Supersolutions for a class of semilinear heat 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 $\Omega \subseteq {\mathbb{R}}^n$. 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 $f(s)=s^p$, $\varphi \in L^q(\Omega )$: 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.

arXiv commentExpanded version of the previous submission arXiv:1111.0258v1. 14 pages