We consider the following reaction-diffusion equation: $$\left(\mathrm{KS}\right)\left\{\begin{array}{cc}{u}_t=\nabla ·\left(\nabla u^m-u^{q-1}\nabla v\right),\hfill & x\in {ℝ}^N,0<t<\infty ,\hfill \\ 0=\Delta v-v+u,\hfill & x\in {ℝ}^N,0<t<\infty ,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {ℝ}^N,\hfill \end{array}\right.$$ 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 $L^p\left({ℝ}^N\right)$ 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) 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 $m>1$.

DOI : https://doi.org/10.1051/m2an:2006025
Classification:  35B40,  35K45,  35K55,  35k65
Keywords: degenerate parabolic system, chemotaxis, Keller-Segel model, drift term, decay property, asymptotic behavior, Fujita exponent, porous medium equation, Barenblatt solution

@article{M2AN_2006__40_3_597_0,
     author = {Luckhaus, Stephan and Sugiyama, Yoshie},
     title = {Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {3},
     year = {2006},
     pages = {597-621},
     doi = {10.1051/m2an:2006025},
     zbl = {1113.35028},
     mrnumber = {2245322},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_3_597_0}
}

Luckhaus, Stephan; Sugiyama, Yoshie. Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 3, pp. 597-621. doi : 10.1051/m2an:2006025. http://www.numdam.org/item/M2AN_2006__40_3_597_0/