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

TL;DR

This paper studies composition operators on Herz-type Triebel-Lizorkin spaces and applies the findings to analyze local and global solutions of semilinear parabolic equations with initial data in these spaces.

Contribution

It establishes conditions for composition operators on Herz-type Triebel-Lizorkin spaces and applies these results to semilinear parabolic equations with initial data in these spaces.

Findings

Conditions for G to preserve Herz-type Triebel-Lizorkin spaces

Existence of solutions to semilinear parabolic equations in these spaces

Extension of known results to more general function spaces

Abstract

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.