Nonuniqueness of an indefinite nonlinear diffusion problem in population genetics

Kimie Nakashima, Linlin Su

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We study the following Neumann problem in one-dimension space arising from population genetics: {ut=duxx+h(x)u2(1−u)in(−1,1)×(0,∞),0≤u≤1in(−1,1)×(0,∞),u(−1,t)=u(1,t)=0in(0,∞), where h changes sign in (−1,1) and d is a positive parameter. Lou and Nagylaki (2002) [6] conjectured that if ∫−11h(x)dx≥0, then this problem has at most one nontrivial steady state (i.e., a time-independent solution which is not identically equal to zero or one). Nakashima (2018) [15] proved this uniqueness under some additional conditions on h(x). Unexpectedly, in this paper, we discover 3 nontrivial steady states for some h(x) satisfying ∫−11h(x)dx≥0. Moreover, bi-stable phenomenon occurs in this scenario: one with two layers is stable; two with one layer each are ordered with the smaller one being stable and the larger one being unstable.

Original languageEnglish
Pages (from-to)4643-4682
Number of pages40
JournalJournal of Differential Equations
Issue number6
Publication statusPublished - 2020 Sept 5
Externally publishedYes


  • Complete dominance
  • Indefinite weight
  • Layers
  • Reaction-diffusion equation
  • Singular perturbation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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