# On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R}) + H^1(\mathbb{T})$

## TL;DR

This paper establishes local well-posedness for a class of nonlinear Schrödinger equations with power nonlinearities in a mixed space setting, and proves global well-posedness for the quadratic case using Strichartz estimates and Gronwall's inequality.

## Contribution

It extends well-posedness results to the combined space $L^2(
) + H^1( )$ for nonlinear Schrödinger equations, including a global result for the quadratic case.

## Key findings

- Local well-posedness for $|u|^{eta - 1} u$ with $eta
eq 2$
- Global well-posedness for quadratic nonlinearity ($eta=2$)
- Use of Strichartz estimates and Gronwall's inequality

## Abstract

We study the one dimensional nonlinear Schr\"odinger equation with power nonlinearity $|u|^{\alpha - 1} u$ for $\alpha \in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($\alpha = 2$) we obtain global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb R) + H^1(\mathbb T))$ via Gronwall's inequality.