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
.