A Widder's type Theorem for the heat equation with nonlocal diffusion

TL;DR

This paper proves a Widder's type theorem for non-negative solutions of the fractional heat equation, establishing a representation formula that guarantees uniqueness and extends classical results to nonlocal diffusion models.

Contribution

It extends Widder's classical theorem to the fractional heat equation, providing a representation formula for solutions with nonlocal diffusion.

Findings

Every non-negative strong solution can be represented via a convolution with the fundamental solution.

The representation formula guarantees uniqueness of solutions in the non-negative class.

The result generalizes classical heat equation theorems to fractional, nonlocal operators.

Abstract

The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-\Delta)^{\alpha/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<\alpha<2, $$ can be written as $$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$ where $$P_{t}(x)=\frac{1}{t^{n/\alpha}}P\left(\frac{x}{t^{1/\alpha}}\right), $$ and $$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdot\xi-|\xi|^{\alpha}}d\xi}. $$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder in \cite{W0} to the nonlocal diffusion framework.