Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term

TL;DR

The paper addresses the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term by deriving decay estimates for the fractional heat semigroup in uniformly local Zygmund spaces and applying a real interpolation framework. This approach yields sharp small-data solvability criteria in the critical case $p=p_*$ and the supercritical regime $p>p_*$, formulated in terms of the uniformly local Zygmund spaces and their weak variants. The main contributions are precise linear and nonlinear estimates for the inhomogeneous term and the nonlinearity, combined with contraction mappings in tailored function spaces to obtain local (and sometimes global) solutions. The methodology advances understanding of how singular inhomogeneities affect solvability in fractional parabolic problems and provides sharp, scale-appropriate conditions for existence.

Abstract

We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces. Furthermore, we apply the real interpolation method in uniformly local Zygmund spaces to obtain sharp integral estimates on the inhomogeneous term and the nonlinear term. This enables us to find sharp sufficient conditions for the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term.