具非局部源反应扩散方程组解的同时爆破与不同时爆破

本文讨论具齐次Dirichlet边界条件非局部源反应扩散方程组的爆破解,给出四类同时爆破与不同时爆破现象的判定指标,这四类爆破现象包含:(i)存在不同时爆破;(ii)同时爆破与不同时爆破共存;(iii)任意爆破必是同时爆破;(iv)任意爆破必是不同时爆破.     

具有非线性吸收项和边界流的m,p-Laplacian抛物方程研究（英文）

本文研究了一类在■中的m,p-Laplacian抛物方程(p2,m1),其具有非线性内部吸收项(-λu~κ)和非线性边界流u~q.当qq~*时,任意解都是整体存在的.当qq~*时,根据初值的选取,爆破解和整体解都可能存在.在临界情形q=q~*,吸收项系数的大小在决定解的整体存在和爆破现象方面发挥决定性作用.当κ≤1时,所有解整体存在.当1κm(p-1)+p时,对于任意初值,大的λ可以导致解发生有限时刻爆破,即Fujita爆破,而小的λ可以导致解整体存在.而且,我们给出了系数大小的定量估计.当κm(p-1)+p时,爆破解和整体解都是可以存在的.     

一类带调和势的非线性Schr(O)dinger方程在RN中的整体解存在的门槛

本文考虑一类带调和势的非线性Schrodinger方程iψt=-△ψ+|x|2ψ-μ|ψ|p-1ψ-λ|ψ|q-1ψ,x∈RN,t≥0,其中μ＞0,λ＞0.当N=1,2时,1＜p＜q＜∞;当N≥3时,1＜p＜q＜N+2/N-2.运用精巧的变分方法、势井方法和凸方法,得到了方程的整体解和爆破解存在的门槛.进一步回答了:当q＞p＞1+4/N时,方程的Cauchy问题的初值小到什么程度,其整体解存在?.     

一类带调和势的非线性Schrödinger方程在RN中的整体解存在的门槛

本文考虑一类带调和势的非线性Schrodinger方程iψt=-△ψ+|x|2ψ-μ|ψ|p-1ψ-λ|ψ|q-1ψ,x∈RN,t≥0,其中μ＞0,λ＞0.当N=1,2时,1＜p＜q＜∞;当N≥3时,1＜p＜q＜N+2/N-2.运用精巧的变分方法、势井方法和凸方法,得到了方程的整体解和爆破解存在的门槛.进一步回答了当q＞p＞1+4/N时,方程的Cauchy问题的初值小到什么程度,其整体解存在?.     

一类非线性抛物方程组解的爆破时间上下界估计

本文研究了一类非线性抛物方程组uj/t=△uj+fj(u)解的爆破时间的估计问题.通过构造恰当的辅助函数和建立一系列微分不等式,获得了此类非线性抛物方程组解的爆破时间上下界的估计.从而将单个方程的结论推广到了方程组的情形.     

带有梯度项的退化抛物方程的爆破

考虑了带有梯度项和变指标项的非线性退化抛物方程u_t=△u~m+μ|▽u|~(p(x))(μ0)非负解的爆破性质.使用特征函数方法和不等式技巧,得到了其齐次Dirichlet问题非负解在有限时刻爆破的充分条件.     

非局部的退化抛物型方程组的解的爆破和整体存在性

该文采用弱上下解方法以及正则化的技巧,研究了一类非局部的退化的抛物型方程组的解的爆破和整体存在性,给出了方程组的解的爆破指标p_c=(p₁+p₂)(q₁+q₂)-mn,证得当p_c<0时,对任意的初值,方程组的解整体存在;当p_c>0时,对充分大的初值,解在有限时刻爆破,对充分小的初值,解整体存在;当p_c=0时,若区域充分小,则方程组存在非负整体解,若区域包含了一个充分大的球, 则解在有限时刻爆破.     

具有Neumann边界条件热方程组的爆破速度

本文考虑具有 Ｎｅｕｍａｎｎ边界条件 ｕ／ｎ＝ｅｖ, ｖ／η＝ｅｕ在 ＳＲ ×（０,Ｔ）热方程组。ｕｔ＝△ｕ,ｖｔ=△ｖ在ＢＲ×（０,Ｔ）解的爆破性质·我们给出了爆破速度估计并证明了爆破仅在边界上发生．     

具有较高级算子的两组分Camassa-Holm方程的柯西问题

讨论了具有高阶算子-(1-_x~2+_x~4)的两组分Camassa-Holm方程在Sobolev空间H~s,s>5/2中的柯西问题,利用构造逼近解方法证明了该方程解的局部适定性问题,同时应用能量估计及嵌入定理等方法得到了在给定初值条件下的爆破准则,并且进一步给出了具体的爆破速率.     

有关非线性扩散方程组初值问题的一个注记（英文）

本文考虑一类非线性扩散方程组的初值问题ut=(um)xx+up11vp12,vt=(vn)xx+up21vp22,m,n≥1,(x,t)∈R×(0,T).通过采用伸缩变换的方法,研究得到解的两个分量发生同时和不同时爆破的完全指标分类,该结果完善了J.Math.Anal.Appl.,2005,308:92-104中的相关结果.另外,得到了所有情形下的同时爆破速率.     

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.     

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.     

一类双重退化抛物型不等式问题解的Liouville型定理

该文对一类双重退化抛物型不等式问题建立了弱解的Liouville型定理.不同于通常的上下解方法,这里采用更为简洁的适当选取试验函数与能量估计的方法证明整体解的不存在性.     

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.     

Non-simultaneous Blow-up Criteria for Localized Parabolic Equations

This paper deals with blow-up solutions for parabolic equations coupled via localized exponential sources, subject to homogeneous Dirichlet boundary con- ditions. The criteria are proposed to identify simultaneous and non-simultaneous blow-up solutions. The related classification for the four nonlinear parameters in the model is optimal and complete.     

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.     

Quenching Versus Blow-up

This paper is concerned with the semilinear heat equation u_t = Δu - u^{-q} in Ω × (0, T) under the nonlinear boundary condition \frac{∂u}{∂v} = u^p on ∂Ω × (0, T). Criteria for finite time quenching and blow-up are established, quenching and blow-up sets are discussed, and the rates of quenching and blow-up are 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.     

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.