Vol. 11, No. 5, 2018

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Well-posedness and smoothing effect for generalized nonlinear Schrödinger equations

Pierre-Yves Bienaimé and Abdesslam Boulkhemair

Vol. 11 (2018), No. 5, 1241–1284
Abstract

We improve the result obtained by one of the authors, Bienaimé (2014), and establish the well-posedness of the Cauchy problem for some nonlinear equations of Schrödinger type in the usual Sobolev space ${H}^{s}\left({ℝ}^{n}\right)$ for $s>\frac{n}{2}+2$ instead of $s>\frac{n}{2}+3$. We also improve the smoothing effect of the solution and obtain the optimal exponent.

Keywords
Cauchy problem, well-posedness, smoothing effect, nonlinear equation, Schrödinger, paradifferential, pseudodifferential, operator, paralinearization
Mathematical Subject Classification 2010
Primary: 47G20, 47G30
Milestones
Received: 20 June 2017
Revised: 26 November 2017
Accepted: 2 January 2018
Published: 11 April 2018
Authors
 Pierre-Yves Bienaimé Laboratoire de Mathématiques Jean Leray CNRS UMR6629, Université de Nantes Nantes France Abdesslam Boulkhemair Laboratoire de Mathématiques Jean Leray CNRS UMR6629, Université de Nantes Nantes France