Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems

Abstract

We consider the Keller-Segel system of degenerate type (KS)${}_m$ with $m>1$ below. We establish a uniform estimate of ${\partial }_x^2{u}^{m-1}$ from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)${}_m$. In addition, we find that the set $D(t):=\{x\in \mathbb{R};u(x,t)>0\}$ of positive region to the solution $u$ is monotonically non-decreasing with respect to $t$.