On the Cauchy problem for a semi-linear hyperdissipative heat equation

TL;DR

This paper investigates the existence, uniqueness, and well-posedness of solutions to a semi-linear hyperdissipative heat equation within Besov and Triebel-Lizorkin spaces, leveraging the properties of the associated semi-group.

Contribution

It establishes local well-posedness results for the semi-linear hyperdissipative heat equation using advanced functional analysis techniques.

Findings

Proves local existence and uniqueness of solutions

Demonstrates well-posedness in specific function spaces

Utilizes semi-group smoothing properties for analysis

Abstract

The paper is concerned with the Cauchy problem for a semi-linear hyperdissipative heat equation in Besov and Triebel-Lizorkin spaces which is related to the generalized Gauss-Weierstrass semi-group via Duhamel's principle. Using caloric smoothing properties of the semi-group we prove existence and uniqueness of mild and strong solutions which are local in time. Moreover, we study well-posedness of the problem.