Abstract
In this short note, we prove that if u solves \((\partial _t - \Delta )^s u = Vu\) in \({\mathbb {R}}^n_x \times {\mathbb {R}}_t\), and vanishes to infinite order at a point \((x_0, t_0)\), then \(u \equiv 0\) in \({\mathbb {R}}^n_x \times {\mathbb {R}}_t\). This sharpens (and completes) our earlier result that proves \(u(\cdot , t) \equiv 0\) for \(t \le t_0\) if it vanishes to infinite order at \((x_0, t_0)\).
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A. Banerjee is supported in part by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520. N. Garofalo is supported in part by a BIRD grant: “Aspects of nonlocal operators via fine properties of heat kernels”, Univ. of Padova, 2022.
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Banerjee, A., Garofalo, N. On the forward in time propagation of zeros in fractional heat type problems. Arch. Math. 121, 189–195 (2023). https://doi.org/10.1007/s00013-023-01886-7
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DOI: https://doi.org/10.1007/s00013-023-01886-7