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

Abstract

Let G : R → R be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such thatHere Kα p,q F s β 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 equationswith initial data in Herz-type Triebel-Lizorkin spaces. Our results cover the results obtained with initial data in some know function spaces such us fractional Sobolev spaces. Some limit cases are given. MSC classification (2010): 46E35, 47H30, 35K45, 35K55.