Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces

We shall show existence of global strong solution to the semi-linear Keller-Segel system in Rⁿ, n ≥ 3, of parabolic-parabolic type with small initial data u₀ ∈ H^{frac(n, r) - 2, r} (Rⁿ) and v₀ ∈ H^{frac(n, r), r} (Rⁿ) for max {1, n / 4} < r < n / 2. Our method is based on the perturbation of linealization 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 prove the decay property of solutions as the time goes to infinity.