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 and for max{1,n/4}<r<n/2. Our method is based on the perturbation of linealization together with the Lp-Lq 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.