An indefinite nonlinear diffusion problem in population gentics,I: Existence and limiting profiles
An indefinite nonlinear diffusion problem in population gentics,I: Existence and limiting profiles
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DOI：
10.3934/dcds.2010.27.617
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发表时间：
2010-02
影响因子：
1.1
通讯作者：
K. Nakashima;W. Ni;Linlin Su
中科院分区：
文献类型：
--
作者：
K. Nakashima;W. Ni;Linlin Su
我们研究以下诺依曼问题 $ d\Delta u+g(x)u^{2}(1-u)=0 \ $ in Ω 、 $ 0\leq u\leq 1 $in Ω 和 $ \frac{\partial u}{\partial\nu}=0 $ on ∂Ω ，其中 $\Delta$ 是拉普拉斯算子，$\Omega$ 是 $\mathbb{R}^{N}$ 中的有界光滑域以 $\nu$ 为边界 $\partial\Omega$ 上的向外法线单位，并且 $g$ 在 $\Omega$ 上改变符号。该方程模拟了两个等位基因群体遗传学中的“完全优势”情况。我们证明，扩散速率 $d$ 和积分 $\int_{\Omega}g\ \d x$ 对于稳定非平凡解的存在起着重要作用，并且 $g(x)$ 的符号决定了当 $d$ 趋向于 $0$ 时解的极限轮廓。特别是Nagylaki和Lou的猜想已基本得到解决。我们的结果和方法涵盖了比 $u^{2}(1-u)$ 更广泛的非线性类别，并且对于 Dirichlet 和 Robin 边值问题也获得了类似的结果。
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.