Global structure of steady-states to the full cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model

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Abstract

In a previous paper [10], the author studied the asymptotic behavior of coexistence steady-states to the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. As a result, he proved that the asymptotic behavior can be characterized by a limiting system that consists of a semilinear elliptic equation and an integral constraint. This paper studies the set of solutions of the limiting system. The first main result gives sufficient conditions for the existence/nonexistence of nonconstant solutions to the limiting system by a topological approach using the Leray-Schauder degree. The second main result exhibits a bifurcation diagram of nonconstant solutions to the one-dimensional limiting system by analysis of a weighted time-map and a nonlocal constraint.

Original languageEnglish
Pages (from-to)103-143
Number of pages41
JournalJournal of Differential Equations
Volume333
DOIs
Publication statusPublished - 2022 Oct 5

Keywords

  • Bifurcation
  • Cross-diffusion
  • Integral constraint
  • Limiting system
  • Nonlinear elliptic equations
  • The Leray-Schauder degree

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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