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April 2024 Asymptotics of solutions to the fractional nonlinear Schrödinger equation with $\alpha \gt \frac{5}{2}$
Nakao Hayashi, Naumkin Pavel I., Isahi Sánchez-suárez
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Osaka J. Math. 61(2): 163-193 (April 2024).


We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional nonlinear Schrödinger equation \begin{equation*}\left\{ \begin{array}{c}i\partial _{t}u-\frac{1}{\alpha }\left\vert \partial _{x}\right\vert^{\alpha }u=\lambda \left\vert u\right\vert ^{\alpha }u,\text{ }t\gt 0,\text{ } x\in \mathbb{R}\mathbf{,} \ u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in \mathbb{R}\mathbf{,}\end{array}\right.\end{equation*}where $\lambda \gt 0,$ the fractional derivative $\left\vert \partial_{x}\right\vert ^{\alpha }=\mathcal{F}^{-1}\left\vert \xi \right\vert^{\alpha }\mathcal{F},$ $\alpha \gt \frac{5}{2}.$ This paper is a sequel to our previous papers [17] for $2\lt \alpha \lt \frac{5}{2}$ and [36] for $\alpha =\frac{5}{2}$. We show that solutions decay in time at the rate $t^{-\frac{1}{\alpha }}\left( \log t\right) ^{-\frac{1}{\alpha }}$, namely that the nonlinearity acts as a dissipative term, when $\lambda \gt 0$. This phenomena does not occur for the cubic problem \begin{equation*}\left\{ \begin{array}{c}i\partial _{t}u-\frac{1}{\alpha }\left\vert \partial _{x}\right\vert^{\alpha }u=\lambda \left\vert u\right\vert ^{2}u,\text{ }t\gt 0,\text{ }x\in \mathbb{R}\mathbf{,} \ u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in \mathbb{R}\mathbf{,}\end{array}\right.\end{equation*}with $0 \lt \alpha \leq 2.$


We would like to thank the referee for useful suggestions on the draft. The work of N.H. is partially supported by JSPS KAKENHI Grant Numbers JP20K03680, JP19H05597. The work of P.I.N. is partially supported by CONACYT and PAPIIT project IN103221.


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Nakao Hayashi. Naumkin Pavel I.. Isahi Sánchez-suárez. "Asymptotics of solutions to the fractional nonlinear Schrödinger equation with $\alpha \gt \frac{5}{2}$." Osaka J. Math. 61 (2) 163 - 193, April 2024.


Received: 5 October 2022; Revised: 10 January 2023; Published: April 2024
First available in Project Euclid: 7 April 2024

MathSciNet: MR4729040

Primary: 35B40
Secondary: 35Q92

Rights: Copyright © 2024 Osaka University and Osaka Metropolitan University, Departments of Mathematics

Vol.61 • No. 2 • April 2024
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