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  derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady-states. Recently, a formal observation by Kan-on  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 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.
Cite this article
Kousuke Kuto, Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 6, pp. 1943–1959DOI 10.1016/J.ANIHPC.2021.02.006