Criteria for finite time blow up for a system of Klein–Gordon equations

References

• Segal IE. Nonlinear partial differential equations in quantum field theory. In: Proceedings of Symposia in Applied Mathematics. Vol. XVII. Providence (RI): American Mathematical Society; 1965. p. 210–226.
   Google Scholar
• Georgiev V. Global solution of the system of wave and Klein–Gordon equations. Math Z. 1990;203(1):683–698. doi: 10.1007/BF02570764
   Google Scholar
• Ionescu AD, Pausader B. On the global regularity for a wave-Klein–Gordon coupled system. Acta Math Sin (Engl Ser). 2019;35(6):933–986. doi: 10.1007/s10114-019-8413-6
   Web of Science ®Google Scholar
• Khusnutdinova KR. Coupled Klein–Gordon equations and energy exchange in two-component systems. Eur Phys J: Spec Top. 2007;147(1):45–72. doi: 10.1140/epjst/e2007-00202-0.
   Google Scholar
• Kutev N, Kolkovska N, Dimova M. Sign-preserving functionals and blow-up to Klein–Gordon equation with arbitrary high energy. Appl Anal. 2016;95(4):860–873. doi: 10.1080/00036811.2015.1038994
   Web of Science ®Google Scholar
• Nakanishi K, Schlag W. Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. J Differ Equ. 2011;250(5):2299–2333. doi: 10.1016/j.jde.2010.10.027
   Web of Science ®Google Scholar
• Nakanishi K, Schlag W. Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption. Arch Ration Mech Anal. 2012;203(3):809–851. doi: 10.1007/s00205-011-0462-7
   Web of Science ®Google Scholar
• Sirakov B. Least energy solitary waves for a system of nonlinear Schrdinger equations in Rn. Commun Math Phys. 2007;271(1):199–221. doi: 10.1007/s00220-006-0179-x
   Web of Science ®Google Scholar
• Chen Z, Zou W. Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch Ration Mech Anal. 2012;205(2):515–551. doi: 10.1007/s00205-012-0513-8
   Web of Science ®Google Scholar
• Chen Z, Zou W. Existence and symmetry of positive ground states for a doubly critical Schrödinger system. Trans Am Math Soc. 2015;367(5):3599–3646. doi: 10.1090/tran/2015-367-05
   Web of Science ®Google Scholar
• Peng S, Wang Q, Wang Z-Q. On coupled nonlinear Schrödinger systems with mixed couplings. Trans Am Math Soc. 2019;371(11):7559–7583. doi: 10.1090/tran/2019-371-11
   Web of Science ®Google Scholar
• Soave N, Tavares H. New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms. J Differ Equ. 2016;261(1):505–537. doi: 10.1016/j.jde.2016.03.015
   Web of Science ®Google Scholar
• Tavares H, Terracini S, Verzini G, et al. Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems. Commun Partial Differ Equ. 2011;36(11):1988–2010. doi: 10.1080/03605302.2011.574244
   Web of Science ®Google Scholar
• Wei J, Wu Y. Ground states of nonlinear Schrödinger systems with mixed couplings. J Math Pures Appl (9). 2020;141:50–88. doi: 10.1016/j.matpur.2020.07.012
   Web of Science ®Google Scholar
• Wu S-T. Global existence, blow-up and asymptotic behavior of solutions for a class of coupled nonlinear Klein–Gordon equations with damping terms. Acta Appl Math. 2012;119(1):75–95. doi: 10.1007/s10440-011-9662-2
   Web of Science ®Google Scholar
• Lian W, Xu R. Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv Nonlinear Anal. 2019;9(1):613–632. doi: 10.1515/anona-2020-0016
   Web of Science ®Google Scholar
• Xu R. Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein–Gordon equation with dissipative term. Math Methods Appl Sci. 2010;33(7):831–844. doi: 10.1002/mma.1405
   Web of Science ®Google Scholar
• Xu G. Dynamics of some coupled nonlinear Schrödinger systems in ℝ3. Math Methods Appl Sci. 2014;37(17):2746–2771. doi: 10.1002/mma.v37.17
   Web of Science ®Google Scholar
• Xu R, Lian W, Niu Y. Global well-posedness of coupled parabolic systems. Sci China Math. 2020;63(2):321–356. doi: 10.1007/s11425-017-9280-x
   Web of Science ®Google Scholar
• Nakanishi K, Schlag W. Invariant manifolds and dispersive Hamiltonian evolution equations. Zürich: European Mathematical Society (EMS); 2011. (Zurich lectures in advanced mathematics).
   Google Scholar
• Payne LE, Sattinger DH. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math. 1975;22(3-4):273–303. doi: 10.1007/BF02761595
   Web of Science ®Google Scholar
• Levine HA. Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=−Au+F(u). Trans Amer Math Soc. 1974;192:1–21. doi: 10.2307/1996814.
   Web of Science ®Google Scholar
• Berestycki H, Lions PL. Nonlinear scalar field equations, I existence of a ground state. Arch Ration Mech Anal. 1983;82(4):313–345. doi: 10.1007/BF00250555
   Web of Science ®Google Scholar
• Lieb EH, Loss M. Analysis. 2nd ed. American Mathematical Society; 2001.
   Google Scholar
• Struwe M. Variational methods. 2nd rev. and exp. ed. Berlin: Springer; 1996.
   Google Scholar
• An Y, Chern J-L, Shi J. Uniqueness of positive solutions to some coupled cooperative variational elliptic systems. Trans Am Math Soc. 2018;370(7):5209–5243. doi: 10.1090/tran/2018-370-07
   Web of Science ®Google Scholar
• Ikoma N. Uniqueness of positive solutions for a nonlinear elliptic system. NoDEA Nonlinear Differ Equ Appl. 2009;16(5):555–567. doi: 10.1007/s00030-009-0017-x
   Web of Science ®Google Scholar
• Lin TC, Wei J. Ground state of N coupled nonlinear Schrödinger equations in Rn, n≤3. Comm Math Phys. 2005;255(3):629–653. doi: 10.1007/s00220-005-1313-x
   Web of Science ®Google Scholar
• Lin TC, Wei J. Erratum: ‘ground state of n coupled nonlinear Schrödinger equations in Rn, n≤3’ [Comm Math Phys 2005;255(3):629–653]. Comm Math Phys. 2008;(2):573–576. doi: 10.1007/s00220-007-0365-5
   Web of Science ®Google Scholar