Blow-up of rough solutions to the fourth-order nonlinear Schrödinger equation

• Published in: Nonlinear analysis Vol. 74; no. 17; pp. 6186 - 6201
• Main Authors: Zhu, Shihui, Yang, Han, Zhang, Jian
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.12.2011; Elsevier
• Subjects: [formula omitted]-concentration; [formula omitted]-method; Exact sciences and technology; Limiting profile; Mathematical analysis; Mathematics; Nonlinear Schrödinger equation; Partial differential equations; Rough blow-up solution; Sciences and techniques of general use; I -method; Limiting profile; 35B44; 35Q55; Rough blow-up solution; Nonlinear Schrödinger equation; L 2 -concentration; Nonlinear analysis; Fourth order equation; concentration; L; Schrödinger equation; Non linear equation; I-method; Mathematical model
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.05.096

This paper deals with the formation of singularities of rough blow-up solutions to the fourth-order nonlinear Schrödinger equation. The limiting profile and L 2 -concentration of the rough blow-up solutions are obtained in H s ( R 4 ) with s > s 0 , where s 0 ≤ 9 + 721 20 ≈ 1.793 . The new ingredient relies on the refined compactness result developed by Zhu et al. [S.H. Zhu, J. Zhang, H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ. 7 (2010) 187–205].