Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems (Q616600)

The author considers the Keller-Segel system of degenerate type \((KS)_m\) with \(m>1\). It is established a uniform estimate of \(\partial^2_x u^{m-1}\) from below in order to prove the optimal Hölder continuity of weak solutions of \((KS)_m\). The corresponding estimate to the porous medium equation is known as an Aronson-Bénilan type. In addition it is 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 \(t\).