On the Cauchy problem for the Schrödinger-Hartree equation

Binhua  Feng; Xiangxia  Yuan

doi:10.3934/eect.2015.4.431

Article Contents

2015, Volume 4, Issue 4: 431-445. Doi: 10.3934/eect.2015.4.431

This issue Previous Article On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions Next Article Mathematics of nonlinear acoustics

On the Cauchy problem for the Schrödinger-Hartree equation

Binhua Feng^1, and
Xiangxia Yuan^1,

1.
Department of Mathematics, Northwest Normal University, Lanzhou 730070

Received: July 31, 2015

Revised: September 30, 2015

Published: November 2015

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Abstract

In this paper, we undertake a comprehensive study for the Schrödinger-Hartree equation \begin{equation*} iu_t +\Delta u+ \lambda (I_\alpha \ast |u|^{p})|u|^{p-2}u=0, \end{equation*} where $I_\alpha$ is the Riesz potential. Firstly, we address questions related to local and global well-posedness, finite time blow-up. Secondly, we derive the best constant of a Gagliardo-Nirenberg type inequality. Thirdly, the mass concentration is established for all the blow-up solutions in the $L^2$-critical case. Finally, the dynamics of the blow-up solutions with critical mass is in detail investigated in terms of the ground state.

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Mathematics Subject Classification: 35Q51, 35Q55.

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