Supersolutions for a class of semilinear heat equations

James C. Robinson; Mikolaj Sierzega

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arXiv · 1111.0258

Supersolutions for a class of semilinear heat equations

James C. Robinson·Mikolaj Sierzega

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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.

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James C. Robinson, Mikolaj Sierzega. 2011-11-01. Supersolutions for a class of semilinear heat equations. https://arxiv.org/abs/1111.0258

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