Concentration for blow-up solutions of the Davey–Stewartson system in R3. (December 2015)

• Record Type: Journal Article
• Authors: Feng, Binhua; Cai, Yuan
• Is Part Of: Nonlinear analysis. Volume 26(2015)
• Page Start: 330
• Page End: 342
• Publication Date: 2015-12
• Subjects: Davey–Stewartson system -- Blow-up solutions -- Concentration; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.06.003
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: This paper is devoted to the analysis of blow-up solutions for the Davey–Stewartson system i u t + Δ u + | u | p u + E ( | u | 2 ) u = 0, ( t, x ) ∈ [ 0, T ) × R 3 where 0 < p < 4 . This equation appears in the description of the evolution of surface water waves. By using the profile decomposition of bounded sequences in H ̇ s c ∩ H ̇ 1, we give some new variational characteristics for the generalized Gagliardo–Nirenberg inequalities. Then, under the assumption that H ̇ s c -norm of the blow-up solution is bounded, we prove that the H ̇ s c -norm of the blow-up solution concentrates at some point and its L p c -norm concentrates as well.