Supersolutions for a class of semilinear heat equations

Abstract

A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative measurable initial data 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 then applied to the model case $f(s)=s^{p}$ with initial data in $L^{q}(\Omega )$, for which an extension of the monotonicity-based existence argument is offered for the critical case ($n(p-1)/2=q>1$) in all dimensions. New sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical ($n(p-1)/2 q>1$) range of parameters. Some possible generalisations of the method to a broader class of equations are discussed.