Cauchy problem of nonlinear Schr\"odinger equation with Cauchy problem of nonlinear Schr\"odinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$

Abstract

In this paper, we consider in $R^n$ the Cauchy problem for nonlinear Schr\"odinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $W^{s,p}$ for $\frac{2n}{n+1}<p<2$ and $s>n(1-\frac{1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $L^p$ for any $1<p<2$.