Sharp asymptotic behavior of solutions to damped nonlinear Schr\"{o}dinger equations

TL;DR

This paper refines the understanding of the decay rates of solutions to damped nonlinear Schrödinger equations, establishing sharp asymptotic convergence rates for solutions in specific function spaces, thus addressing open problems in scattering theory.

Contribution

It provides a refined and sharp characterization of the asymptotic decay rates of solutions, advancing the understanding of scattering behavior in damped nonlinear Schrödinger equations.

Findings

Convergence rate of solutions is sharper and optimal under certain initial data conditions.

The results partially solve open problems on the decay rates in scattering theory.

Solutions asymptotically behave like linear solutions with refined decay estimates.

Abstract

We consider large time asymptotics for damped nonlinear Schr\"{o}dinger equations. It is known that the nonlinear solution asymptotically behaves like a linear solution when time $t$ tends to infinity in the energy space. We prove that its convergence rate can be refined and the obtained rate is sharp if initial data belong to certain function spaces. This result partially solves open problems concerning the optimal decay rate of scattering.