Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces (Q1022319)

The authors consider the Keller-Segel system \((KS)\) of chemotaxis modeling the aggregation process of amoebae, cellular slime molds. The purpose of this paper is to show the time global existence of strong solutions to \((KS)\) in \(\mathbb R^n (n \geq 3)\) for small initial data in the scaling invariant class. The uniqueness of strong solutions and the decay property in \(L^p (\mathbb R^n )\) as \(t \rightarrow \infty\) are established. The employed 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.