# Blowup solutions for the nonlinear Schr\"odinger equation with complex coefficient

## TL;DR

This paper constructs finite-time blow-up solutions for a nonlinear Schrödinger equation with complex coefficients, extending previous results to broader parameter ranges and higher dimensions, using compactness methods.

## Contribution

It generalizes the existence of blow-up solutions to include complex coefficients and higher space dimensions, expanding the understanding of solution behaviors.

## Key findings

- Constructed finite-time blow-up solutions with complex coefficients
- Extended the parameter range for power and coefficient beyond prior work
- Applied Aubin-Lions lemma for convergence and compactness

## Abstract

We construct a finite time blow up solution for the nonlinear Schr\"odinger equation with the power nonlinearity whose coefficient is complex number. We generalize the range of both the power and the complex coefficient for the result of Cazenave, Martel and Zhao \cite{CMZ}. As a bonus, we may consider the space dimension $5$. We show a sequence of solutions closes to the blow up profile which is a blow up solution of ODE. We apply the Aubin-Lions lemma for the compactness argument for its convergence.