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

Yoshie Sugiyama
Source: Rev. Mat. Iberoamericana Volume 26, Number 3 (2010), 891-913.

#### 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 $t$.

First Page:
Primary Subjects: 35K65, 35K55, 35B57, 35K45
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Permanent link to this document: http://projecteuclid.org/euclid.rmi/1282913825
Mathematical Reviews number (MathSciNet): MR2789369

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