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May  2010, 27(2): 617-641. doi: 10.3934/dcds.2010.27.617

An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles

1. 

Tokyo University of Marine Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo 108-8477

2. 

School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

3. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  October 2009 Revised  February 2010 Published  February 2010

We study the following Neumann problem

$ d\Delta u+g(x)u^{2}(1-u)=0 \ $ in Ω ,
$ 0\leq u\leq 1 $in Ω and $ \frac{\partial u}{\partial\nu}=0 $ on ∂Ω,

where $\Delta$ is the Laplace operator, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ with $\nu$ as its unit outward normal on the boundary $\partial\Omega$, and $g$ changes sign in $\Omega$. This equation models the "complete dominance" case in population genetics of two alleles. We show that the diffusion rate $d$ and the integral $\int_{\Omega}g\ \d x$ play important roles for the existence of stable nontrivial solutions, and the sign of $g(x)$ determines the limiting profile of solutions as $d$ tends to $0$. In particular, a conjecture of Nagylaki and Lou has been largely resolved.
   Our results and methods cover a much wider class of nonlinearities than $u^{2}(1-u)$, and similar results have been obtained for Dirichlet and Robin boundary value problems as well.

Citation: Kimie Nakashima, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 617-641. doi: 10.3934/dcds.2010.27.617
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