Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space 𝑊^{𝑠,𝑝} for 𝑝<2

Zhou, Yi

doi:10.1090/S0002-9947-10-05055-5

Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$
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Trans. Amer. Math. Soc. 362 (2010), 4683-4694 Request permission

Abstract:

In this paper, we consider in $R^n$ the Cauchy problem for the nonlinear Schrödinger equation with initial data in the 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 $1<p<2$.

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