Concentration for blow-up solutions of the Davey–Stewartson system in [formula omitted

Abstract

This paper is devoted to the analysis of blow-up solutions for the Davey–Stewartson system iut+Δu+|u|pu+E(|u|2)u=0, (t,x)∈[0,T)×R3 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 Ḣsc∩Ḣ1, we give some new variational characteristics for the generalized Gagliardo–Nirenberg inequalities. Then, under the assumption that Ḣsc-norm of the blow-up solution is bounded, we prove that the Ḣsc-norm of the blow-up solution concentrates at some point and its Lpc-norm concentrates as well.