  Let $G:\mathbb{R\rightarrow R}$ 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.
