On the $L^{2}$-critical nonlinear Schr\"odinger Equation with a nonlinear damping

TL;DR

This paper investigates the $L^{2}$-critical nonlinear Schrödinger equation with nonlinear damping, establishing conditions for global existence or finite-time blowup with log-log speed based on the damping's power.

Contribution

It provides new results on the global behavior and blowup dynamics of solutions depending on the damping term's power in the $L^{2}$-critical nonlinear Schrödinger equation.

Findings

Global existence under certain damping conditions

Finite time blowup with log-log speed for specific damping powers

Characterization of solution behavior based on damping strength

Abstract

We consider the Cauchy problem for the $L^{2}$-critical nonlinear Schr\"{o}dinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.