TY - JOUR

T1 - Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

AU - Kuto, Kousuke

N1 - Funding Information:
This research was partially supported by JSPS KAKENHI Grant Number 19K03581 .
Publisher Copyright:
© 2021 L'Association Publications de l'Institut Henri Poincaré

PY - 2021/11/1

Y1 - 2021/11/1

N2 - 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.

AB - 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.

KW - A priori estimate

KW - Bifurcation

KW - Cross-diffusion

KW - Limiting system

KW - Maximum principle

KW - Nonlinear elliptic system

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U2 - 10.1016/j.anihpc.2021.02.006

DO - 10.1016/j.anihpc.2021.02.006

M3 - Article

AN - SCOPUS:85101299999

VL - 38

SP - 1943

EP - 1959

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 6

ER -