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

Let $G:\mathbb{R}\to \mathbb{R}$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on $G$ such that \begin{equation*} {G(f):f\in \dot{K}{p,q}^{\alpha }F{\beta }^{s}}\subset \dot{K}{p,q}^{\alpha }F{\beta }^{s} \end{equation*} holds. Here ${\dot{K}}_{p,q}^{\alpha }{F}_{\beta }^s$ 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 \begin{equation*} \partial _{t}u-\Delta u=G(u) \end{equation*} with 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.