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An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
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Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program
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 |
$ 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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