Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

Abstract

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ | u | p − 1 u , where 1 < p < 1 + 2 / n , n is the space dimension and λ is a complex constant satisfying Im λ < 0 . We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n ⩽ 3 , p is sufficiently close to 1 + 2 / n and the initial data is sufficiently small.