# Well-posedness of the Two-dimensional Nonlinear Schr\"odinger Equation with Concentrated Nonlinearity

## TL;DR

This paper establishes local well-posedness and conservation laws for a 2D nonlinear Schrödinger equation with concentrated nonlinearity, demonstrating global existence in the defocusing case and potential blow-up in the focusing case.

## Contribution

It proves well-posedness and conservation laws for the 2D nonlinear Schrödinger equation with concentrated nonlinearity, including global results in the defocusing case.

## Key findings

- Local well-posedness proven for both focusing and defocusing cases
- Energy and mass conservation established
- Global existence in defocusing case, blow-up possible in focusing case

## Abstract

We consider a two-dimensional nonlinear Schr\"odinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.