Global uniqueness for the fractional semilinear Schrödinger equation

Lai, Ru-Yu; Lin, Yi-Hsuan

doi:10.1090/proc/14319

Global uniqueness for the fractional semilinear Schrödinger equation
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by Ru-Yu Lai and Yi-Hsuan Lin

Proc. Amer. Math. Soc. 147 (2019), 1189-1199

DOI: https://doi.org/10.1090/proc/14319

Published electronically: November 16, 2018

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Abstract:

We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-\Delta )^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide an $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.

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