Supersolutions for a class of semilinear heat equations

A semilinear heat equation $u_t=\mathrm{\Delta }u+f(u)$ with nonnegative initial data in a subset of $L^1(\mathrm{\Omega })$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$. 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 $f(s)=s^p$, $\varphi \in L^q(\mathrm{\Omega })$: 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.