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: January 2017

Revised: October 2017

Published: July 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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References

	H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Non-Linear Partial Differential Equations, A Series of Comprehensive Studies in Mathematics 343, Springer-Verlag, Berlin, 2011.
	J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976.
	T. Cazenave and F. B. Weissler , The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990) , 807-836. doi: 10.1016/0362-546X(90)90023-A.
	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, AMS, 2003.
	A. Choffrut and O. Pocovnicu , Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not., 2016 (2016) , 1-40. doi: 10.1093/imrn/rnw246.
	M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv:math/0311048.
	M. Christ and I. Weinstein , Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991) , 87-109. doi: 10.1016/0022-1236(91)90103-C.
	J. Colliander , M. Keel , G. Staffilani , H. Takaoka and T. Tao , Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008) , 767-865. doi: 10.4007/annals.2008.167.767.
	V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, preprint, arXiv:1609.06181.
	A. Elgart and B. Schlein , Mean field dynamics of boson stars, Commum. Pure Appl. Math., 60 (2007) , 500-545. doi: 10.1002/cpa.20134.
	J. Fröhlich and E. Lenzmann , Blowup for nonlinear wave equations describing boson stars, Commum. Pure Appl. Math., 60 (2007) , 1691-1705. doi: 10.1002/cpa.20186.
	K. Fujiwara, V. Georgiev and T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases, preprint, arXiv:1611.09674.
	K. Fujiwara and T. Ozawa , Remarks on global solutions to the Cauchy problem for semirelativistic equations with power type nonlinearity, Int. J. Math. Anal., 9 (2015) , 2599-2610. doi: 10.12988/ijma.2015.58211.
	J. Ginibre and G. Velo , The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985) , 487-505. doi: 10.1007/BF01168155.
	L. Grafakos and S. Oh , The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014) , 1128-1157. doi: 10.1080/03605302.2013.822885.
	Y. Hong and Y. Sire , On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015) , 2265-2282. doi: 10.3934/cpaa.2015.14.2265.
	A. D. Ionescu and F. Pusateri , Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014) , 139-176. doi: 10.1016/j.jfa.2013.08.027.
	T. Kato , On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995) , 281-306. doi: 10.1007/BF02787794.
	M. Keel and T. Tao , Endpoint Strichartz estimates, Amer. J. Math., 120 (1998) , 955-980. doi: 10.1353/ajm.1998.0039.
	J. Krieger , E. Lenzmann and P. Raphaël , Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Rational Mech. Anal., 209 (2013) , 61-129. doi: 10.1007/s00205-013-0620-1.
	O. Pocovnicu , First and second order approximations for a nonlinear wave equation, J. Dynam. Differential Equations, 25 (2013) , 305-333. doi: 10.1007/s10884-013-9286-5.
	G. Staffilani, The Initial Value Problem for Some Dispersive Differential Equations, Ph. D thesis, University of Chicago, 1995.
	M. Taylor, Tool for PDE Pseudodifferential Operators, Paradifferential Operators and Layer Potentials, Mathematical Surveys and Monographs 81, AMS, 2000.
	H. Triebel, Theory of Function Spaces, Basel: Birkhäuser, 1983.