# Morawetz estimates and spacetime bounds for quasilinear Schr\"{o}dinger equations with critical Sobolev exponent

## TL;DR

This paper investigates the behavior of solutions to a quasilinear Schrödinger equation with critical Sobolev exponent, deriving conditions for blowup or global existence, and establishing Morawetz estimates and spacetime bounds to aid scattering theory.

## Contribution

It introduces Morawetz estimates and spacetime bounds for quasilinear Schrödinger equations with critical Sobolev exponent, utilizing pseudoconformal conservation law for the first time in this context.

## Key findings

- Derived sufficient conditions for finite-time blowup.
- Established global existence criteria.
- Developed Morawetz estimates and spacetime bounds.

## Abstract

In this paper, we study the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u + 2uh'(|u|^2)\Delta h(|u|^2) + F(|u|^2)u\mp A[h(|u|^2]^{2^*-1} h'(|u|^2)u,\ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x), \quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, $h(s)\geq 0$ and $h'(s)\geq 0$ for $s\geq 0$, $N\geq 3$, $A>0$. Besides obtaining sufficient conditions on the blowup in finite time and global existence of the solution, we establish Morawetz estimates and spacetime bounds for the global solution based on pseudoconformal conservation law, which is an important tool to construct scattering operator on the energy space.