Well-posedness issues for some critical coupled non-linear Klein-Gordon equations

Radhia Ghanmi; Tarek Saanouni

doi:10.3934/cpaa.2019030

Article Contents

2019, Volume 18, Issue 2: 603-623. Doi: 10.3934/cpaa.2019030

This issue Previous Article Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition Next Article On Positive solutions of integral equations with the weighted Bessel potentials

Well-posedness issues for some critical coupled non-linear Klein-Gordon equations

Radhia Ghanmi and
Tarek Saanouni

University Tunis El Manar, Faculty of Sciences of Tunis, 2092, Tunis, Tunisia

Received: November 2017

Revised: July 2018

Published: October 2018

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

The initial value problem for some coupled non-linear wave equations is investigated. In the defocusing case, global well-posedness and ill-posedness results are obtained. In the focusing sign, the existence of global and non global solutions are discussed via the potential-well theory. Finally, strong instability of standing waves are established.

Keywords:

Mathematics Subject Classification: Primary: 35L05; Secondary: 53B44.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	R. A. Adams, Sobolev Spaces, Academic, New York, 1975.
	J. M. Arnaudiès and H. Fraysse, Cours de Mathématiques, Dunod, 1996.
	A. Atallah Baraket , Local existence and estimations for a semilinear wave equation in two dimension space, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2004) , 1-21.
	M. Christ, J. Colliander and T. Tao, Ill- posedness for nonlinear Schrödinger and wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2005.
	J. Ferreira and G. Perla Menzala , Decay of solutions of a system of nonlinear Klein Gordon equations, Internat. J. Math. Math. Sci., 9 (1986) , 417-483. doi: 10.1155/S0161171286000601.
	J. Ginibre and G. Velo , The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z., 189 (1985) , 487-505. doi: 10.1007/BF01168155.
	M. Grillakis , Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Annal. of Math., 132 (1990) , 485-509. doi: 10.2307/1971427.
	S. Ibrahim , N. Masmoudi and K. Nakanishi , Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis and PDE, 4 (2011) , 405-460. doi: 10.2140/apde.2011.4.405.
	M. Keel and T. Tao , Endpoint Strichartz estimates, American Journal of Mathematics, 120 (1998) , 955-980.
	E. Kenig and F. Merle , Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, American Journal of Mathematics, 133 (2011) , 1029-1065. doi: 10.1353/ajm.2011.0029.
	C. Kenig , G. Ponce and L. Vega , Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. Appl. Math., 46 (1993) , 527-620. doi: 10.1002/cpa.3160460405.
	M. O. Korpusov , Blow up the solution of a nonlinear system of equations with positive energy, Theoretical and Mathematical Physics, 171 (2012) , 725-738. doi: 10.1007/s11232-012-0070-1.
	G. Lebeau , Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège., 70 (2001) , 267-306.
	G. Lebeau , Perte de régularité pour l'équation des ondes surcritique, Bull. Soc. Math. France., 133 (2005) , 145-157.
	M. M. Miranda and L. A. Medeiros , Weak solutions for a system of nonlinear KleinGordon equations, Annali di Matematica pura ed applicata CXLVI, (1987) , 173-183. doi: 10.1007/BF01762364.
	M. M. Miranda and L. A. Medeiros , On the existence of global solutions of a coupled nonlinear KleinGordon equations, Funkcialaj Ekvacjoj, 30 (1987) , 147-161.
	O. Mahouachi and T. Saanouni , Global well posedness and linearization of a semilinear wave equation with exponential growth, Georgian Math. J., 17 (2010) , 543-562.
	O. Mahouachi and T. Saanouni , Well and ill-posedness issues for a class of 2D wave equation with exponential growth, J. Partial Diff. Eqs., 24 (2011) , 361-384. doi: 10.4208/jpde.v24.n4.7.
	L. A. Medeiros and G. Perla Menzala , On a mixed problem for a class of nonlinear Klein Gordon equations, Acta Math. Hung, 52 (1988) , 61-69. doi: 10.1007/BF01952481.
	M. Nakamura and T. Ozawa , Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z., 231 (1999) , 479-487. doi: 10.1007/PL00004737.
	M. Nakamura and T. Ozawa , The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order, Discrete and Continuous Dynamical Systems, 5 (1999) , 215-231.
	E. Y. Ovcharov, Global Regularity of Nonlinear Dispersive Equations and Strichartz Estimates, Ph. D thesis, University of Edinburgh, 2009.
	E. Pişkin , Uniform decay and blow-up of solutions for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Math. Meth. Appl. Sci., 37 (2014) , 3036-3047. doi: 10.1002/mma.3042.
	E. Pişkin , Blow-up of solutions for coupled nonlinear Klein-Gordon equations with weak damping terms, Math. Sci. Letters, 3 (2014) , 189-191.
	T. Saanouni , A note on coupled focusing nonlinear Schrödinger equations, Applicable Analysis, 95 (2016) , 2063-2080. doi: 10.1080/00036811.2015.1086757.
	I. Segal , Nonlinear partial differential equations in Quantum Field Theory, Proc. Symp. Appl. Math. A.M.S., 17 (1965) , 210-226.
	M. Struwe , Semilinear wave equations, Bull. Amer. Math. Soc, N.S., 26 (1992) , 53-85. doi: 10.1090/S0273-0979-1992-00225-2.
	M. Struwe , Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann., 350 (2011) , 707-719. doi: 10.1007/s00208-010-0567-6.
	J. Shatah and M. Struwe , Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, 7 (1994) , 303-309. doi: 10.1155/S1073792894000346.
	Y. Wang , Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA Journal of Applied Mathematics, 74 (2009) , 392-415. doi: 10.1093/imamat/hxp004.
	S. T. Wu , Blow-up results for systems of nonlinear Klein-Gordon equation with arbitrary positive initial energy, Electronic Journal of Differential Equations, 92 (2012) , 1-13.
	W. Xiao and Y. Ping , Global solutions and finite time blow up for some system of nonlinear wave equations, Applied Mathematics and Computation, 219 (2012) , 3754-3768. doi: 10.1016/j.amc.2012.10.005.