Modified scattering for the critical nonlinear Schrodinger equation

We consider the nonlinear Schrodinger equation iu(t) + Delta u = lambda broken vertical bar u broken vertical bar(2/N) u in all dimensions N >= 1, where lambda is an element of C and (sic)lambda <= 0. We construct a class of initial values for which the corresponding solution is global and decays as t -> infinity, like t(-N/2) if (sic)lambda = 0 and like (t log t)(-N/2) if (sic)lambda < 0. Moreover, we give an asymptotic expansion of those solutions as t -> infinity. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at u = 0. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents. (c) 2017 Elsevier Inc. All rights reserved.