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The Cauchy problem for the nonlinear Schrödinger equation in H1

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Abstract

We consider the initial value problem for the nonlinear Schrödinger equation in H1(Rn). 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, anda priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.

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Authors and Affiliations

  1. Analyse Numérique, Université Pierre et Marie Curie, 4, Place Jussieu, 75252, Paris Cedex 05, France

    Thierry Cazenave & Fred B. Weissler

  2. Department of Mathematics, Texas A&M University, 77843, College Station, TX, USA

    Fred B. Weissler

Authors
  1. Thierry Cazenave
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  2. Fred B. Weissler
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Research supported by NSF grants DMS 8201639 and DMS 8703096.

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Cazenave, T., Weissler, F.B. The Cauchy problem for the nonlinear Schrödinger equation in H1 . Manuscripta Math 61, 477–494 (1988). https://doi.org/10.1007/BF01258601

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