Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

Chao Yang

doi:10.3934/dcdss.2021136

Article Contents

2021, Volume 14, Issue 12: 4631-4642. Doi: 10.3934/dcdss.2021136 open access

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Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

Chao Yang^,

College of Mathematical Sciences, Harbin Engineering University, No. 145 Nantong Street, Harbin 150001, China

^* Corresponding author: Chao Yang
^* Corresponding author: Chao Yang

Received: August 31, 2021

Revised: September 30, 2021

Early access: October 2021

Published: December 2021

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Abstract

This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term $ V(x)|\varphi|^{p-1}\varphi $ in $ \mathbb{R}^n $. For the case $ p > 1+\frac{4(1+\varepsilon_0)}{n} (0 < \varepsilon_0 < \frac{2}{n-2}) $, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case $ p < 1+\frac{4}{n} $, we obtain the global existence of solution for any initial data in $ H^1 (\mathbb{R}^n) $.

Keywords:

Inhomogeneous nonlinear Schrödinger equation,
potential well,
global existence,
blowup.

Mathematics Subject Classification: Primary: 35Q55, 35G25.

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References

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