Blow-up criteria for linearly damped nonlinear Schrödinger equations

Van Duong Dinh

doi:10.3934/eect.2020082

Article Contents

2021, Volume 10, Issue 3: 599-617. Doi: 10.3934/eect.2020082

This issue Previous Article On finite Morse index solutions of higher order fractional elliptic equations Next Article Results on controllability of non-densely characterized neutral fractional delay differential system

Blow-up criteria for linearly damped nonlinear Schrödinger equations

Van Duong Dinh^1,2, ,

1.
Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2.
Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

^* Corresponding author: Van Duong Dinh
^* Corresponding author: Van Duong Dinh

Received: January 31, 2020

Revised: April 30, 2020

Early access: July 2020

Published: September 2021

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Abstract

We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations

$ i\partial_t u + \Delta u + i a u = \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb R^N, $

where $ a>0 $ and $ \alpha>0 $. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up $ H^1 $ solutions to the focusing problem in the mass-critical and mass-supercritical cases.

Keywords:

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q44.

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References

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