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

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 $L^1(\mathbb{R}^N)$ 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.