IUMJ

Title: Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases

Authors: Stephan Luckhaus and Yoshie Sugiyama

Issue: Volume 56 (2007), Issue 3, 1279-1298

Abstract:

We consider the following reaction-diffusion equation: \begin{equation}\label{KS}\begin{cases} u_t = \nabla \cdot ( \nabla u^m - u^{q-1} \nabla v), & x \in \mathbb{R}^N, \ 0 < t < \infty, \\ 0 = \Delta v - v + u, & x \in \mathbb{R}^N, \ 0 < t < \infty,\\ u(x,0) = u_0(x), & x \in \mathbb{R}^N, \end{cases} \tag{KS}\end{equation} where $N \ge 1$, $m \ge 1$, and $q \ge m + 2/N$ with $q > \frac{3}{2}$.\par In our previous work [S. Luckhaus and Y. Sugiyama, \emph{Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems}, Math. Model. Numer. Anal. \textbf{40} (2006), 597--621], in the case of $m > 1$, $q \ge 2$, $q > m + 2/N$, we showed that a solution $u$ to the first equation in \eqref{KS} behaves like "the Barenblatt solution" asymptotically as $t \to \infty$, where the Barenblatt solution is well known as the exact solution to $u_t = Delta u^m$ ($m > 1$). In this paper, we improve the result obtained in S. Luckhaus and Y. Sugiyama [\emph{op.~cit.}] and establish the optimal convergence rate for the asymptotic profile. In particular, our new result covers the critical case when \[ q = m + \frac2N . \] We also consider the semilinear case of $m = 1$ and prove that $u$ behaves like "the heat kernel" asymptotically as $t \to \infty$ when $q \ge 1 + 2/N$.