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On the global Cauchy problem for some nonlinear Schrödinger equations. (English) Zbl 0569.35070

The authors study the Cauchy problem for a class of nonlinear Schrödinger equations in space time dimension \(n+1\). They look for solutions which belong to the class \(C({\mathbb{R}},H^ k({\mathbb{R}}^ n))\), \(k>n/2\). Under some suitable assumptions on the nonlinearity the authors prove the existence of a (global) solution for \(n\leq 7\). The global existence proof breaks down for \(n\geq 8\).
Reviewer: N.Jacob

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35G25 Initial value problems for nonlinear higher-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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