Supersolutions for a class of semilinear heat equations

TL;DR

This paper introduces a simple supersolution-based method for establishing existence and regularity of solutions to semilinear heat equations, providing new conditions especially in critical and subcritical cases.

Contribution

It presents a novel, straightforward technique for constructing supersolutions to prove existence and regularity, extending to broader classes of equations.

Findings

New sufficient conditions for local and global solutions in critical/subcritical ranges

A simple supersolution construction technique demonstrated

Potential for generalization to other semilinear equations

Abstract

A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative initial data in a subset of $L^{1}(\Omega)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\Omega\subseteq \R^{n}$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case $f(s)=s^{p}$, $\phi\in L^{q}(\Omega)$: new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of parameters. Some possible generalisations of the method to a broader class of equations are discussed.