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

We consider the following reaction-diffusion equation: KS = {(u_t = ▽ · (▽u^m - u^q-1 ▽v), x ∈ ℝ^N, 0 < t < ∞, 0 = ▽v - v + u, x ∈ ℝ^N, 0 < t < ∞, u(x, 0) = u₀(x), x ∈ ℝ^N, where N ≥ ≥ 1, m ≥ 1, and q ≥ m + 2/N with q > 3/2. In our previous work [14], in the case of m > 1, q ≥ 2, q > m + 2/N, we showed that a solution u to the first equation in (KS) behaves like "the Barenblatt solution" asymptotically as t → ∞, where the Barenblatt solution is well known as the exact solution to u_t = Δu^m (m > 1). In this paper, we improve the result obtained in [14] and establish the optimal convergence rate for the asymptotic profile. In particular, our new result covers the critical case when We also consider the semilinear case of m = 1 and prove that u behaves like "the heat kernel" asymptotically as t → ∞ when q ≥1 + 2/N. Indiana University Mathematics Journal