Revista Matemática Iberoamericana


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Volume 26, Issue 3, 2010, pp. 891–913
DOI: 10.4171/RMI/620

Published online: 2010-12-31

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

Yoshie Sugiyama[1]

(1) Tsuda University, Tokyo, Japan

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 $t$.

Keywords: Parabolic system of degenerate type, Keller-Segel, porous medium, Aronson- Bénilan estimate, interface, optimal Hölder continuity

Sugiyama Yoshie: Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems. Rev. Mat. Iberoamericana 26 (2010), 891-913. doi: 10.4171/RMI/620