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Let $G:\mathbb{R\rightarrow R}$ be a continuous function. In the first partof 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 $\dot{K}_{p,q}^{\alpha }F_{\beta }^{s}$ areHerz-type Triebel-Lizorkin spaces. These spaces unify and generalize manyclassical function spaces such as Lebesgue spaces of power weights, Sobolev andTriebel-Lizorkin spaces of power weights. In the second part of this paper wewill study local and global Cauchy problems for the semilinear parabolicequations  \begin{equation*} \partial _{t}u-\Delta u=G(u)  \end{equation*} with initial data in Herz-type Triebel-Lizorkin spaces. Ourresults cover the results obtained with initial data in some know functionspaces such us fractional Sobolev spaces. Some limit cases are given.

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