非线性边界条件下具有变系数的热量方程解的存在性及爆破现象

考虑了定义在Ω上的有变系数的热量方程,其中Ω?RN(N≥2)是一个有界的凸区域,并且方程具有非线性边界条件.利用微分不等式技术,首先推导了爆破一定发生的条件,并确定了爆破时间的上界.同时,通过对非线性项做一定的限制,得到了解的全局存在性.当爆破发生时,确定了爆破时间的下界.     

带Robin边界条件的拟线性抛物方程解的整体存在性与爆破

主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界.     

一类反应扩散方程的爆破时间下界估计

该文讨论了一类反应项为非线性非局部热源且热汇具有时间系数的反应扩散方程,分别在Dirichlet、Neu-mann或Robin边界条件下,在有界区域中的爆破行为.若解可能在有限时间发生爆破,通过构造合适的辅助函数,对时间系数给出适当的条件,利用Sobolev、H?lder不等式及Payne和Schaefer积分不等式等...     

一类含非局部源的非线性退化扩散方程解的爆破性质

研究了一类带非局部源的非线性退化抛物型方程.在一定条件下,证得方程的解在有限时刻爆破且爆破点集为整个区域.积分方法被用来研究解的爆破性质.     

齐次Neumann边界条件非线性抛物方程解的爆破时间的下界

研究了非线性抛物方程具有齐次Neumann边界条件问题解的爆破.在对问题中的f,ρ和g作出适当的假设的前提下,推导出了上述问题解的爆破时间的下界.同时,也得到了问题的解不发生爆破的条件.     

带色散项的周期可积Hunter-Saxton方程的波的破裂

本文考虑一个带色散项的可积Hunter-Saxton方程的周期柯西问题.首先,我们得出该模型的强解的一个精确的爆破机制.其次,利用守恒量和特征方法,分别建立了有限时间内强解的爆破发生的充分条件.最后,给出了强解的爆破率.     

一个非线性抛物型方程正解的性质

考虑一个具有非线性吸收项和非线性边界流的拟线性抛物型方程正解的性质．得到了解整体存在的充要条件．此外，借助于Chasseigne和Vázquez的结论以及比较原理，导出了爆破解只可能在区域的边界¶Ω上发生爆破．对于有界的Lipschitz型区域Ω, 还估计了在a=0时爆破解的爆破速率.     

一类带记忆项的非经典热方程的爆破问题

本文考虑了一类带记忆项的非经典热方程,证明解会在有限时间爆破,而且爆破只会发生在边界.主要结论是:首先利用Green函数与Banach压缩映射定理,建立了问题的经典解;其次,利用经典解,证明了解是有限时间爆破的;最后,证明了一个关于非经典热方程解的性质,利用这个性质,证明了解是在边界上爆破的.     

非局部反应扩散方程的一致爆破速率

该文研究了带有非局部源的扩散方程的爆破速率．关于这类方程,作者证得该类方程的解整体爆破且其爆破速率在区域的任一紧子区域内是一致的．在各种情形下,当t趋向于爆破时刻T*时,|u(t)|∞的爆破速率可精确确定．     

非线性抛物方程在非线性边界条件下解的爆破时间的下界

研究了非线性抛物方程在非线性边界条件下的解的爆破问题,通过构造一个能量表达式,运用微分不等式的方法,得到该能量表达式所满足的微分不等式,然后通过积分得到当爆破发生时解在非线性边界条件下的爆破时间的下界.     

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.     

Remarks on Blow-Up Phenomena in p-Laplacian Heat Equation with Inhomogeneous Nonlinearity

We investigate the $p$-Laplace heat equation $u_t-\Delta_p u=ζ(t)f(u)$ in a bounded smooth domain. Using differential-inequality arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial datum $u_0$. We also give an upper bound for the blow-up time in each case.     

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.     

Blow-Up Phenomena for Some Pseudo-Parabolic Equations with Nonlocal Term

We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term. We establish a lower bound for the blow-up time if blow-up does occur. Also both the upper bound for $T$ and blow up rate of the solution are given when $J(u_0)<0$. Moreover, we establish the blow up result for arbitrary initial energy and the upper bound for $T$. As a product, we refine the lifespan when $J(u_0)<0.$     

Blow-up in parabolic problems under Robin boundary conditions

By means of a first-order differential inequality technique, sufficient conditions are determined which imply that blow-up of the solution does occur or does not occur for the semilinear heat equation under Robin boundary conditions. In addition, a lower bound on blow-up time is obtained when blow-up does occur.     

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.     

通过移动媒质避免爆破

This paper deals with two parabolic initial-boundary value problems in multidimensional domain. The first problem describes the situation where the spherical medium is static and the nonlinear reaction takes place only at a single point. We show that under some conditions, the solution blows up in finite time and the blow-up set is the whole spherical medium. When the spherical medium is allowed to move in a special space, we investigate another parabolic initial-boundary value problem. It is proved that the blow-up can be avoided if the acceleration of the motion satisfies certain conditions.     

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.     

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.     

Local solvability and a priori estimates for classical solutions to an equation of Benjamin-Bona-Mahony-Bürgers type

We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary-value problems on the half-line for a nonlinear equation similar to Benjamin-Bona-Mahony-Bürgers-type equation. We also derive an a priori estimate that implies sufficient blow-up conditions for the second boundary-value problem. We obtain analytically an upper bound of the blow-up time and refine it numerically using Richardson effective accuracy order technique.