AAM  >> Vol. 5 No. 4 (November 2016)

    一类四阶非线性Schro¨dinger方程的爆破准则
    Blow-Up Criteria for a Kind of Fourth Order Nonlinear Schro¨dinger Equations

  • 全文下载: PDF(560KB) HTML   XML   PP.672-682   DOI: 10.12677/AAM.2016.54079  
  • 下载量: 907  浏览量: 2,919  

作者:  

米彩莲,卢美虹,杨晗:西南交通大学数学学院,四川 成都

关键词:
非线性Schro¨dinger方程初值问题B-G型不等式整体解爆破准则Nonlinear Schro¨dinger Equation Initial Value Problem B-G Type Inequality Global Solution Blow-Up Criteria

摘要:

本文通过改进的B-G型不等式研究了一类四阶非线性Schro¨dinger方程的初边值问题。首先借助半群理论得到初值问题局部解的存在唯一性,其次利用B-G型不等式得到了初值问题经典解整体存在的一个新判定准则,即整体解是否存在可由其H2范数是否爆破决定。

The initial-boundary value problem for a kind of fourth order nonlinear Schro¨dinger equations is studied in this paper. Firstly, with the help of the semi-group theory, the existence and uniqueness of local solution of initial value problem is obtained. Secondly, a new global existence criterion for the classical solution is given by using B-G inequality, namely, that whether the solution globally exists is determined by whether its H2 norm blows up.

文章引用:
米彩莲, 卢美虹, 杨晗. 一类四阶非线性Schro¨dinger方程的爆破准则[J]. 应用数学进展, 2016, 5(4): 672-682. http://dx.doi.org/10.12677/AAM.2016.54079

参考文献

[1] Brézis, H. and Gallouët, T. (1980) Nonlinear Schrödinger Evolution Equations. Nonlinear Analysis: Theory, Methods and Applications, 4, 677-681.
https:/doi.org/10.1016/0362-546X(80)90068-1
[2] Tsutsumi, M. (1989) On Smooth Solutions to the Initial-Boundary Value Problem for the Nonlinear Schrödinger Equation in Two Space Dimensions. Nonlinear Analysis: Theory, Methods and Applications, 13, 1051-1056.
https:/doi.org/10.1016/0362-546X(89)90094-1
[3] Ozawa, T. and Visciglia, N. (2015) An Improvement on the Brézis-Gallouët Technique for 2D NLS and 1D Half-Wave Equation. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 33, 1069-1079.
[4] Karpman, V.I. and Shagalov, A.G. (2000) Stability of Solitons Described by Nonlinear Schrödinger-Type Equations with Higher-Order Dispersion. Physica D: Nonlinear Phenomena, 144, 194-210.
https:/doi.org/10.1016/S0167-2789(00)00078-6
[5] Zhu, S., Yang, H. and Zhang, J. (2011) Blow-Up of Rough Solutions to the Fourth-Order Nonlinear Schrödinger Equation. Nonlinear Analysis: Theory, Methods and Applications, 74, 6186-6201.
https:/doi.org/10.1016/j.na.2011.05.096
[6] Zheng, S.M. (2004) Nonlinear Evolution Equations. Chapman & Hall/CRC, London, 56-57.
https:/doi.org/10.1201/9780203492222