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: { ∂_tu- △u^α + div(u▽ψ) = 0, t > 0, x ε ℝⁿ,-△ψ + ψ = u, t> 0, x ε ℝⁿ, u(0,x) = u₀(x) ≥ 0, x ε ℝⁿ, where α > 1. It is known when the exponent a = 2-2/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 U(t) - (1 + t) ^-n/σ(A- |x|²/(1 + t)^1/(σ-1)^ in L¹ such as ||u(t)-u(t)||₁ ≤C(1 + t)^-v, where v > 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 [12] and Ogawa [47]. 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 [18, 19].