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Some remarks on the nonlinear Schrödinger equation in the subcritical case. (English) Zbl 0699.35217
New methods and results in nonlinear field equations, Proc. Conf., Bielefeld/FRG 1987, Lect. Notes Phys. 347, 59-69 (1989).
[For the entire collection see Zbl 0686.00010.]
This paper is devoted to the study of the initial value problem for the nonlinear Schrödinger equation in \({\mathbb{R}}^ N:\) \(iu_ t+\Delta u=g(u)\). The authors establish local existence and uniqueness for a wide class of subcritical nonlinearities g. the proofs make use of a truncation argument, space-time integrability properties of the linear equation, and a priori estimates derived from the conservation of energy. In particular, no differentiability property of the nonlinearity with respect to x is needed.
Reviewer: T.Cazenave

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35K55 Nonlinear parabolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics