Composition operators on Herz-type Triebel–Lizorkin spaces with application to semilinear parabolic equations

Let G : R → R be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such that { G ( f ) : f ∈ K ˙ p , q α F β s } ⊂ K ˙ p , q α F β s holds. Here K ˙ p , q α F β s are Herz-type Triebel–Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel–Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabolic equations ∂ t u - Δ u = G ( u ) with initial data in Herz-type Triebel–Lizorkin spaces. Our results cover the results obtained with initial data in some known function spaces such us fractional Sobolev spaces. Some limit cases are given.