JournalsrmiVol. 26, No. 3pp. 891–913

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

Abstract

We consider the Keller-Segel system of degenerate type (KS)m_m with m>1m > 1 below. We establish a uniform estimate of x2um1\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_m. In addition, we find that the set D(t):={xR;u(x,t)>0}D(t):=\{ x \in \mathbb{R}; u(x,t) > 0\} of positive region to the solution uu is monotonically non-decreasing with respect to tt.

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