Modified scattering for the critical nonlinear Schrödinger equation

Original · EN

We consider the nonlinear Schrödinger equation iuₜ + Δu= λ|u|²/ⁿ u in all dimensions N≥ 1, where λ∈ C and λ≤ 0. We construct a class of initial values for which the corresponding solution is global and decays as t→ ∞, like t⁻ ⁿ/² if λ=0 and like (t t)⁻ ⁿ/² if λ<0. Moreover, we give an asymptotic expansion of those solutions as t→ ∞. 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.