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

Kimie Nakashima; Wei-Ming Ni; Linlin Su

doi:10.3934/dcds.2010.27.617

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2010, Volume 27, Issue 2: 617-641. Doi: 10.3934/dcds.2010.27.617

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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

Received: September 30, 2009

Revised: January 31, 2010

Published: June 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35B25.

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