Spacetime estimates and scattering theory for quasilinear Schr\"{o}dinger equations in arbitrary space dimension

Abstract

In this paper, we consider the following Cauchy problem of \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2\delta_huh'(|u|^2)\Delta h(|u|^2)+V(x)u+F(|u|^2)u+(W*|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 $\delta_h$ is a constant, $N\geq 1$, $h(s)$, $F(s)$, $V(x)$ and $W(x)$ are some real functions, $W(x)$ is even. Besides obtaining some sufficient conditions on global existence of the solution, we establish pseudoconformal conservation law and give Morawetz type estimates, spacetime bounds and asymptotic behaviors for the global solution. We bring two ideas to establish scattering theory, one is that we take different admissible pairs in Strichartz estimates for different terms on the right side of Duhamel's formula in order to keep each term independent, another is that we factitiously let a continuous function be the sum of two piecewise functions and chose different admissible pairs in Strichartz estimates for the terms containing these functions. Basing on the two ideas, we provide the direct and simple proofs of classic scattering theories in $L^2(\mathbb{R}^N)$ and $\Sigma$ for any space dimension($N\geq 1$) under certain assumptions. Here $$ \Sigma=\{u\in H^1(\mathbb{R}^N),\quad |xu|\in L^2(\mathbb{R}^N)\}. $$