Blow-up analysis for a nonlinear diffusion system

This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.Supported by the National Natural Science Foundation of China.Received: December 12, 2001; revised: May 6, December 3, 2002     

Blow-up rate for a nonlinear diffusion equation

In this work we study the blow-up rate for a nonlinear diffusion equation with an inner source and a nonlinear boundary flux, which is equivalent to a porous medium equation with convection. Depending upon the sign of a parameter included, the source can be positive or negative (absorption). By the scaling method, we obtain that the blow-up rate is independent of a negative source, while for the situation with a positive source, the blow-up rate is determined by the interaction between the inner source and the boundary flux. Comparing with the previous results for the porous medium model without convection, we observe that the gradient term included here does not affect the blow-up rates of solutions.     

一类非线性扩散方程解的blow up速率估计

利用F riedm an-M cleod方法和变动尺度方法研究了一类具有非线性边界条件的非线性扩散方程解的b low up问题,证明了解在有限时间b low up,并且得到了b low up速率估计.     

Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources

This paper deals with the global existence and blow-up of nonnegative solution of the degenerate reaction-diffusion system with nonlinear localized sources involved a product with local terms. We investigate the influence of localized sources and local terms on global existence and blow up for this system. Moreover, we establish the precise blow-up estimates. Finally, for the special case p₁=p₂=0, we show the blow-up set is whole region and the uniform blow-up profiles are obtained. These extend a resent work of Chen and Xie in [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. Comput. 159 (2004) 79-93], which considered the single equation with localized sources.     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

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.     

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

This paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball.     

Blow-up solutions to a system of nonlinear Volterra equations

A system of nonlinear Volterra integral equations with convolution kernels is considered. Estimates are given for the blow-up time when conditions are such that the solution is known to become unbounded in finite time. For two examples that arise in combustion problems, numerical estimates of blow-up time are presented. Additionally, the asymptotic behavior of the blow-up solution in the key limit is established for the power-law and exponential nonlinearity cases.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

具有非局部源的退化奇异抛物方程组解的爆破

研究了一类新的包含幂函数和指数函数相耦合的具有非局部源的抛物方程组.用正则化的方法证明了局部解的存在唯一性,用上下解方法得到了整体存在和在有限时刻爆破的充分条件.     

Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities

This paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. As the results of the interaction among the multi-coupled nonlinearities in the system, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.     

Blow-up phenomena for some nonlinear parabolic problems

We consider the blow-up of solutions of equations of the form

_ut=div(ρ(|∇u|²) grad u)+f(u)$u_t=\text{div}\left(\rho \left({\left|\nabla u\right|}^2\right)\text{ grad }u\right)+f\left(u\right)$

Blow-up set for a semilinear heat equation with small diffusion

We consider the blow-up problem for a semilinear heat equation,

     

Blow-up problems for a semilinear heat equation with large diffusion

We consider the blow-up problem of a semilinear heat equation,

     

Blow-up for doubly degenerate nonlinear parabolic equations

In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations. The project is supported by the Natural Science Foundation of Fujian Province of China (No. Z0511048)