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 $L^{\infty }$ 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 the limiting system.

本文研究了 Shigesada-Kawasaki-Teramoto 模型共存稳态的渐近行为，因为两个交叉扩散系数都以相同的速率趋于无穷大。在两个交叉扩散系数中的任何一个趋于无穷大的情况下，Lou 和 Ni [18] 推导出了几个限制系统，它们表征了共存稳态的渐近行为。最近，Kan-on [10] 的正式观察暗示存在一个包括非平稳问题的限制系统，因为两个交叉扩散系数都以相同的速率趋于无穷大。这篇论文给出了他对平稳问题的观察的严格证明。作为证明的关键要素，我们建立了一个统一的${升}^{\infty }$所有稳态的估计。由于这个先验估计，我们表明共存稳态的渐近轮廓可以通过限制系统的解来表征。