Upper and lower $ L^2 $-decay bounds for a class of derivative nonlinear Schrödinger equations

Chunhua Li; Yoshinori Nishii; Yuji Sagawa; Hideaki Sunagawa

doi:10.3934/dcds.2022129

Article Contents

2022, Volume 42, Issue 12: 5893-5908. Doi: 10.3934/dcds.2022129

This issue Previous Article Anderson localization for multi-frequency quasi-periodic Jacobi operators Next Article Infinitely many positive solutions for a class of semilinear elliptic equations

Upper and lower $ L^2 $-decay bounds for a class of derivative nonlinear Schrödinger equations

1.
Department of Mathematics, College of Science, Yanbian University, 977 Gongyuan Road, Yanji, Jilin 133002, China
2.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
3.
Department of Mathematics, Chiba Institute of Technology, 2-1-1 Shibazono, Narashino, Chiba 275-0023, Japan
4.
Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

^*Corresponding author
^*Corresponding author

Received: April 2022

Revised: August 2022

Early access: September 2022

Published: December 2022

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Abstract

We consider the initial value problem for cubic derivative nonlinear Schrödinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $ O((\log t)^{-1/4}) $ in $ L^2 $ as $ t\to +\infty $. Furthermore, we find that this $ L^2 $-decay rate is optimal by giving a lower estimate of the same order.

Keywords:

Derivative nonlinear Schrödinger equation,
weakly dissipative structure,
optimal $ L^2 $-decay rate.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.

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