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

### Yoshie Sugiyama

Tsuda University, Tokyo, Japan

## 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 $t$.

## Cite this article

Yoshie Sugiyama, Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 891–913

DOI 10.4171/RMI/620