数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (3): 1181-1192.doi: 10.1016/S0252-9602(12)60090-3

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LIMITING BEHAVIOR OF BLOW-UP SOLUTIONS OF THE NLSE WITH A STARK POTENTIAL

朱世辉|张健   

  1. 1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China;
    2. Visual Computing and Virtual Reality Key Laboratory, Sichuan Normal University, Chengdu 610066, China
  • 收稿日期:2010-10-27 出版日期:2012-05-20 发布日期:2012-05-20
  • 基金资助:

    Supported by National Science Foundation of China (11071177) and Excellent Youth Foundation of Sichuan Province (2012JQ0011).

LIMITING BEHAVIOR OF BLOW-UP SOLUTIONS OF THE NLSE WITH A STARK POTENTIAL

 ZHU Shi-Hui, ZHANG Jian   

  1. 1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China;
    2. Visual Computing and Virtual Reality Key Laboratory, Sichuan Normal University, Chengdu 610066, China
  • Received:2010-10-27 Online:2012-05-20 Published:2012-05-20
  • Supported by:

    Supported by National Science Foundation of China (11071177) and Excellent Youth Foundation of Sichuan Province (2012JQ0011).

摘要:

This article is concerned with blow-up solutions of the Cauchy problem of critical nonlinear Schr¨odinger equation with a Stark potential. By using the variational characterization of corresponding ground state, the limiting behavior of blow-up solutions with critical and small super-critical mass are obtained in the natural energy space ∑ ={u H1; ∫RN |x|2|u|2dx < +∞}. Moreover, an interesting concentration property of the blow-up solutions with critical mass is gotten, which reads that |u(t, x)|2 → ||Q||2 Lδx=x1 as tT.

关键词: Nonlinear Schr¨odinger equation, Stark potential, blow-up solution, limiting behavior, mass concentration

Abstract:

This article is concerned with blow-up solutions of the Cauchy problem of critical nonlinear Schr¨odinger equation with a Stark potential. By using the variational characterization of corresponding ground state, the limiting behavior of blow-up solutions with critical and small super-critical mass are obtained in the natural energy space ∑ ={u H1; ∫RN |x|2|u|2dx < +∞}. Moreover, an interesting concentration property of the blow-up solutions with critical mass is gotten, which reads that |u(t, x)|2 → ||Q||2 Lδx=x1 as tT.

Key words: Nonlinear Schr¨odinger equation, Stark potential, blow-up solution, limiting behavior, mass concentration

中图分类号: 

  • 35Q55