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An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
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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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