Concentration for blow-up solutions of the Davey–Stewartson system in R3. (December 2015) - Feng, Binhua; Cai, Yuan
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.