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

Luckhaus, Stephan ; Sugiyama, Yoshie

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006), p. 597-621 / Harvested from Numdam

Résumé

We consider the following reaction-diffusion equation: ( KS )u t =∇·∇ u m - u q-1 ∇ v,x∈ℝ N ,0<t<∞,0=Δv-v+u,x∈ℝ N ,0<t<∞,u(x,0)=u 0 (x),x∈ℝ N , where $N \ge 1, \ m > 1, \ q \ge \max \lbrace m+\frac{2}{N},2\rbrace $. 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 \lbrace m+\frac{2}{N},2\rbrace $, 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 \lbrace m+\frac{2}{N},2\rbrace $ 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) and it decays to 0 as $t$ tends to $\infty $ and (ii) a solution $u$ of the first equation in (KS) behaves like the Barenblatt solution asymptotically as $t$ tends to $\infty $, where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation $u_t = \Delta u^m$ with $m>1$.

Détail

Publié le : 2006-01-01

MR 2245322 | Zbl 1113.35028

DOI : https://doi.org/10.1051/m2an:2006025
Classification:  35B40,  35K45,  35K55,  35k65