Blowup for the nonlinear Schr\"odinger equation with an inhomogeneous damping term in the $L^2$ critical case

TL;DR

This paper investigates the blowup behavior of solutions to the $L^2$-critical nonlinear Schrödinger equation with an inhomogeneous damping term, demonstrating blowup phenomena in the energy space using advanced analytical tools.

Contribution

It introduces the analysis of blowup for the inhomogeneously damped $L^2$-critical nonlinear Schrödinger equation, extending existing methods to this new setting.

Findings

Proves existence of blowup solutions in $H^1( )$.

Utilizes tools developed by Merle and Raphael.

Establishes blowup phenomena in the presence of inhomogeneous damping.

Abstract

We consider the nonlinear Schr\"odinger equation with $L^2$-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space $H^1(\mathbb{R})$.