Concentration for blow-up solutions of the Davey-Stewartson system in R3

This paper is devoted to the analysis of blow-up solutions for the Davey-Stewartson system iu(t) + Delta u + vertical bar u vertical bar(p)u + E(vertical bar u vertical bar(2))u = 0, (t, x) is an element of [0, T) x 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) over dot(Sc) boolean AND (H) over dot(1), we give some new variational characteristics for the generalized Gagliardo-Nirenberg inequalities. Then, under the assumption that (H) over dot(Sc)-norm of the blow-up solution is bounded, we prove that the (H) over dot(Sc)-norm of the blow-up solution concentrates at some point and its L-Pc-norm concentrates as well. (C) 2015 Elsevier Ltd. All rights reserved.