BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE∗

doi:10.1007/s10473-023-0423-x

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1852-1864.doi: 10.1007/s10473-023-0423-x

BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗

Qianqian BAI¹, Xiaoguang LI², Li ZHANG^2,*

1. School of Mathematics Science and V.C. & V.R. Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610068, China;
2. V.C. & V.R. Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610068, China

收稿日期:2022-02-23 发布日期:2023-08-08
通讯作者: † Li ZHANG, E-mail: lizhang-hit@163.com
作者简介:Qianqian BAI, E-mail: 1370733971@qq.com; Xiaoguang LI, E-mail: Lixgmath@163.com
基金资助:
*National Natural Science Foundation of China (11771314), the Sichuan Science and Technology Program (2022JDTD0019) and the Guizhou Province Science and Technology Basic Project (QianKeHe Basic[2020]1Y011).

BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗

Qianqian BAI¹, Xiaoguang LI², Li ZHANG^2,*

1. School of Mathematics Science and V.C. & V.R. Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610068, China;
2. V.C. & V.R. Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610068, China

Received:2022-02-23 Published:2023-08-08
Contact: † Li ZHANG, E-mail: lizhang-hit@163.com
About author:Qianqian BAI, E-mail: 1370733971@qq.com; Xiaoguang LI, E-mail: Lixgmath@163.com
Supported by:
*National Natural Science Foundation of China (11771314), the Sichuan Science and Technology Program (2022JDTD0019) and the Guizhou Province Science and Technology Basic Project (QianKeHe Basic[2020]1Y011).

摘要/Abstract

摘要： We consider the blow-up solutions to the following coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &{\rm i}u_{t}+\Delta u+(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u=0,\\ &{\rm i}v_{t}+\Delta v+(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v=0,\\ &u(0,x)=u_{0}(x),\ \ \ \ v(0,x)=v_{0}(x),\ \ x\in \mathbb{R}^{N},\ t\geq0. \end{aligned} \right. \end{equation*} On the basis of the conservation of mass and energy, we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions. These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance ($(|x|u_{0},|x|v_{0})\in L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$).

关键词: Schrö, dinger equations, radial symmetry, blow-up, virial identity

Abstract: We consider the blow-up solutions to the following coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &{\rm i}u_{t}+\Delta u+(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u=0,\\ &{\rm i}v_{t}+\Delta v+(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v=0,\\ &u(0,x)=u_{0}(x),\ \ \ \ v(0,x)=v_{0}(x),\ \ x\in \mathbb{R}^{N},\ t\geq0. \end{aligned} \right. \end{equation*} On the basis of the conservation of mass and energy, we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions. These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance ($(|x|u_{0},|x|v_{0})\in L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$).

Key words: Schrö, dinger equations, radial symmetry, blow-up, virial identity

Qianqian BAI, Xiaoguang LI, Li ZHANG. BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1852-1864.

Qianqian BAI, Xiaoguang LI, Li ZHANG. BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1852-1864.

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BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗

BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗

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