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

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 (1999) derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady states. Recently, a formal observation by Kan-on (2020) 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 $L^{\infty}$ 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 either of two limiting systems.