Patterns in the Keller–Segel system with density cut-off

Abstract

The Patlak–Keller–Segel system with logistic sensitivity has been widely advocated as a model which avoids overcrowding and generates complex patterns. Here we also consider the general case of a nonlinear diffusion of porous medium type with exponent $m$. The complexity of the observed patterns makes it complicated to understand the processes at work. Here, we analyze the pattern formation ability of such a system, which depends highly on $m$ and three different analyzes are needed for $m=1$ (linear diffusion), for $1<m<2$ and for $m\ge 2$. Within these regimes, the sensitivity also plays a crucial role, as does the conserved total mass. Typically, small mass patterns exist for $m>2$ but not for $m<2$. We focus specifically on the conditions for long-term convergence to the constant solution, uniqueness of the steady state and on the contrary, existence of increasing steady solutions in dimension one. Our method is based on several tools such as energy functional, reduction to a single equation, and reduction to a first-order equation. A major difficulty, in contrast to the case $m=1$, is that solutions can vanish locally when $m>1$.