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The Cauchy problem for the nonlinear Schrödinger equation in $$H^ 1$$. (English) Zbl 0696.35153
Summary: We consider the initial value problem for the nonlinear Schrödinger equation in $$H^ 1({\mathbb{R}}^ n)$$. We establish local existence and uniqueness for a wide class of subcritical nonlinearities. 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, we do not need any differentiability property of the nonlinearity with respect to x.

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 47J25 Iterative procedures involving nonlinear operators
##### Keywords:
nonlinear Schrödinger equation; a priori estimates
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##### References:
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