An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles
Kimie Nakashima, Wei‐Ming Ni, Linlin Su · Discrete and Continuous Dynamical Systems · 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 u}=0 $ on ∂Ω, where $\Delta$ is the Laplace operator, $\Omega$ is a boundedsmooth domain in $\mathbb{R}^{N}$ with $ u$ as its unit outwardnormal 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 thediffusion rate $d$ and the integral $\int_{\Omega}g\ \d x$ playimportant roles for the existence of stable nontrivial solutions, and the sign of $g(x)$ determines thelimiting 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 beenobtained for Dirichlet and Robin boundary value problems as well.