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

Abstract

We shall show existence of global strong solution to the semi-linear Keller–Segel system in R n , n ⩾ 3 , of parabolic–parabolic type with small initial data u 0 ∈ H n r − 2 , r ( R n ) and v 0 ∈ H n r , r ( R n ) 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.