The Keller-Segel system of parabolic-parabolic type with initial data in weak L^n/2(ℝⁿ) and its application to self-similar solutions

We shall show the existence of a global strong solution to the semilinear Keller-Segel system in ℝⁿ, n ≥ 3 of parabolic-parabolic type with small initial data u₀ ∈ L_w^n/2 (ℝⁿ) and v₀ ∈ BMO. Our method is based on the perturbation of linearization together with the L^p -L ^q-estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall construct a self-similar solution and prove the smoothing effect. Furthermore, the stability problem on our strong solutions will be also discussed.