Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

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Abstract

This paper studies the asymptotic behavior of coexistence steady-states of the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. In the case when either one of two cross-diffusion coefficients tends to infinity, Lou and Ni [18] derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady-states. Recently, a formal observation by Kan-on [10] implied the existence of a limiting system including the nonstationary problem as both cross-diffusion coefficients tend to infinity at the same rate. This paper gives a rigorous proof of his observation as far as the stationary problem. As a key ingredient of the proof, we establish a uniform L estimate for all steady-states. Thanks to this a priori estimate, we show that the asymptotic profile of coexistence steady-states can be characterized by a solution of the limiting system.

Original languageEnglish
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
DOIs
Publication statusAccepted/In press - 2021

Keywords

  • A priori estimate
  • Bifurcation
  • Cross-diffusion
  • Limiting system
  • Maximum principle
  • Nonlinear elliptic system

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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