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Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 8
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Research Article

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

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Pages 1513-1539 | Received 15 Mar 2023, Accepted 27 Aug 2023, Published online: 10 Sep 2023

References

  • Herman F, Felix V. Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles. Rev Mod Phys. 1958;30(1):24–45. doi:10.1103/RevModPhys.30.102
  • Drazin PG, Johnson RS. Solitons: an introduction. Cambridge (MA): Cambridge University Press; 1989.
  • Temam R. Infinite-Dimensional dynamical systems in mechanics and physics. New York: Springer Press; 1997.
  • Cazenave T. Uniform estimates for solutions of nonlinear Klein-Gordon equations. J Funct Anal. 1985;60(1):36–55. doi:10.1016/0022-1236(85)90057-6
  • Zhang J. Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nonlinear Anal Theory. 2002;48(2):191–207. doi:10.1016/S0362-546X(00)00180-2
  • Wang YJ. A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy. Proc Am Math Soc. 2008;136(10):3477–3482. doi:10.1090/S0002-9939-08-09514-2
  • Huang WY, Zhang J. Instability of the standing waves for nonlinear Klein-Gordon equations with damping term. Appl Math Comput. 2009;213(2):522–528. doi:10.1016/j.cam.2008.08.037
  • Xu RZ. 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
  • Côte R, Martel Y, Yuan X. Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation. Arch Ration Mech Anal. 2021;239:1837–1874. doi:10.1007/s00205-020-01605-4
  • Avrin JD. Convergence of the strongly damped nonlinear Klein-Gordon equation in rn with radial symmetry. Proc R Soc Edinb A. 1987;107(1–2):169–174. doi:10.1017/S0308210500029437
  • Aviles P, Sandefur J. Nonlinear second order equations with applications to partial differential equations. J Differ Equ. 1985;58(3):404–427. doi:10.1016/0022-0396(85)90008-7
  • Xu RZ, Ding YH. Global solutions and finite time blow up for damped Klein-Gordon equation. Acta Math Sci. 2013;33B(3):643–652. doi:10.1016/S0252-9602(13)60027-2
  • Pang Y, Yang YB. A note on finite time blowup for dissipative Klein-Gordon equation. Nonlinear Anal Theory Methods Appl. 2020;195:111729. doi:10.1016/j.na.2019.111729
  • Barrow JD, Parsons P. Inflationary models with logarithmic potentials. Phys Rev D. 1995;52:5576–5587. doi:10.1103/PhysRevD.52.5576
  • Enqvist K, McDonald J. Q-balls and baryogenesis in the MSSM. Phys Lett B. 1998;425(3–4):309–321. doi:10.1016/S0370-2693(98)00271-8
  • Bialynicki-Birula I, Mycielski J. Wave equations with logarithmic nonlinearities. Bull Acad Pol Sci Ser Sci Math Astron Phys. 1975;23(4):461–466.
  • Górka P. Logarithmic Klein-Gordon equation. Acta Phys Polon B. 2009;40(1):59–66.
  • Bartkowski K, Górka P. One-dimensional Klein-Gordon equation with logarithmic nonlinearities. J Phys A Math Theor. 2008;41(35):355201. doi:10.1088/1751-8113/41/35/355201
  • Ye YJ, Li LL. Global existence and blow-up of solutions for logarithmic Klein-Gordon equation. AIMS Math. 2021;6(7):6898–6914. doi:10.3934/math.2021404
  • Hiramatsu T, Kawasaki M, Takahashi F. Numerical study of q-ball formation in gravity mediation. J Cosmol Astropart Phys. 2010;2010(6):8. doi:10.1088/1475-7516/2010/06/008
  • Han XS. Global existence of weak solutions for a logarithmic wave equation arising from q-ball dynamics. Bull Korean Math Soc. 2013;50(1):275–283. doi:10.4134/BKMS.2013.50.1.275
  • Zhang HW, Liu GW, Hu QY. Exponential decay of energy for a logarithmic wave equation. J Partial Differ Equ. 2015;28(3):269–277. doi:10.4208/jpde
  • Ye YJ. Global solution and blow-up of logarithmic Klein-Gordon equation. Bull Korean Math Soc. 2020;57(2):281–294.
  • Kafini M, Messaoudi S. Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay. Appl Anal. 2020;99(3):530–547. doi:10.1080/00036811.2018.1504029
  • Liu GW. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electron Res Arch. 2020;28(1):263–289. doi:10.3934/era.2020016
  • Cheng Y, Chu Y. A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms. Electron Res Arch. 2021;29(6):3867–3887. doi:10.3934/era.2021066
  • Chen YX, Xu RZ. Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity. Nonlinear Anal Theory Methods Appl. 2020;192:111664. doi:10.1016/j.na.2019.111664
  • Lian W, Xu RZ. Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv Nonlinear Anal. 2020;9(1):613–632. doi:10.1515/anona-2020-0016
  • Levine HA. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+F(u). Arch Rational Mech Anal. 1973;51:371–386. doi:10.1007/BF00263041
  • Simon J. Compact sets in the space Lp(0,T;B). Ann Mat Pura Appl. 1987;146(4):65–96.
  • Lions JLQuelque méthodes de résolution des problemes aux limites non linéaires. Paris: Dunod; 1969.

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