# Local Cauchy theory for the nonlinear Schr\"odinger equation in spaces of infinite mass

## TL;DR

This paper establishes local well-posedness for the nonlinear Schrödinger equation with initial data in certain infinite mass spaces, expanding understanding of solution behavior in these function spaces.

## Contribution

It introduces a local Cauchy theory for NLS in spaces combining homogeneous Sobolev and Lebesgue spaces, addressing infinite mass initial data.

## Key findings

- Proves local well-posedness for a broad range of p in the initial data space.
- Discusses conditions under which global well-posedness can be achieved.
- Extends the theory of NLS to new function spaces with infinite mass.

## Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^d$, where the initial data is in $\dot{H}^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$. We prove local well-posedness for large ranges of $p$ and discuss some global well-posedness results.