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

Pages: 1 - 29, Volume 14, Issue 1, January 2006

doi:10.3934/dcds.2006.14.1       Abstract        Full Text (350.4K)       Related Articles

Yihong Du - School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia (email)
Zongming Guo - Department of Mathematics, Donghua University, Shanghai, 200051, China (email)

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.

Received: August 2004;      Revised: February 2005;      Published: October 2005.