带非局部源的退化半线性抛物型方程解的爆破

该文研究带Dirichlet边界条件的退化半线性抛物型方程:xqut-uxx=∫0af(u)dx,这里q>0.作者证明了局部解的存在唯一性并且得到当初值充分大时解在有限时刻爆破.进而,证明解的爆破点集是整个区间[0,a],这与具有局部源的方程解的性质不同.     

带非局部源的退化奇异半线性抛物方程的爆破

本文研究带齐次Dirichlet边界条件的非局部退化奇异半线性抛物方程ut-(xαux)x=∫0af(u)dx在(0,a)×(0,T)内正解的爆破性质,建立了古典解的局部存在性与唯一性.在适当的假设条件下,得到了正解的整体存在性与有限时刻爆破的结论.本文还证明了爆破点集是整个区域,这与局部源情形不同.进而,对于特殊情形:f(u)=up,p>1及,f(u)=eu,精确地确定了爆破的速率.     

具有非局部源的退化半线性抛物型方程组解的爆破

本文讨论具有非局部源退化半线性抛物型方程组的初边值问题 .证明了局部解的存在唯一性并且得到当初值充分大时解在有限时刻爆破 .     

有非局部源退化抛物方程组解的爆破

本文讨论一类具有非局部源退化抛物方程组．通过利用上下解方法得到解的全局存在和有限时刻爆破，给出爆破集是整个区域，而且得到了解的爆破率．     

一类含非局部源的非线性退化抛物型方程

The global existence and finite time blow up of the positive solution for a nonlinear degenerate parabolic equation with non-local source are studied.     

非局部退化拟线性抛物型方程组解的爆破和整体存在性

该文采用弱上下解方法和正则化技巧,研究了一类非局部退化抛物型方程组解的爆破和整体存在性,给出了爆破指标,并对非退化情形m=n=1,p_1=q_1=0,p_2q_21给出了一致爆破速率.     

半线性抛物型方程的单点爆破

本文考虑一般的半线性抛物型方程ｕｔ＝ｕｘｘ＋ｆ（ｕ，ｕｘ，ｔ）的爆破点集，证明在一定条件下，爆破只能在一点发生     

具非局部源退化抛物型方程组的一致爆破模式

考虑带有齐次Dirichlet边界条件且具有非局部源项的退化抛物型方程组正解的爆破性质. 在适当条件下, 建立了该问题解的局部存在性并证明解在有限时刻爆破, 此外,还导出了解的两个分量同时爆破的必要条件, 并得到了该问题解的一致爆破模式.     

非局部边界条件的退化抛物方程组解的爆破和整体存在性

主要讨论具有非局部源与非局部边界条件的退化抛物型方程组,借助于上解与下解的技术,给出了该系统整体解的存在与有限时刻爆破的条件.此结果不仅扩充了已有的结论~([8]),而且表明,系数a,b和边界条件中的权重函数g_1(x,y),g_2(x,y),以及常数l_1,l_2在决定系统解的爆破与否中起着关键的作用.     

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.     

Uniform blow-up rate for a nonlocal degenerate parabolic equations

In this paper, we introduce a new method for investigating the rate of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of _|u(t)|∞${\left|u\left(t\right)\right|}_{\infty }$ is precisely determined.     

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose L^∞$L^{\infty }$-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique.     

Uniform blow-up profiles and boundary layer for a parabolic system with localized sources

This paper deals with the Dirichlet problem for a parabolic system with localized sources. We first obtain some sufficient conditions for blow-up in finite time, and then deal with the possibilities of simultaneous blow-up under suitable assumptions. Moreover, when simultaneous blow-up occurs, we also establish the uniform blow-up profiles in the interior and estimate the boundary layer.     

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

In this paper, we investigate the initial-boundary problem of a degenerate parabolic system with nonlinear localized sources. We classify the blow-up solutions into global blow-up cases and single-point blow-up cases according to the values of m,n,p_i,q_i. Furthermore, we obtain the uniform blow-up profiles of solutions for the global blow-up case. Finally, we give some numerical examples to verify the results. These extend and generalize a recent work of one of the authors [L. Du, Blow-up for a degenerate reaction-diffusion systems with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320], which only considered uniform blow-up profiles under the special case p₁=p₂=0.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory

We show the existence of monotone in time solutions for a semilinear parabolic equation with memory. The blow-up rate estimate of the solution is known to be a consequence of the monotonicity property.     

Rates of convergence to zero for a semilinear parabolic equation with a critical exponent

We study nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren. We establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate.     

Properties of non-simultaneous blow-up solutions in nonlocal parabolic equations

This paper deals with blow-up solutions in parabolic equations coupled via nonlocal nonlinearities, subject to homogeneous Dirichlet conditions. Firstly, some criteria on non-simultaneous and simultaneous blow-up are given, including four kinds of phenomena: (i) the existence of non-simultaneous blow-up; (ii) the coexistence of non-simultaneous and simultaneous blow-up; (iii) any blow-up must be simultaneous; (iv) any blow-up must be non-simultaneous. Next, total versus single point blow-up are classified completely. Moreover, blow-up rates are obtained for both non-simultaneous and simultaneous blow-up solutions.     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.