Bounds for blow-up time in nonlinear parabolic problems

A first order differential inequality technique is used on suitably defined auxiliary functions to determine lower bounds for blow-up time in initial-boundary value problems for parabolic equations of the form

ut=div(ρ(u)gradu)+f(u)     

Infinite time blow-up for superlinear parabolic problems with localized reaction

We consider the nonlocal diffusion equation

on the space interval , with Dirichlet boundary conditions. It is known that if the curve remains in a compact subset of for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on is sharp: for a large class of functions approaching the boundary as , blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered.

     

On the global existence and finite time blow-up of shadow systems

Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we study the global existence and blow-up phenomena for shadow systems. Our results show that even for these fundamental aspects, there are serious discrepancies between the dynamics of the reaction-diffusion systems and that of their corresponding shadow systems.     

Global existence and blow-up problems for reaction diffusion model with multiple nonlinearities

In this paper, we study the following reaction diffusion model,

     

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

Uniform blow-up rate for diffusion equations with localized nonlinear source

In this short paper, we investigate blow-up rate of solutions of reaction–diffusion equations with localized reactions. We prove that the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.     

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.     

Classification of blow-up with nonlinear diffusion and localized reaction

We study the behaviour of nonnegative solutions of the reaction-diffusion equation

     

Existence of positive solutions for some problems with nonlinear diffusion

In this paper we study the existence of positive solutions for problems of the type

where is the -Laplace operator and . If , such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases , and , respectively. Also, some systems of equations are considered.

     

On boundedness of solutions of reaction-diffusion equations with nonlinear boundary conditions

We give conditions on the nonlinearities of a reaction-diffusion equation with nonlinear boundary conditions that guarantee that any solution starting at bounded initial data is bounded locally around a certain point of the boundary, uniformly for all positive time. The conditions imposed are of a local nature and need only to hold in a small neighborhood of the point .

     

一类非局部非线性扩散方程解的全局爆破

主要研究在Dirichlet边界条件或Neumann边界条件下的一类非局部非线性的扩散方程问题.在适当的假设下,证明解的存在性、唯一性、比较原则、以及解对初边值条件的连续依赖性,并就给定的初边值条件,证明解在有限时刻全局爆破.     

Existence results for quasilinear elliptic exterior problems involving convection term and nonlinear Robin boundary conditions

In this paper, the authors establish the existence of solutions for a class of elliptic exterior problems involving convection terms and nonlinear Robin boundary conditions. The proof of the result is made by combining Galerkin method with a priori estimates for this kind of problem.     

The blow-up rates for systems of heat equations with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained.     

Global solution and blow-up solution for a nonlinear damped beam with source term

A nonlinear damped system with boundary input and output, which also has source term, is studied in this paper. It is proved that under some conditions the system has global solution and blow-up solution.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

A parabolic–elliptic Keller–Segel system $\left\{\begin{array}{c}{u}_t=\Delta u-\chi \nabla \cdot (uf\left(|\nabla v|\right)\nabla v),\hfill \\ 0=\Delta v-M+u,\hfill \end{array}\right.$with homogeneous Neumann boundary condition is considered in a radially symmetric domain $\Omega =B_R\left(0\right)\subset {\mathbb{R}}^N(N\ge 3),$ where $f\left(\xi \right)=\left({\xi }^{p-2}{\left(1+{\xi }^p\right)}^{\frac{q-p}{p}}\right),\xi \ge 0,p\ge 2,1<q\le p<\infty ,$and $B_R\left(0\right)$ is a $N$-dimensional ball of radius $R>0.$ We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when $\frac{N}{N-1}<q<2.$