A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

Abstract

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem

$$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$

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

$$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$

This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.