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 {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$ 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.

References in corpus (1)

• The influence of fractional diffusion in Fisher-KPP equations

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