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March  2019, 18(2): 603-623. doi: 10.3934/cpaa.2019030

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

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

Received  November 2017 Revised  July 2018 Published  October 2018

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.

Citation: Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic, New York, 1975. [2] J. M. Arnaudiès and H. Fraysse, Cours de Mathématiques, Dunod, 1996. [3] 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. [4] M. Christ, J. Colliander and T. Tao, Ill- posedness for nonlinear Schrödinger and wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2005. [5] 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. [6] J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155. [7] 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. [8] 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. [9] M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics, 120 (1998), 955-980. [10] 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. [11] 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. [12] 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. [13] G. Lebeau, Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège., 70 (2001), 267-306. [14] G. Lebeau, Perte de régularité pour l'équation des ondes surcritique, Bull. Soc. Math. France., 133 (2005), 145-157. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] E. Y. Ovcharov, Global Regularity of Nonlinear Dispersive Equations and Strichartz Estimates, Ph. D thesis, University of Edinburgh, 2009. [23] 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. [24] E. Pişkin, Blow-up of solutions for coupled nonlinear Klein-Gordon equations with weak damping terms, Math. Sci. Letters, 3 (2014), 189-191. [25] T. Saanouni, A note on coupled focusing nonlinear Schrödinger equations, Applicable Analysis, 95 (2016), 2063-2080. doi: 10.1080/00036811.2015.1086757. [26] I. Segal, Nonlinear partial differential equations in Quantum Field Theory, Proc. Symp. Appl. Math. A.M.S., 17 (1965), 210-226. [27] M. Struwe, Semilinear wave equations, Bull. Amer. Math. Soc, N.S., 26 (1992), 53-85. doi: 10.1090/S0273-0979-1992-00225-2. [28] 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. [29] 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. [30] 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. [31] 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. [32] 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.

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic, New York, 1975. [2] J. M. Arnaudiès and H. Fraysse, Cours de Mathématiques, Dunod, 1996. [3] 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. [4] M. Christ, J. Colliander and T. Tao, Ill- posedness for nonlinear Schrödinger and wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2005. [5] 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. [6] J. Ginibre and G. Velo, The Global Cauchy problem for nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155. [7] 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. [8] 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. [9] M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics, 120 (1998), 955-980. [10] 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. [11] 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. [12] 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. [13] G. Lebeau, Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège., 70 (2001), 267-306. [14] G. Lebeau, Perte de régularité pour l'équation des ondes surcritique, Bull. Soc. Math. France., 133 (2005), 145-157. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] E. Y. Ovcharov, Global Regularity of Nonlinear Dispersive Equations and Strichartz Estimates, Ph. D thesis, University of Edinburgh, 2009. [23] 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. [24] E. Pişkin, Blow-up of solutions for coupled nonlinear Klein-Gordon equations with weak damping terms, Math. Sci. Letters, 3 (2014), 189-191. [25] T. Saanouni, A note on coupled focusing nonlinear Schrödinger equations, Applicable Analysis, 95 (2016), 2063-2080. doi: 10.1080/00036811.2015.1086757. [26] I. Segal, Nonlinear partial differential equations in Quantum Field Theory, Proc. Symp. Appl. Math. A.M.S., 17 (1965), 210-226. [27] M. Struwe, Semilinear wave equations, Bull. Amer. Math. Soc, N.S., 26 (1992), 53-85. doi: 10.1090/S0273-0979-1992-00225-2. [28] 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. [29] 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. [30] 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. [31] 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. [32] 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.
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