## Regularity and Asymptotic Behavior for the Keller-Segel System of Degenerate Type with Critical Nonlinearity

J. Math. Sci. Univ. Tokyo
Vol. 20 (2013), No. 3, Page 375–433.

Mizuno, Masashi; Ogawa, Takayoshi
Regularity and Asymptotic Behavior for the Keller-Segel System of Degenerate Type with Critical Nonlinearity
We discuss the large time behavior of a weak solution of the Keller-Segel system of degenerate type: \begin{equation}\notag \left\{ \begin{aligned} \partial_tu&-\Delta u^\alpha+\Div(u\nabla\psi)=0, &\quad t>0\,,\,x\in\Bbb R^n, \\ &-\Delta\psi+\psi=u, &\quad t>0\,,\,x\in\Bbb R^n, \\ &\quad u(0,x)=u_0(x)\geq0, &\quad x\in\Bbb R^n, \end{aligned} \right. \end{equation} where $\alpha>1$. It is known when the exponent $\alpha=2-\frac{2}{n}$, then the problem shows the critical situation. In this case, we show that the small data global solution decays and its asymptotic profile converges to the Barenblatt-Pattle solution $\mathcal{U}(t)=(1+t)^{-n/\sigma}\big(A|x|^2/(1+t)^{1/\sigma}\big)_+^{1/(\alpha-1)}$ in $L^1$ such as $\|u(t)-\mathcal{U}(t)\|_1\le C(1+t)^{-\nu},$ where $\nu>0$ is depending on $n$ and the regularity of the solution. To show this, we employ the forward self-similar transform and use the entropy dissipation term to derive the asymptotic profile due to Carrillo-Toscani \cite{C-T} and Ogawa \cite{Og2}. The Hölder continuity of the weak solution for the forward self-similar equation plays a crucial role. We derive the uniform Hölder continuity by using the rescaled alternative selection originated by DiBenedetto-Friedman~\cite{Db-F1,Db-F2}.