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June  2013, 33(6): 2389-2401. doi: 10.3934/dcds.2013.33.2389

On the stability of standing waves of Klein-Gordon equations in a semiclassical regime

Marco Ghimenti 1, , Stefan Le Coz 2, and  Marco Squassina 3,

1. 

Dipartimento di Matematica Applicata, Università di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy
2. 

Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France
3. 

Dipartimento di Informatica, Università di Verona, Strada Le Grazie 15, 37134 Verona, Italy

Received  January 2012 Revised  September 2012 Published  December 2012

We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
Keywords: stability, semi-classical limit., Klein-Gordon equations.
Mathematics Subject Classification: 35D99, 35J62, 58E05, 35J7.
Citation: Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389
References:
[1]

L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Comm. Partial Differential Equations, 36 (2011), 797-818. doi: 10.1080/03605302.2010.534684.

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.

[3]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006.

[4]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.

[5]

M. Beals and W. Strauss, $L^p$ estimates for the wave equation with a potential, Comm. Partial Differential Equations, 18 (1993), 1365-1397. doi: 10.1080/03605309308820977.

[6]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168.

[7]

H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sé. I Math., 293 (1981), 489-492.

[8]

F. A. Berezin and M. A. Shubin, "The Schrödinger Equation," 66 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.

[9]

T. Cazenave, "Semilinear Schrödinger Equations," New York University - Courant Institute, New York, 2003.

[10]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.

[11]

S. Cuccagna, On the local existence for the Maxwell-Klein-Gordon system in $\R^{3+1}$, Comm. Partial Differential Equations, 24 (1999), 851-867. doi: 10.1080/03605309908821449.

[12]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.

[13]

E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations, Phys. D, 18 (1986), 371-373. Solitons and coherent structures (Santa Barbara, Calif., 1985). doi: 10.1016/0167-2789(86)90201-0.

[14]

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, "Solitons and Nonlinear Wave Equations," Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982.

[15]

M. Grillakis, J. Shatah and W. A. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.

[16]

______, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348.

[17]

I. Ianni and S. Le Coz, Orbital stability of standing waves of a semiclassical nonlinear Schrödinger-Poisson equation, Adv. Differential Equations, 14 (2009), 717-748.

[18]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.

[19]

______, Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., 361 (2009), 5401-5416.

[20]

M. Keel, T. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621. doi: 10.3934/dcds.2011.30.573.

[21]

S. Le Coz, Standing waves in nonlinear Schrödinger equations, in "Analytical and Numerical Aspects of Partial Differential Equations" Walter de Gruyter, Berlin, (2009), 151-192.

[22]

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Phys. D, 237 (2008), 1103-1128. doi: 10.1016/j.physd.2007.12.004.

[23]

T.-C. Lin and J. Wei, Orbital stability of bound states of semiclassical nonlinear Schrödinger equations with critical nonlinearity, SIAM J. Math. Anal., 40 (2008), 365-381. doi: 10.1137/070683842.

[24]

Y. Liu, M. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 539-548. doi: 10.1016/j.anihpc.2006.03.005.

[25]

E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law, Rev. Math. Phys., 21 (2009), 459-510. doi: 10.1142/S0129055X09003669.

[26]

Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr\"odinger equations with potentials, Comm. Math. Phys., 121 (1989), 11-33.

[27]

M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322. doi: 10.1137/050643015.

[28]

______, Instability of standing waves for nonlinear Klein-Gordon equation and related system, in "Nonlinear Dispersive Equations" 26 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkötosho, Tokyo, (2006), 189-200.

[29]

______, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931 (electronic).

[30]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.

[31]

J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190.

[32]

C. D. Sogge, Lectures on non-linear wave equations, International Press, Boston, MA, second ed., (2008).

[33]

C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399. doi: 10.1007/s00032-008-0089-9.

[34]

D. M. A. Stuart, Modulational approach to stability of non-topological solitons in semilinear wave equations, J. Math. Pures Appl. (9), 80 (2001), 51-83. doi: 10.1016/S0021-7824(00)01189-2.

[35]

T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33. doi: 10.1090/S0273-0979-08-01228-7.

[36]

G. Vaira, Semiclassical states for the nonlinear Klein-Gordon-Maxwell system, J. Pure Appl. Math. Adv. Appl., 4 (2010), 59-95.

[37]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567. 

[38]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.

[39]

Y. Yu, Vortex dynamics for the nonlinear Maxwell-Klein-Gordon equation, Arch. Ration. Mech. Anal., 201 (2011), 743-776. doi: 10.1007/s00205-011-0422-2.

show all references

References:
[1]

L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Comm. Partial Differential Equations, 36 (2011), 797-818. doi: 10.1080/03605302.2010.534684.

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.

[3]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006.

[4]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.

[5]

M. Beals and W. Strauss, $L^p$ estimates for the wave equation with a potential, Comm. Partial Differential Equations, 18 (1993), 1365-1397. doi: 10.1080/03605309308820977.

[6]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168.

[7]

H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sé. I Math., 293 (1981), 489-492.

[8]

F. A. Berezin and M. A. Shubin, "The Schrödinger Equation," 66 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3154-4.

[9]

T. Cazenave, "Semilinear Schrödinger Equations," New York University - Courant Institute, New York, 2003.

[10]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.

[11]

S. Cuccagna, On the local existence for the Maxwell-Klein-Gordon system in $\R^{3+1}$, Comm. Partial Differential Equations, 24 (1999), 851-867. doi: 10.1080/03605309908821449.

[12]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.

[13]

E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations, Phys. D, 18 (1986), 371-373. Solitons and coherent structures (Santa Barbara, Calif., 1985). doi: 10.1016/0167-2789(86)90201-0.

[14]

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, "Solitons and Nonlinear Wave Equations," Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982.

[15]

M. Grillakis, J. Shatah and W. A. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.

[16]

______, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348.

[17]

I. Ianni and S. Le Coz, Orbital stability of standing waves of a semiclassical nonlinear Schrödinger-Poisson equation, Adv. Differential Equations, 14 (2009), 717-748.

[18]

L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.

[19]

______, Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., 361 (2009), 5401-5416.

[20]

M. Keel, T. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621. doi: 10.3934/dcds.2011.30.573.

[21]

S. Le Coz, Standing waves in nonlinear Schrödinger equations, in "Analytical and Numerical Aspects of Partial Differential Equations" Walter de Gruyter, Berlin, (2009), 151-192.

[22]

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Phys. D, 237 (2008), 1103-1128. doi: 10.1016/j.physd.2007.12.004.

[23]

T.-C. Lin and J. Wei, Orbital stability of bound states of semiclassical nonlinear Schrödinger equations with critical nonlinearity, SIAM J. Math. Anal., 40 (2008), 365-381. doi: 10.1137/070683842.

[24]

Y. Liu, M. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 539-548. doi: 10.1016/j.anihpc.2006.03.005.

[25]

E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law, Rev. Math. Phys., 21 (2009), 459-510. doi: 10.1142/S0129055X09003669.

[26]

Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr\"odinger equations with potentials, Comm. Math. Phys., 121 (1989), 11-33.

[27]

M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322. doi: 10.1137/050643015.

[28]

______, Instability of standing waves for nonlinear Klein-Gordon equation and related system, in "Nonlinear Dispersive Equations" 26 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkötosho, Tokyo, (2006), 189-200.

[29]

______, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931 (electronic).

[30]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.

[31]

J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190.

[32]

C. D. Sogge, Lectures on non-linear wave equations, International Press, Boston, MA, second ed., (2008).

[33]

C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399. doi: 10.1007/s00032-008-0089-9.

[34]

D. M. A. Stuart, Modulational approach to stability of non-topological solitons in semilinear wave equations, J. Math. Pures Appl. (9), 80 (2001), 51-83. doi: 10.1016/S0021-7824(00)01189-2.

[35]

T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33. doi: 10.1090/S0273-0979-08-01228-7.

[36]

G. Vaira, Semiclassical states for the nonlinear Klein-Gordon-Maxwell system, J. Pure Appl. Math. Adv. Appl., 4 (2010), 59-95.

[37]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567. 

[38]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.

[39]

Y. Yu, Vortex dynamics for the nonlinear Maxwell-Klein-Gordon equation, Arch. Ration. Mech. Anal., 201 (2011), 743-776. doi: 10.1007/s00205-011-0422-2.

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