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Applied Analysis
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
|---|
2008/11/13
16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
杉山 由恵 (津田塾大学・学芸学部・数学科)
Aronson-Benilan type estimate and the optimal Hoelder continuity of weak solutions for the 1D degenerate Keller-Segel systems
杉山 由恵 (津田塾大学・学芸学部・数学科)
Aronson-Benilan type estimate and the optimal Hoelder continuity of weak solutions for the 1D degenerate Keller-Segel systems
[ Abstract ]
We consider the Cauchy problem for the 1D Keller-Segel system of degenerate
type (KS)_m with $m>1$:
u_t= \\partial_x^2 u^m - \\partial_x (u^{q-2} \\partial_x v),
-\\partial_x^2 v + v - u=0.
We establish a uniform estimate from below of $\\partial_x^2 u^{m-1}$.
The corresponding estimate to the porous medium equation is well-known
as an Aronson-Benilan type.
As an application of our Aronson-Benilan type estimate,
we prove the optimal Hoelder continuity of the weak solution $u$ of (KS)_m.
In addition, we find that the positive region $D(t):=\\{x \\in \\R; u(x,t)>0\\}$
of $u$ is monotonically non-decreasing with respect to the time $t$.
We consider the Cauchy problem for the 1D Keller-Segel system of degenerate
type (KS)_m with $m>1$:
u_t= \\partial_x^2 u^m - \\partial_x (u^{q-2} \\partial_x v),
-\\partial_x^2 v + v - u=0.
We establish a uniform estimate from below of $\\partial_x^2 u^{m-1}$.
The corresponding estimate to the porous medium equation is well-known
as an Aronson-Benilan type.
As an application of our Aronson-Benilan type estimate,
we prove the optimal Hoelder continuity of the weak solution $u$ of (KS)_m.
In addition, we find that the positive region $D(t):=\\{x \\in \\R; u(x,t)>0\\}$
of $u$ is monotonically non-decreasing with respect to the time $t$.
