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January  2006, 14(1): 1-29. doi: 10.3934/dcds.2006.14.1

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, Australia
2. 

Department of Mathematics, Donghua University, Shanghai, 200051, China

Received  August 2004 Revised  February 2005 Published  October 2005

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: 35k2.
Citation: Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1
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