Three asymptotic regimes for a Keller--Segel system with volume-filling effect

Abstract

We study three asymptotic regimes for a parabolic--elliptic Keller--Segel system with porous-medium diffusion and the volume-filling sensitivity function $u(1-u)$. First, when $D=\epsilon^2\to0$, $\delta=1$, $1\le m<2$, and the chemotactic sensitivity coefficient is below an explicit $m$-dependent parabolicity threshold, we prove strong convergence to a scalar nonlinear diffusion equation. Second, as $\delta=\epsilon\to0$ with $D=1$, we obtain a hyperbolic--elliptic Keller--Segel limit by means of a kinetic reduction argument. Third, under the scaling $\chi=\epsilon^{-1}$, $D=\epsilon^2$, $\delta=1$, and for characteristic initial data of finite perimeter, we prove convergence on fixed time intervals to the time-independent initial patch in $BV(\Omega;\{0,1\})$. Because the sensitivity function is nonlinear in the density, weak convergence alone is insufficient to identify the chemotactic flux. Strong compactness of the density is therefore required in all three regimes and is obtained, respectively, through energy--entropy estimates, kinetic reduction, and $BV$ compactness.