Blow-up criteria for linearly damped nonlinear Schr\"odinger equations

TL;DR

This paper investigates the effects of linear damping on the nonlinear Schrödinger equation, establishing conditions for global existence, scattering, and finite-time blow-up across different critical regimes.

Contribution

It provides new blow-up criteria and global existence results for damped nonlinear Schrödinger equations in various critical cases.

Findings

Global existence and scattering for large damping in energy-critical case

Existence of finite-time blow-up solutions in mass-critical and supercritical cases

Conditions under which damping prevents or allows blow-up

Abstract

We consider the Cauchy problem for linearly damped nonlinear Schr\"odinger equations \[ i\partial_t u + \Delta u + i a u= \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $\alpha>0$. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up $H^1$ solutions to the focusing problem in the mass-critical and mass-supercritical cases.