Decay at infinity for solutions to some fractional parabolic equations

TL;DR

This paper proves a decay at infinity result for solutions to certain fractional parabolic equations, showing that under specific decay conditions, the solution must be identically zero.

Contribution

It establishes a unique continuation property for fractional parabolic equations with bounded potential, extending classical results to nonlocal operators.

Findings

Solutions decay exponentially at infinity under certain conditions

If the solution's spatial average decays faster than a polynomial rate, then the solution is trivial

The result applies to fractional powers of parabolic operators with bounded potentials

Abstract

For $s \in [1/2, 1)$, let $u$ solve $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^{n} \times [-T, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb R^n \times [-T, 0])} < \infty$. We show that if for some $0< c< T$ and $\epsilon>0$ $$\frac{1}{c} \int_{[-c,0]} u^2(x, t) dt \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb R^n,$$ then $u \equiv 0$ in $\mathbb R^{n} \times [-T, 0]$.