Existence and uniqueness of positive solutions for some quasilinear elliptic problems

This expository
article concerns a system of semilinear parabolic partial differential equations that describes
the evolution of the gene frequencies at a single locus under the joint action
of migration and selection.
We shall review mathematical techniques suited for the models under investigation;
 discuss some of the main mathematical results, including most recent developments;
 and also propose some open problems.

http://cpde.ecnu.edu.cn/papers/fgyseILqhJBrk6lpxTDw.pdf

An introduction to migration-selection pde models

In these notes, we consider a scalar parabolic equation which is either autonomous or time dependent. We give a survey of some of the results that deal with the qualitative properties of the dynamics dened by general equations as well as specic results for particular equations. Many of the ideas for a scalar equation have served as a guide to the study of interesting phenomena in systems. We make some remarks about systems, but space limits a detailed discussion of this case. The table of contents describes well the essential plan of the notes. 1. Fundamental concepts If we consider an evolutionary equation in a Banach space X and assume that the vector eld does not depend upon time, then the solution through a point x2 X at time 0 may be represented as T (t)x. If each solution is dened for all t 0 and there a unique solution through each point, then the family of operators T (t);t 0;satises the following properties:

/pdf/dynamics-of-a-scalar-parabolic-equation-puwqdh4tu8.pdf

Dynamics of a Scalar Parabolic Equation

A nonlocal dispersal equation arising from a selection-migration model in genetics

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.

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

/pdf/stability-and-uniqueness-of-positive-solutions-for-a-semi-5khro62d2f.pdf

K. J. Brown

Papers

Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem