Optimal Existence and Uniqueness Theory for the Fractional Heat Equation

TL;DR

This paper develops a comprehensive theory for the existence, uniqueness, and regularity of solutions to the fractional heat equation in the whole space, accommodating data with controlled growth at infinity.

Contribution

It introduces an optimal framework for solutions with measure data, establishing equivalence between data and solutions, and deriving new estimates and inequalities.

Findings

Established existence and uniqueness for measure data solutions.

Proved optimal pointwise estimates for solutions.

Derived new Harnack inequalities for the fractional heat equation.

Abstract

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-\Delta)^s u=0$, $0<s<1$, posed in the whole space $\mathbb{R}^N$ with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions, which is given in one direction by the representation formula, in the other one by the initial trace. We review many of the typical properties of the solutions, in particular we prove optimal pointwise estimates and new Harnack inequalities.