Sharp asymptotic of solutions to some nonlocal parabolic equations

Agnid Banerjee; Abhishek Ghosh

doi:10.3934/dcds.2024130

Article Contents

2025, Volume 45, Issue 4: 1297-1334. Doi: 10.3934/dcds.2024130

This issue Previous Article Standing waves of fractional Schrödinger equations with critical nonlinearities and decaying potentials Next Article Normalized solutions for Kirchhoff equations with Choquard nonlinearity

Sharp asymptotic of solutions to some nonlocal parabolic equations

Agnid Banerjee^1, , and
Abhishek Ghosh^{1, 2, 1,}

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe
2.
Institute of Mathematics, Polish Academy of Sciences, Ul. Śniadeckich 8, 00-656 Warsaw, Poland

^*Corresponding author: Agnid Banerjee
^*Corresponding author: Agnid Banerjee

¹A.G. was funded in part by the National Science Center, Poland, grant 2021/43/D/ST1/00667

Received: March 2024

Revised: September 2024

Early access: October 10, 2024

Published online: October 10, 2024

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Abstract

We show that if $ u $ solves the fractional parabolic equation $ (\partial_t - \Delta )^s u = Vu $ in $ B_5 \times (-25, 0] $ ($ 0<s<1 $) such that $ u(\cdot, 0) \not\equiv 0 $, then the maximal vanishing order of $ u $ in space-time at $ (0, 0) $ is upper bounded by $ C\left(1+\|V\|_{C^{1}_{(x, t)}}^{1/2s}\right) $. As $ s \to 1 $, it converges to the sharp maximal order of vanishing due to Donnelly-Fefferman and Bakri. This quantifies a space-like strong unique continuation result recently proved in [3]. The proof is achieved by means of a new quantitative Carleman estimate that we derive for the corresponding extension problem combined with a quantitative monotonicity in time result and a compactness argument.

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Mathematics Subject Classification: 35A02, 35B60, 35K05.

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