Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data

Abstract

We study the asymptotic behavior in time of solutions to the Cauchy problem of nonlinear Schrödinger equations with a long-range dissipative nonlinearity given by $\lambda |u{|}^{p-1}u$ in one space dimension, where (namely, $p$ is a critical or subcritical exponent) and $\lambda$ is a complex constant satisfying Im and $((p-1)/2\sqrt{p})\left|Re\lambda \right|\le \left|Im\lambda \right|$ . We present the time decay estimates and the large-time asymptotics of the solution for arbitrarily large initial data, when “$p=3$ ” or “ and $p$ is suitably close to $3$ ”.