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

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 $m>1$.