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

Let $$G:\mathbb {R}\rightarrow \mathbb {R}$$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such that $$\begin{aligned} \{G(f):f\in {\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset {\dot{K}} _{p,q}^{\alpha }^{s} \end{aligned}$$ holds. Here $${\dot{K}}_{p,q}^{\alpha }^{s}$$ are Herz-type Triebel–Lizorkin spaces. These spaces unify and generalize many classical function as Lebesgue power weights, Sobolev weights. second paper will study local global Cauchy problems for semilinear parabolic equations \partial _{t}u-\Delta u=G(u) with initial data in Our results cover obtained some known us fractional Some limit cases given.