Local solvability and dilation-critical singularities of supercritical fractional heat equations

2023/08/09 by Yohei Fujishima, Kotaro Hisa, Fujishima, Yohei +5 · 5 citations
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Fractional Differential Equations Solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.05240

Abstract

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

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