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

We consider the following reaction-diffusion equation: (KS){ u_t = Δ · (Δ_u^m - u^q-1 Δ_v), x ∈ ℝ^N, 0 < t < ∞, 0 = Δ_v - v + u, x ∈ ℝ, 0 < t < ∞, (u(x,0}=u₀(x), x ∈ ℝ^N, where N ≥ 1, m > 1, q ≥ max{m + 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 ≥ max{m + 2/N, 2}, the above problem (KS) is solvable globally in time for "small L^N(q-m)/2 data". Moreover, the decay of the solution (u, v) in L^p(ℝ^N) was proved. In this paper, we consider the case of "q ≥ max{m+ 2/N, 2} and small L^ℓ data" with any fixed ℓ ≥ ^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 ∞ and (ii) a solution u of the first equation in (KS) behaves like the Barenblatt solution asymptotically as t tends to ∞, where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation u_t = Δ_u^m with m > 1.