Blowup results and concentration in focusing Schrödinger-Hartree equation

Yingying Xie; Jian Su; Liquan Mei

doi:10.3934/dcds.2020209

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2020, Volume 40, Issue 8: 5001-5017. Doi: 10.3934/dcds.2020209

This issue Previous Article Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times Next Article

$ L^p $

Neumann problems in homogenization of general elliptic operators

Blowup results and concentration in focusing Schrödinger-Hartree equation

School of Mathematics and Statistics, Jiaotong University, Xi'an, Shanxi 710049, China

^* Corresponding author: Liquan Mei
^* Corresponding author: Liquan Mei

Received: September 30, 2019

Published: May 2020

The work is partially supported by Science Challenge Project, No. TZ2016002

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Abstract

This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in $ \dot{H}^1(\mathbb{R}^N)\cap\dot{H}^{S_c}(\mathbb{R}^N) $ and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded $ \dot{H}^{S_c} $ norm definitely have concentration properties related to a fixed $ \dot{H}^{S_c} $ norm of certain standing waves.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 74H35.

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Figure 1. Comparison of energy-mass

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Figure 2. Comparison of $ H^1 $ norm

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