Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space \(W^{s,p}\) for \(p<2\). (English) Zbl 1198.35264
Summary: We consider in \(\mathbb R^n\) 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 \(1<p<2\).
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
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