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

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.