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

Robert Laister; Mikolaj Sierzega

doi:10.1016/j.anihpc.2019.12.001

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arXiv · 1911.10530

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

Robert Laister·Mikolaj Sierzega

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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 $L^1$.

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Robert Laister, Mikolaj Sierzega. 2019-11-24. Well-posedness of Semilinear Heat Equations in $L^1$. https://doi.org/10.1016/j.anihpc.2019.12.001

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