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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 $\mathbb {R}^{n} \times [-T,\, 0]$

Summary

We address the decay and the quantitative uniqueness properties for solutions of the elliptic equation with a gradient term, $$\Delta u=W\cdot \nabla u$$. We prove that there exists a solution in a complement of the unit ball which satisfies $$|u(x)|\le C\exp (-C^{-1}|x|^2)$$ where $$W$$ is a certain function bounded by a constant. Next, we revisit the quantitative uniquenessfor the equation$$-\Delta u= W \cdot \nabla u$$ and provide an example of a solution vanishing at a point with the rate$${\rm const}\Vert W\Vert_{L^\infty}^2$$.We also review decay and vanishing results for the equation $$\Delta u= V u$$.

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Decay at infinity for solutions to some fractional parabolic equations

4 - On localization and quantitative uniqueness for elliptic partial differential equations

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Decay at infinity for solutions to some fractional parabolic equations

4 - On localization and quantitative uniqueness for elliptic partial differential equations

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