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 $t$.
Publié le : 2010-09-15
Classification:
parabolic system of degenerate type,
Keller-Segel,
porous medium,
Aronson-Bénilan estimate,
interface,
optimal Höder continuity,
35K65,
35K55,
35B57,
35K45
@article{1282913825,
author = {Sugiyama
,
Yoshie},
title = {Aronson-B\'enilan type estimate and the optimal H\"older continuity of
weak solutions for the 1-D degenerate Keller-Segel systems},
journal = {Rev. Mat. Iberoamericana},
volume = {26},
number = {1},
year = {2010},
pages = { 891-913},
language = {en},
url = {http://dml.mathdoc.fr/item/1282913825}
}
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, Tome 26 (2010) no. 1, pp. 891-913. http://gdmltest.u-ga.fr/item/1282913825/