The degenerate logistic model and a singularly mixed boundary blow-up problem

Yihong Du; Zongming Guo

doi:10.3934/dcds.2006.14.1

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2006, Volume 14, Issue 1: 1-29. Doi: 10.3934/dcds.2006.14.1

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The degenerate logistic model and a singularly mixed boundary blow-up problem

Yihong Du^1, and
Zongming Guo^2,

1.
School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351
2.
Department of Mathematics, Donghua University, Shanghai, 200051

Received: July 31, 2004

Revised: January 31, 2005

Published: February 2006

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Abstract

We study the degenerate logistic model described by the equation $ u_t - $Δ$ u=au-b(x)u^p$ with standard boundary conditions, where $p>1$, $b$ vanishes on a nontrivial subset $\Omega_0$ of the underlying bounded domain $\Omega\subset R^N$ and $b$ is positive on $\Omega_+=\Omega\setminus \overline{\Omega}_0$. We consider the difficult case where $\partial\Omega_0\cap \partial \Omega$≠$\emptyset$ and $\partial\Omega_+\cap \partial \Omega$≠$\emptyset$, and examine the asymptotic behaviour of the solutions. By a detailed study of a singularly mixed boundary blow-up problem, we obtain some basic results on the dynamics of the model.

Keywords:

Logistic equation asymptotic behavior,
boundary blow-up.

Mathematics Subject Classification: Primary: 35J25,35J65; Secondary: 35k20.

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