一类带非线性边界条件的非经典扩散方程解的爆破

In this article,under suitable conditions on the internal term f(·)and the bound-ary nonlinear term g(·),we prove blow-up results for a class of nonclassical diffusion equations with nonlinear boundary condition.     

关于半线性抛物型方程组解的短破率

本文将对某一类半线性抛物型议程组在解的单点爆破情况下，估计当点邻近短爆点时，解的爆破率。     

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.     

Critical Exponents for a System of Heat Equations Coupled in a Non-linear Boundary Condition

In this paper we consider a system of heat equations u_t = Δu, v_t = Δv in an unbounded domain Ω⊂ℝ^N coupled through the Neumann boundary conditions u_v = v^p, v_v = u^q, where p>0, q>0, pq>1 and ν is the exterior unit normal on ∂Ω. It is shown that for several types of domain there exists a critical exponent such that all of positive solutions blow up in a finite time in subcritical case (including the critical case) while there exist positive global solutions in the supercritical case if initial data are small.     

Global Existence and Blow-up for a Parabolic System with Nonlinear Boundary Conditions

This paper deals with the global existence and blow-up of positive solutions to the systems: u_t = ∇(u^∇u) + u¹ + v^a v_t = ∇(v^n∇v) + u^b + v^k in B_R × (0, T) \frac{∂u}{∂η} = u^αv^p, \frac{∂v}{∂η} = u^qv^β on S_R × (0, T) u(x, 0) = u_0(x), v(x, 0} = v_0(x) in B_R We prove that there exists a global classical positive solution if and only if l ≤ l, k ≤ 1, m + α ≤ 1, n + β ≤ 1, pq ≤ (1 - m - α)(1 - n - β),ab ≤ 1, qa ≤ (1 - n - β) and pb ≤ (1 - m - α).     

The Blow-up Rate of Positive Solution of a Parabolic System

This paper deals with the blow-up properties of solutions to a system of semilinear heat equations with the boundary conditions u(x,t)=v(x,t)=0 on SR×[0,T).The exact estimates on blow-up rate of solutions are established.     

具有吸收和非线性边界流的非线性扩散方程的爆破估计

本文研究一类具有非线性吸收和非线性边界流的非线性扩散方程，建立了解的爆破速率估计，所得结果依赖于模型中三种非线性机制之间的相互作用。     

Blow-up Sets to a Coupled Heat System

This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. In a previous paper of the authors, a complete result on the multiple blow-up rates was obtained. In the present paper, we continue to consider the blow-up sets to the system via a complete classification for the nonlinear parameters. That is the discussion on single point versus total blow-up of the solutions. It is mentioned that due to the influence of the localized sources, there is some substantial difficulty to be overcomed there to deal with the single point blow-up of the solutions.     

一类半线性抛物型方程组解的爆破速率估计

考虑一类带有齐次Dirichlet边界条件且反应项分别为指数形式和幂函数形式的半线性抛物型方程组,利用比较原理得到了方程解爆破的充分条件,由数学分析原理和最大值原理得到了爆破解的爆破速率估计.     

一个局部波动方程组的解的全局存在性与爆破

我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p, (e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x), (e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率.     

带有非线性边界条件的非线性抛物型方程组的爆破速率估计

In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered: We will prove that there exist two positive constants such that: where l_1= l_(21)α／α_2 l_(22),r=α_1／α_2＞1,α_1≤α_2＜0.     

Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source

This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source.Conditions for the existence of global or bl...     

Blow-up for Stochastic Reaction-Diffusion Equations with Jumps

In this paper, we focus on stochastic reaction-diffusion equations with jumps. By a new auxiliary function, we investigate non-negative property of the local strong (variational) solutions, which applies to stochastic reaction-diffusion equations with highly nonlinear noise terms. As a byproduct, we study the problem of non-existence of global strong solutions by imposing appropriate conditions on the drift terms, which can cover many more models than the existing literature. Moreover, we also investigate the subject of Lévy-type noise-induced explosion by bringing some plausible assumptions to bear on the noise terms, which, however, need not guarantee local strong (variational) solutions to enjoy the non-negative property. Meanwhile, several examples are presented to illustrate the theory established.     

Critical Quenching Exponents for Heat Equations Coupled with Nonlinear Boundary Flux

We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case the simultaneous quenching can happen for any solution.     

抛物型边界热源方程的爆破解及其上下界

讨论了一类典型的半线性抛物型方程,其在物理上对应具有边界热源的热传导问题,证明了非平凡解发生爆破的充分条件,并讨论了爆破速率之上界与下界.     

具阻尼和源项的非线性粘弹热方程组解的爆破

考虑一个具阻尼和源项的非线性粘弹热方程组的初边值问题.建立一个正初始能量下方程组解的爆破结果.     

On the Blow-up for Quasilinear Parabolic Equations

This article is concerned with the position of blow-up points, blow up rate and an isoperimetric problem for the equation u_t = Δu^m + u^p(p > m ≥ 1) in a convex bounded domain.     

一个带非线性边界条件的强耦合抛物方程组的整体存在性与爆破

本文处理带非线性边界条件 u n=uα, v n=vβ ,(x ,t) ∈ Ω× (0 ,T)的抛物方程组ut =vpΔu ,vt=uqΔv ,(x ,t) ∈Ω× (0 ,T) ,其中Ω RN 为一个有界区域 ,p ,q>0和α ,β≥ 0为常数 .研究了上述问题正解的整体存在性和爆破 ,建立了整体存在和爆破的新标准 .证明了当max{p+β,q+α}≤ 1时正解 (u ,v)整体存在 ,当min{p+β ,q+α}>1且max{α ,β}<1时正解 (u ,v)在有限时刻爆破     

Solution Blow-up in a Nonlinear System of Equations with Positive Energy in Field Theory

A problem for a nonlinear system of electromagnetic equations in the Coulomb calibration with allowance for sources of free-charge currents is considered. The local-in-time solvability in the weak sense of the corresponding initial–boundary value problem is proved by applying the method of a priori estimates in conjunction with the Galerkin method. A modified Levine method is used to prove that, for an arbitrary positive initial energy, under a certain initial condition on the functional \(\Phi (t) = \int\limits_\Omega {|A{|^2}dx} \), where A(x) is a vector potential, the solution of the initial–boundary value problem blows up in finite time. An upper bound for the blow-up time is obtained.