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Decay at infinity for solutions to some fractional parabolic equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 14 March 2024, pp. 1-37
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- Article
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For $s\in [\tfrac {1}{2},\, 1)$
, 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 $0<\mathfrak {K} < T$
and $\epsilon >0$
\[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \]then $u \equiv 0$
in $\mathbb {R}^{n} \times [-T,\, 0]$
.
