On a Fujita critical time-fractional semilinear heat equation in the uniformly local weak Zygmund type space

2024/08/27 by Mizuki Kojima, Kojima, Mizuki
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2408.14897

openalex publication_date 2024/08/27 · openalex created_date 2024/09/22 · openalex updated_date 2026/07/28

Abstract

In this paper, we derive sufficient conditions on initial data for the local-in-time solvability of a time-fractional semilinear heat equation with the Fujita exponent in a uniformly local weak Zygmund type space. It is known that the time-fractional problem with the Fujita exponent in the scale critical space L¹(ℝ^N) exhibits the local-in-time solvability in contrast to the unsolvability of the Fujita critical classical semilinear heat equation. Our new sufficient conditions take into account the fine structure of singularities of the initial data, in order to show a natural correspondance between the time-fractional and the classical case for the local-in-time solvability. We also apply our arguments to life span estimates for some typical initial data.

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