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Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition
Well-posedness issues for some critical coupled non-linear Klein-Gordon equations
University Tunis El Manar, Faculty of Sciences of Tunis, 2092, Tunis, Tunisia |
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.
References:
[1] | |
[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] | |
[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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