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

TL;DR

This paper investigates the existence of solutions to a fractional semilinear heat equation with a singular inhomogeneous term, using decay estimates, interpolation methods, and sharp integral conditions in specialized function spaces.

Contribution

It introduces new decay estimates and sharp integral conditions in Zygmund spaces, advancing understanding of solutions with singular inhomogeneous terms.

Findings

Established decay estimates of the fractional heat semigroup in Zygmund spaces

Derived sharp integral estimates for the inhomogeneous and nonlinear terms

Provided sufficient conditions for the existence of solutions with singular inhomogeneous terms

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.