May  2021, 41(5): 2411-2445. doi: 10.3934/dcds.2020370

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

Teacher Training Courses, Faculty of Education, Kagawa University, Takamatsu, Kagawa, 760-8522, Japan

Received  April 2020 Revised  September 2020 Published  May 2021 Early access  November 2020

Fund Project: The author is supported by JSPS KAKENHI Grant Numbers 19K14580 and the Overseas Research Fellowship Program by National Institute of Technology

This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know.

In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [27]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.

Citation: Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

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 Sér. I Math., 293 (1981), 489-492. 

[3]

H. Brezis and E. H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys., 96 (1984), 97–113. http://projecteuclid.org/euclid.cmp/1103941720. doi: 10.1007/BF01217349.

[4]

J. ByeonL. Jeanjean and M. Mariș, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36 (2009), 481-492.  doi: 10.1007/s00526-009-0238-1.

[5]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.

[6]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[7]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.

[8]

V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp. doi: 10.1016/j.na.2019.111589.

[9]

V. D. Dinh, Strong instability of standing waves for a system NLS with quadratic interaction, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 515-528.  doi: 10.1007/s10473-020-0214-6.

[10]

D. Garrisi, On the orbital stability of standing-wave solutions to a coupled non-linear Klein-Gordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.

[11]

M. GrillakisJ. Shatah and W. 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.

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.

[13]

M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, https://arXiv.org/abs/1805.12245.

[14]

N. HayashiM. Ikeda and P. I. Naumkin, Wave operator for the system of the Dirac-Klein-Gordon equations, Math. Methods Appl. Sci., 34 (2011), 896-910.  doi: 10.1002/mma.1409.

[15]

N. HayashiT. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.  doi: 10.1016/j.anihpc.2012.10.007.

[16]

L. Jeanjean and M. Squassina, Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1701-1716.  doi: 10.1016/j.anihpc.2008.11.003.

[17]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567.  doi: 10.1016/j.jde.2011.04.001.

[18]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.  doi: 10.1007/BF01394245.

[19]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[20]

Y. LiuM. 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.

[21]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.

[22]

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147–1164. http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.5merle.pdf. doi: 10.1353/ajm.2003.0033.

[23]

H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan, 46 (1994), 557-586.  doi: 10.2969/jmsj/04640557.

[24]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.

[25]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.  doi: 10.2307/2048340.

[26]

M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.  doi: 10.3934/dcds.2005.12.315.

[27]

M. Ohta and G. Todorova, 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.  doi: 10.1137/050643015.

[28]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[29]

H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.  doi: 10.1007/BF01181697.

[30]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys., 91 (1983), 313–327, http://projecteuclid.org/euclid.cmp/1103940612. doi: 10.1007/BF01208779.

[31]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.  doi: 10.1090/S0002-9947-1985-0792821-7.

[32]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173–190, http://projecteuclid.org/euclid.cmp/1103943442. doi: 10.1007/BF01212446.

[33]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149–162, http://projecteuclid.org/euclid.cmp/1103900983. doi: 10.1007/BF01626517.

[34]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325.  doi: 10.1016/S0022-0396(03)00125-6.

[35]

Y. Tsutsumi, Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., 46 (2003), 41-62.  doi: 10.1619/fesi.46.41.

[36]

B. Wang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in $H^s$, Nonlinear Anal., 31 (1998), 573-587.  doi: 10.1016/S0362-546X(97)00424-0.

[37]

Y. Wu, Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension, preprint, https://arXiv.org/abs/1705.04216.

[38]

J. ZhangZ.-h. Gan and B.-l. Guo, Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 427-442.  doi: 10.1007/s10255-010-0008-z.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

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 Sér. I Math., 293 (1981), 489-492. 

[3]

H. Brezis and E. H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys., 96 (1984), 97–113. http://projecteuclid.org/euclid.cmp/1103941720. doi: 10.1007/BF01217349.

[4]

J. ByeonL. Jeanjean and M. Mariș, Symmetry and monotonicity of least energy solutions, Calc. Var. Partial Differential Equations, 36 (2009), 481-492.  doi: 10.1007/s00526-009-0238-1.

[5]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.

[6]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[7]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.

[8]

V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp. doi: 10.1016/j.na.2019.111589.

[9]

V. D. Dinh, Strong instability of standing waves for a system NLS with quadratic interaction, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 515-528.  doi: 10.1007/s10473-020-0214-6.

[10]

D. Garrisi, On the orbital stability of standing-wave solutions to a coupled non-linear Klein-Gordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.

[11]

M. GrillakisJ. Shatah and W. 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.

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.

[13]

M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, https://arXiv.org/abs/1805.12245.

[14]

N. HayashiM. Ikeda and P. I. Naumkin, Wave operator for the system of the Dirac-Klein-Gordon equations, Math. Methods Appl. Sci., 34 (2011), 896-910.  doi: 10.1002/mma.1409.

[15]

N. HayashiT. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.  doi: 10.1016/j.anihpc.2012.10.007.

[16]

L. Jeanjean and M. Squassina, Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1701-1716.  doi: 10.1016/j.anihpc.2008.11.003.

[17]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567.  doi: 10.1016/j.jde.2011.04.001.

[18]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.  doi: 10.1007/BF01394245.

[19]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[20]

Y. LiuM. 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.

[21]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.

[22]

F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147–1164. http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.5merle.pdf. doi: 10.1353/ajm.2003.0033.

[23]

H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan, 46 (1994), 557-586.  doi: 10.2969/jmsj/04640557.

[24]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.

[25]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.  doi: 10.2307/2048340.

[26]

M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.  doi: 10.3934/dcds.2005.12.315.

[27]

M. Ohta and G. Todorova, 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.  doi: 10.1137/050643015.

[28]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[29]

H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.  doi: 10.1007/BF01181697.

[30]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys., 91 (1983), 313–327, http://projecteuclid.org/euclid.cmp/1103940612. doi: 10.1007/BF01208779.

[31]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.  doi: 10.1090/S0002-9947-1985-0792821-7.

[32]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173–190, http://projecteuclid.org/euclid.cmp/1103943442. doi: 10.1007/BF01212446.

[33]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149–162, http://projecteuclid.org/euclid.cmp/1103900983. doi: 10.1007/BF01626517.

[34]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325.  doi: 10.1016/S0022-0396(03)00125-6.

[35]

Y. Tsutsumi, Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., 46 (2003), 41-62.  doi: 10.1619/fesi.46.41.

[36]

B. Wang, On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in $H^s$, Nonlinear Anal., 31 (1998), 573-587.  doi: 10.1016/S0362-546X(97)00424-0.

[37]

Y. Wu, Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension, preprint, https://arXiv.org/abs/1705.04216.

[38]

J. ZhangZ.-h. Gan and B.-l. Guo, Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 427-442.  doi: 10.1007/s10255-010-0008-z.

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