February  2009, 25(1): 123-132. doi: 10.3934/dcds.2009.25.123

On the long-time limit of positive solutions to the degenerate logistic equation

1. 

Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW2351, Australia

2. 

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku 169-8555, Tokyo

Received  November 2006 Revised  December 2007 Published  June 2009

We study the long-time behavior of positive solutions to the problem

$u_t-\Delta u=a u-b(x)u^p \mbox{ in } (0,\infty)\times \Omega, Bu=0 \mbox{ on } (0,\infty)\times \partial \Omega, $

where $a$ is a real parameter, $b\geq 0$ is in $C^\mu(\bar{\Omega})$ and $p>1$ is a constant, $\Omega$ is a $C^{2+\mu}$ bounded domain in $R^N$ ($N\geq 2$), the boundary operator $B$ is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that $\overline\Omega_0$:=$\{x\in\Omega: b(x)=0\}$ has non-empty interior, is connected, has smooth boundary and is contained in $\Omega$, it is shown in [8] that when $a\geq \lambda_1^D(\Omega_0)$, for any fixed $x\in \overline{\Omega}_0$, $\overline{\lim}_{t\to\infty}u(t,x)$=$\infty$, and for any fixed $x\in \overline{\Omega}\setminus \overline{\Omega}_0$,

$\overline{\lim}_{t\to\infty}u(t,x)\leq \overline{U}_a(x), \underline{\lim}_{t\to\infty}u(t,x)\geq \underline{U}_a(x),

where $\underline{U}_a$ and $\overline{U}_a$ denote respectively the minimal and maximal positive solutions of the boundary blow-up problem

$-\Delta u=au-b(x)u^p \mbox{ in} \ \Omega\setminus\overline{\Omega}_0,\ Bu=0 \mbox{ on}\ \partial \Omega,\ \ u=\infty \mbox{ on}\ \partial \Omega_0.$

The main purpose of this paper is to show that, under the above assumptions,

$\lim_{t\to\infty} u(t,x)=\underline U_a(x),\forall x\in \overline\Omega\setminus \overline\Omega_0.$

This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

Citation: Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
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