非線形放物型問題の解の存在･非存在

要約のみTohoku University岡部真也課

Optimal singularities of initial data of a fractional semilinear heat equation in open sets

We consider necessary conditions and sufficient conditions on the solvability of the Cauchy--Dirichlet problem for a fractional semilinear heat equation in open sets (possibly unbounded and disconnected) with a smooth boundary. Our conditions enable us to identify the optimal strength of the admissible singularity of initial data for the local-in-time solvability and they differ in the interior of the set and on the boundary of the set.Comment: 12/20:Fixed some typos. This article draws heavily from arXiv:2209.0639

Solvability of Superlinear Fractional Parabolic Equations

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure

Solvability of superlinear fractional parabolic equations

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure

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

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