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On the standing wave in coupled fractional Klein–Gordon equation

  • Zhenyu Guo EMAIL logo and Xin Zhang
Published/Copyright: November 20, 2023
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Abstract

The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave.

MSC 2020: 35A15; 35R11

Funding statement: Supported by NSFLN (2021-MS-275) and EFLN (LJKQZ2021093).

References

[1] M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Universitext, Springer, London, 2011. 10.1007/978-0-85729-227-8Search in Google Scholar

[2] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), no. 11, 6133–6162. 10.1016/j.jde.2012.02.023Search in Google Scholar

[3] C. Bjorland, L. Caffarelli and A. Figalli, Non-local gradient dependent operators, Adv. Math. 230 (2012), no. 4–6, 1859–1894. 10.1016/j.aim.2012.03.032Search in Google Scholar

[4] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), no. 1, 43–80. 10.4064/sm-123-1-43-80Search in Google Scholar

[5] L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1151–1179. 10.4171/jems/226Search in Google Scholar

[6] H. Chen, S. Lü and W. Chen, A fully discrete spectral method for the nonlinear time fractional Klein–Gordon equation, Taiwanese J. Math. 21 (2017), no. 1, 231–251. 10.11650/tjm.21.2017.7357Search in Google Scholar

[7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330–343. 10.1002/cpa.20116Search in Google Scholar

[8] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[9] W. Feng, M. Stanislavova and A. Stefanov, On the spectral stability of ground states of semi-linear Schrödinger and Klein–Gordon equations with fractional dispersion, Commun. Pure Appl. Anal. 17 (2018), no. 4, 1371–1385. 10.3934/cpaa.2018067Search in Google Scholar

[10] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. 10.1002/cpa.21591Search in Google Scholar

[11] D. Garrisi, Multiple normalized standing-wave solutions to the scalar non-linear Klein–Gordon equation with two competing powers, J. Differential Equations 269 (2020), no. 11, 9189–9223. 10.1016/j.jde.2020.06.038Search in Google Scholar

[12] C. Huang, B. Guo, D. Huang and Q. Li, Global well-posedness of the fractional Klein–Gordon–Schrödinger system with rough initial data, Sci. China Math. 59 (2016), no. 7, 1345–1366. 10.1007/s11425-016-5133-6Search in Google Scholar

[13] H. Khan, A. Khan, W. Chen and K. Shah, Stability analysis and a numerical scheme for fractional Klein–Gordon equations, Math. Methods Appl. Sci. 42 (2019), no. 2, 723–732. 10.1002/mma.5375Search in Google Scholar

[14] S. Klainerman, Q. Wang and S. Yang, Global solution for massive Maxwell–Klein–Gordon equations, Comm. Pure Appl. Math. 73 (2020), no. 1, 63–109. 10.1002/cpa.21864Search in Google Scholar

[15] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P u t t = - A u + ( u ) , Trans. Amer. Math. Soc. 192 (1974), 1–21. 10.2307/1996814Search in Google Scholar

[16] O. H. Miyagaki, E. L. de Moura and R. Ruviaro, Positive ground state solutions for quasicritical the fractional Klein–Gordon–Maxwell system with potential vanishing at infinity, Complex Var. Elliptic Equ. 64 (2019), no. 2, 315–329. 10.1080/17476933.2018.1434625Search in Google Scholar

[17] F. Natali and E. Cardoso, Jr., Orbital stability of periodic standing waves for the logarithmic Klein–Gordon equation, J. Math. Anal. Appl. 484 (2020), no. 2, Article ID 123723. 10.1016/j.jmaa.2019.123723Search in Google Scholar

[18] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3–4, 273–303. 10.1007/BF02761595Search in Google Scholar

[19] S. G. Révész, Minimization of maxima of nonnegative and positive definite cosine polynomials with prescribed first coefficients, Acta Sci. Math. (Szeged) 60 (1995), no. 3–4, 589–608. Search in Google Scholar

[20] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302. 10.1016/j.matpur.2013.06.003Search in Google Scholar

[21] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 587–628. 10.1007/s00205-014-0740-2Search in Google Scholar

[22] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898. 10.1016/j.jmaa.2011.12.032Search in Google Scholar

[23] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), no. 1, 133–154. 10.5565/PUBLMAT_58114_06Search in Google Scholar

[24] R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), no. 1, 67–102. 10.1090/S0002-9947-2014-05884-4Search in Google Scholar

[25] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112. 10.1002/cpa.20153Search in Google Scholar

[26] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63–85. 10.1515/crll.1998.100Search in Google Scholar

[27] G. Xu, C. Mu and D. Li, Global existence and non-existence analyses to a nonlinear Klein–Gordon system with damping terms under positive initial energy, Commun. Pure Appl. Anal. 19 (2020), no. 5, 2491–2512. 10.3934/cpaa.2020109Search in Google Scholar

[28] J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein–Gordon equations, Nonlinear Anal. 48 (2002), no. 2, 191–207. 10.1016/S0362-546X(00)00180-2Search in Google Scholar

[29] J. Zhang, On the standing wave in coupled non-linear Klein–Gordon equations, Math. Methods Appl. Sci. 26 (2003), no. 1, 11–25. 10.1002/mma.340Search in Google Scholar

Received: 2023-03-15
Revised: 2023-05-02
Accepted: 2023-05-18
Published Online: 2023-11-20
Published in Print: 2024-06-01

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