scholarly journals Global well-posedness and scattering for the defocusingH12-subcritical Hartree equation inRd

2009 ◽  
Vol 26 (5) ◽  
pp. 1831-1852 ◽  
Author(s):  
Changxing Miao ◽  
Guixiang Xu ◽  
Lifeng Zhao
2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.


2017 ◽  
Vol 37 (4) ◽  
pp. 941-948 ◽  
Author(s):  
Lingyan YANG ◽  
Xiaoguang LI ◽  
Yonghong WU ◽  
Louis CACCETTA

Author(s):  
Anudeep K. Arora ◽  
Oscar Riaño ◽  
Svetlana Roudenko

We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.


2019 ◽  
Vol 22 (02) ◽  
pp. 1950004 ◽  
Author(s):  
Changxing Miao ◽  
Guixiang Xu ◽  
Jianwei-Urbain Yang

By [Formula: see text]-method, the interaction Morawetz estimate, long-time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation [Formula: see text] where [Formula: see text], and [Formula: see text], [Formula: see text], with radial data in [Formula: see text] for [Formula: see text]. It is a sharp global result except the critical case [Formula: see text], which is a very difficult open problem.


2020 ◽  
Vol 9999 (9999) ◽  
pp. 1-20
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko ◽  
Kai Yang

In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.


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