Decay at infinity for solutions to some fractional parabolic equations

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 $\epsilon >0$ $$\frac{1}{c}{\int }_{[-c,0]}{u}^2(x,t)dt\le C{e}^{-|x{|}^{2+\epsilon }}\forall x\in {\mathbb{R}}^n,$$ then $u\equiv 0$ in ${\mathbb{R}}^n\times [-T,0]$.

arXiv admin note: text overlap with arXiv:2306.00341