We consider the Keller-Segel system of degenerate type (KS) with below. We establish a uniform estimate of 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). In addition, we find that the set of positive region to the solution is monotonically non-decreasing with respect to .
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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. 891–913DOI 10.4171/RMI/620