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

Kimie Nakashima, Wel Ming Ni, Linun Su

Research output: Contribution to journalArticlepeer-review

32 Scopus citations

Abstract

We study the following Neumann problem Math eqution presented whereδ is the Laplace operator, ω is a bounded smooth domain in Rn with νas its unit outward normal on the boundary∂ω and g changes sign in ω. This equation models the "complete dominance" case in population genetics of two alleles. We show that the diffusion rate d and the integral ∫ωg dx 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 u2(1 - u), and similar results have been obtained for Dirichlet and Robin boundary value problems as well.

Original languageEnglish (US)
Pages (from-to)617-641
Number of pages25
JournalDiscrete and Continuous Dynamical Systems
Volume27
Issue number2
DOIs
StatePublished - Jun 2010

Keywords

  • Diffusion equations
  • Indefinite nonlinearity
  • Limiting behavior
  • Variational method

Fingerprint

Dive into the research topics of 'An indefinite nonlinear diffusion problem in population gentics,I: Existence and limiting profiles'. Together they form a unique fingerprint.

Cite this