A porous medium equation with nonlocal boundary condition and a localized source

This article studies the blow-up properties of solutions to a porous medium equation with nonlocal boundary condition and a general localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, it is proved that the unique solution has global blow-up property whenever blow-up occurs. Blow-up rate estimates are also obtained for some special cases.     

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.     

A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u _t = f(u)(Δu + a∫_Ω u(x, t)dx ? u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s ^p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.     

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary (?)Ω. In addition, for a bounded Lipschitz domainΩ, we establish the blow-up rate estimates for the positive solution to this problem with a= 0.     

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.     

Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa–Holm system

We study the Cauchy problem of a weakly dissipative 2-component Camassa–Holm system. We first establish local well-posedness for a weakly dissipative 2-component Camassa–Holm system. We then present a global existence result for strong solutions to the system. We finally obtain several blow-up results and the blow-up rate of strong solutions to the system.     

一类生物捕食模型解的整体存在性和爆破性

研究了一类具有捕食速率的生物捕食模型解整体存在性和爆破性,利用反映扩散方程组的比较原理,通过构造自相似形式的上下解,得到了模型的解的整体存在和爆破条件.     

On the cauchy problem for a family of quasilinear hyperbolic equations

We prove local well-posedness and give some global existence and blow-up criteria for solutions of a family of quasilinear hyperbolic equations arising in shallow water theory.     

两类Chemotaxis模型解的性质

本文讨论了抛物-双曲和抛物-常微两个Chemotaxis模型初边值问题解的性质.利用形式级数展开的方法，得到全局解在微小扰动下，导致解在有限时间内爆破，并对爆破时间进行了估计.因此说明了这种模型空间齐性解的不稳定性。     

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.     

一类具有局部化源和吸收项的抛物系统解的整体存在与爆破

在齐次Dirichlet边界条件研究如下抛物系统其中x_0(t):R⁺→(0,a)是Holder连续函数;常数0≤α,β<1,p_1,p_2,q_1,q_2,k_1,k_2>0.利用正则化方法,在一定的假设条件下证明了经典解的存在性.接着利用比较原理证明了该系统正解的整体存在性和爆破性.最后给出了爆破解的精确爆破速率和爆破模式.     

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.     

Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.     

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.     

具有非线性边界条件半线性热方程组解的爆破性质

本文考虑一类半线性热方程组的解,给出了解爆破的充分必要条件,爆破速率和爆破点的位置。     

Blow-up for a weakly dissipative modified two-component Camassa–Holm system

In this paper, we study the Cauchy problem of a weakly dissipative modified two-component Camassa–Holm (MCH2) system. We first derive the precise blow-up scenario and then give several criteria guaranteeing the blow-up of the solutions. We finally discuss the blow-up rate of the blowing-up solutions.     

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 behavior for a semilinear parabolic equation with localized source

In this paper, we investigate the blow-up behavior of solutions of a parabolic equation with localized reactions. We completely classify blow-up solutions into the total blow-up case and the single point blow-up case, and give the blow-up rates of solutions near the blow-up time which improve or extend previous results of several authors. Our proofs rely on the maximum principle, a variant of the eigenfunction method and an initial data construction method.     

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.