破问题,通过构造恰当的辅助函数和利用一阶微分不等式技术,给出了解在有限时刻爆破的一个充分条件,并在一定条件下得到了爆破时刻的上界和下界.     

A degenerate parabolic system with localized sources and nonlocal boundary condition

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.     

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.     

具有非局部源与加权非局部边界条件的拟线性抛物方程组

In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In addition, the blow-up rate estimate of non-global solutions for a class of weight functions is also obtained, which is found to be independent of nonlinear diffusion parameters m and n.     

一类具有时变系数梯度源项的弱耦合反应-扩散方程组解的爆破分析

该文研究了具有时变系数梯度项的弱耦合反应-扩散方程组齐次Dirichlet初边值问题解的爆破现象.结合修正微分不等式技巧及比较原理,得到了在若干个不同测度意义下解的整体存在性与有限时刻发生爆破的充分条件,并在高维空间中导出了爆破解的爆破时间界的估计.     

Uniform Blow-up Behavior for Degenerate and Singular Parab olic Equations

This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blow-up exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases.     

Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

A lower bound for the blow-up time to a damped semilinear wave equation

This paper studies the blow-up property of weak solutions to an initial and boundary value problem for a nonlinear viscoelastic hyperbolic equation with nonlinear sources. A lower bound for the blow-up time is given.     

Blow-up properties for heat equations coupled via different nonlinearities

This paper deals with heat equations coupled via exponential and power nonlinearities, subject to null Dirichlet boundary conditions. The complete and optimal classification on non-simultaneous and simultaneous blow-ups is proposed by four sufficient and necessary conditions. We find out that, in some exponent region, the blow-up properties of the solutions depend much on the choosing of initial data. Moreover, all kinds of non-simultaneous and simultaneous blow-up rates are obtained.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.