Well-posedness of Semilinear Heat Equations in $L^1$

TL;DR

This paper investigates the fundamental properties of solutions to semilinear heat equations with initial data in L^1, establishing conditions for existence, uniqueness, regularity, and global continuation under integral constraints on the source term.

Contribution

It extends previous work by analyzing solutions with indefinite sign initial data in L^1, providing new results on uniqueness, regularity, and global existence under integral conditions.

Findings

Established uniqueness and regularity of solutions in L^1

Derived conditions for global-in-time continuation for small initial data

Confirmed the necessity of integral conditions on the source term

Abstract

The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term $f$ has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in $L^1$, a necessary and sufficient integral condition on $f$ emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with $L^1$ initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We also establish sufficient conditions for the global-in-time continuation of solutions for small initial data in $L^1$.