Issue |
ESAIM: M2AN
Volume 40, Number 3, May-June 2006
|
|
---|---|---|
Page(s) | 597 - 621 | |
DOI | https://doi.org/10.1051/m2an:2006025 | |
Published online | 22 July 2006 |
Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
1
Departement of mathematics and computer science,
Universität Leipzig,
Leipzig, 04109, Germany. luckhaus@mis.mpg.de
2
Department of Mathematics and Computer Science,
Tsuda College,
2-1-1, Tsuda-chou, Kodaira-shi, Tokyo, 187-8577, Japan.
sugiyama@tsuda.ac.jp
Received:
22
September
2005
Revised:
27
February
2006
We consider the following reaction-diffusion equation:
where .
In [Sugiyama, Nonlinear Anal. 63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)]
it was shown that
in the case of
,
the above problem (KS) is solvable globally in time for “small
data”.
Moreover,
the decay of the solution (u,v) in
was proved.
In this paper, we consider
the case of “
and
small
data” with any fixed
and show that
(i)
there exists a time global solution (u,v) of (KS) and
it decays to 0 as t tends to ∞ and
(ii)
a solution u of the first equation in (KS)
behaves
like the Barenblatt solution asymptotically as t tends to ∞,
where the Barenblatt solution is the exact solution (with self-similarity)
of the porous medium equation
with m>1.
Mathematics Subject Classification: 35B40 / 35K45 / 35K55 / 35k65
Key words: Degenerate parabolic system / chemotaxis / Keller-Segel model / drift term / decay property / asymptotic behavior / Fujita exponent / porous medium equation / Barenblatt solution.
© EDP Sciences, SMAI, 2006
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