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Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

Published online by Cambridge University Press:  22 July 2006

Stephan Luckhaus  and
Yoshie Sugiyama
Stephan Luckhaus
Affiliation:
Departement of mathematics and computer science, Universität Leipzig, Leipzig, 04109, Germany. [email protected]
Yoshie Sugiyama
Affiliation:
Department of Mathematics and Computer Science, Tsuda College, 2-1-1, Tsuda-chou, Kodaira-shi, Tokyo, 187-8577, Japan. [email protected]
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Abstract

We consider the following reaction-diffusion equation:

$$ {\rm (KS)}\left\{\begin{array}{llll}u_t=\nabla \cdot \Big( \nabla u^m - u^{q-1} \nabla v \Big),& x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber0 = \Delta v - v + u, & x \in \mathbb{R}^N, \ 0<t<\infty, \nonumberu(x,0) = u_0(x), & x \in \mathbb{R}^N,\end{array}\right.$$

where $N \ge 1, \ m > 1, \ q \ge \max\{m+\frac{2}{N},2\}$ .
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)]it was shown that in the case of $q \ge \max\{m+\frac{2}{N},2\}$ , the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”.Moreover, the decay of the solution (u,v) in $L^p(\mathbb{R}^N)$ was proved.In this paper, we consider the case of “ $q \ge \max\{m+\frac{2}{N},2\}$ and small $L^{\ell}$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$ and show that (i) there exists a time global solution (u,v) of (KS) andit decays to 0 as t tends to ∞ and(ii) a solution u of the first equation in (KS)behaveslike the Barenblatt solution asymptotically as t tends to ∞,where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation $u_t = \Delta u^m$ with m>1.

Keywords

Degenerate parabolic systemchemotaxisKeller-Segel modeldrift termdecay propertyasymptotic behaviorFujita exponentporous medium equationBarenblatt solution.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 3 , May 2006 , pp. 597 - 621
Copyright
© EDP Sciences, SMAI, 2006

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