# American Institute of Mathematical Sciences

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

 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.
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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