On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

Van Duong Dinh

doi:10.3934/dcds.2018047

Article Contents

2018, Volume 38, Issue 3: 1127-1143. Doi: 10.3934/dcds.2018047

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On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

Van Duong Dinh^,

Institut de Mathématiques de Toulouse UMR5219, Université Toulouse CNRS, 31062 Toulouse Cedex 9, France

^* Corresponding author
^* Corresponding author

Received: December 31, 2016

Revised: September 30, 2017

Published: June 2018

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Abstract

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs are mainly based on the contraction mapping argument using Strichartz estimates. We also apply the technique of Christ-Colliander-Tao in [6] to prove the ill-posedness for the (NLHW) in a certain range of the super-critical case.

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Mathematics Subject Classification: Primary: 35A01, 35E15; Secondary: 35Q55.

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