Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
Departement of mathematics and computer science,
Leipzig, 04109, Germany. email@example.com
2 Department of Mathematics and Computer Science, Tsuda College, 2-1-1, Tsuda-chou, Kodaira-shi, Tokyo, 187-8577, Japan. firstname.lastname@example.org
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