American Institute of Mathematical Sciences

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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

Hayato Miyazaki

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

Received  April 2020 Revised  September 2020 Published  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.

Keywords: Systems of nonlinear Klein-Gordon equations, standing waves, instability, mass resonance condition, variational methods.
Mathematics Subject Classification: Primary: 35L71, 35B35; Secondary: 35A15.
Citation: Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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J. Byeon, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Grillakis, J. 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.  Google Scholar

[12]

M. Grillakis, J. 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.  Google Scholar

[13]

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

[14]

N. Hayashi, M. 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.  Google Scholar

[15]

N. Hayashi, T. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[37]

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

[38]

J. Zhang, Z.-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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

J. Byeon, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Grillakis, J. 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.  Google Scholar

[12]

M. Grillakis, J. 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.  Google Scholar

[13]

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

[14]

N. Hayashi, M. 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.  Google Scholar

[15]

N. Hayashi, T. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[37]

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

[38]

J. Zhang, Z.-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.  Google Scholar

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