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On nonlinear Schrödinger equations. II: $$H^ S$$-solutions and unconditional well-posedness. (English) Zbl 0848.35124
J. Anal. Math. 67, 281-306 (1995); correction ibid. 68, 305 (1996).
The author studies the initial value problem for the nonlinear Schrödinger equation $$\partial_t u= i(\Delta u- F(u))$$, $$t\geq 0$$, $$x\in \mathbb{R}^m$$, under the assumption that $$F\in C^1(\mathbb{C}, \mathbb{C})$$, $$F(0)= 0$$, $$DF(\xi)= O(|\xi |^{k- 1})$$ for some $$k\geq 1$$ as $$|\xi |\to \infty$$. The author proves uniqueness in $$L^\infty((0, T); L^2(\mathbb{R}^m))\cap L^r((0, T); L^q(\mathbb{R}^m))$$, where $$r$$ and $$q$$ depend on $$m$$ and $$k$$. After this, a local existence theorem is proved for $$H^s$$-solutions using Lebesgue-type spaces instead of Besov-type spaces (used by T. Cazenaze and F. B. Weissler). Moreover, the existence of global $$H^s$$-solutions is shown under the assumption that $$F(\xi)= O^{\{s\}}(|\xi|^h)$$ as $$|\xi|\to 0$$, where $$h= 1+ 4/m$$.
Reviewer: E.Minchev (Sofia)

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35G25 Initial value problems for nonlinear higher-order PDEs
##### Keywords:
local and global existence; initial value problem
Full Text:
##### References:
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