European Journal of Applied Mathematics

Table of Contents - June 2009 - Volume 20, Issue 03  

Papers

Asymptotic behaviour for a non-local parabolic problem

LIU QILINa1a2, LIANG FEIa1 and LI YUXIANGa1a3

a1 Department of Mathematics, Southeast University, Nanjing 210018, Jiangsu, PR China email: liuqlseu@yahoo.com.cn

a2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China

a3 Universitée Paris 13, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications, 99, Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

Abstract

In this paper, we consider the asymptotic behaviour for the non-local parabolic problem

\begin{eqnarray} \[ u_{t}=\Delta u+\displaystyle\frac{\lambda f(u)}{(\int_{\Omega}f(u)dx)^{p}},\quad x\in \Omega,\ t>0, \\end{eqnarray}

with a homogeneous Dirichlet boundary condition, where λ > 0, p > 0 and f is non-increasing. It is found that (a) for 0 < p ≤ 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 < p < 2, u(x, t) is globally bounded for any λ > 0; (c) for p = 2, if 0 < λ < 2|∂Ω|2, then u(x, t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution and u(x, t) is a global solution and u(x, t) → ∞ as t → ∞ for all x ∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution and u(x, t) blows up in finite time for all x ∈ Ω; (d) for p > 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* and u0(x) sufficiently large, u(x, t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour of u(x, t) as it blows up are obtained for p ≥ 2.

(Received September 07 2008)

(Revised December 04 2008)

(Online publication February 05 2009)