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

Fujishima, Yohei; Hisa, Kotaro; Ishige, Kazuhiro; Laister, Robert

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.

• Journal Article Type: Article
• Acceptance Date: Feb 19, 2024
• Online Publication Date: May 3, 2024
• Publication Date: Jun 30, 2024
• Deposit Date: Mar 11, 2024
• Publicly Available Date: May 4, 2026
• Journal: Journal des Mathematiques Pures et Appliquees
• Print ISSN: 0021-7824
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 186
• Pages: 150-175
• DOI: https://doi.org/10.1016/j.matpur.2024.04.005
• Keywords: superlinear fractional parabolic equations; solvability; supercritical nonlinear- ity; dilation-critical singularity 1
• Public URL: https://uwe-repository.worktribe.com/output/11752935