Dissipative nonlinear schrödinger equations for large data in one space dimension

Gaku Hoshino

doi:10.3934/cpaa.2020044

Article Contents

2020, Volume 19, Issue 2: 967-981. Doi: 10.3934/cpaa.2020044

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Dissipative nonlinear schrödinger equations for large data in one space dimension

Gaku Hoshino

Tokyo Denki University, Division of Science, Hatoyama, Saitama, 350-0394, Japan

Received: February 2019

Revised: June 2019

Published: October 2019

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Abstract

In this study, we consider the global Cauchy problem for the nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension. In particular, we show the global existence, smoothing effect and asymptotic behavior for solutions to the nonlinear Schrödinger equations with data which belong to $ \mathcal{F}H^\gamma, $ $ 1/2<\gamma\leq 1. $ In the proof of main theorem, we introduce a priori estimate for $ H^\gamma $-type norm and the condition $ \mathcal{F}H^1 $ for data relaxed into $ \mathcal{F}H^\gamma, $ $ 1/2<\gamma\leq1. $

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q40.

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