Cauchy problem of nonlinear Klein-Gordon equations with general nonlinearities. (English) Zbl 1500.35063
Summary: This paper is concerned with the Cauchy problem of nonlinear Klein-Gordon equations with general nonlinearities. We use the potential well and convexity methods to prove the global existence and finite time blow up of solution with low and critical initial energy levels. And a finite time blow up of the solution with arbitrarily positive initial energy level is proved.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L71 | Second-order semilinear hyperbolic equations |
Keywords:
Cauchy problem; Klein-Gordon equations; global solution; finite time blow up; potential well methods; convexity methodsReferences:
| [1] | Drazin, PJ; Johnson, RS, Solitons: An Introduction (1989), Cambridge: Cambridge University Press, Cambridge · Zbl 0661.35001 · doi:10.1017/CBO9781139172059 |
| [2] | Duncany, DB, Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 34, 1742-1760P (1997) · Zbl 0889.65093 · doi:10.1137/S0036142993243106 |
| [3] | Perring, JK; Skyrme, TH, A model unified field equation, Nucl. Phys., 31, 550-555 (1962) · Zbl 0106.20105 · doi:10.1016/0029-5582(62)90774-5 |
| [4] | Keller, JB, Electrodynamics. I. The equilibrium of a charged gas in a container, J. Ration. Mech. Anal., 5, 715-724 (1956) · Zbl 0070.44207 |
| [5] | Wazwaz, AM, The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation, Appl. Math. Comput., 167, 1179-1195 (2005) · Zbl 1082.65584 |
| [6] | Shatah, J., Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91, 313-327 (1983) · Zbl 0539.35067 · doi:10.1007/BF01208779 |
| [7] | Lee, TD, Particle Physics and Introduction to Field Theory (1981), New York: Harwood Academic Publishers, New York · doi:10.1063/1.2914386 |
| [8] | 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., 136, 3477-3482 (2008) · Zbl 1154.35064 · doi:10.1090/S0002-9939-08-09514-2 |
| [9] | Xu, RZ, Global existence, blow up and asymototic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Math. Methods Appl. Sci., 35, 831-844 (2009) |
| [10] | Li, KT; Zhang, QD, Existence and nonexistence of global solutions for global solution for the equation of dislocation of crystals, J. Differ. Equ., 146, 5-21 (1998) · Zbl 0926.35073 · doi:10.1006/jdeq.1998.3409 |
| [11] | Kutev, N.; Kolkovska, N.; Dimova, M., Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy, Electron. Res. Arch., 28, 671-689 (2020) · Zbl 1442.35260 · doi:10.3934/era.2020035 |
| [12] | Kutev, N.; Kolkovska, N.; Dimova, M., Sign-preserving functionals and blow-up to Klein-Gordon equation with arbitrary high energy, Appl. Anal., 95, 860-873 (2016) · Zbl 1338.35072 · doi:10.1080/00036811.2015.1038994 |
| [13] | Liu, YC, On potential and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equ., 192, 155-169 (2013) · Zbl 1024.35078 |
| [14] | Gazzola, F.; Squassina, M., Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 185-207 (2006) · Zbl 1094.35082 · doi:10.1016/j.anihpc.2005.02.007 |
| [15] | Xiang, MQ; Rǎdulescu, VD; Zhang, BL, Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, Nonlinearity, 31, 3228-3250 (2018) · Zbl 1393.35090 · doi:10.1088/1361-6544/aaba35 |
| [16] | Giacomoni, J.; Rǎdulescu, VD; Warnault, G., Quasilinear parabolic problem with variable exponent: qualitative analysis and stabilization, Commun. Contemp. Math., 20, 1750065 (2018) · Zbl 1404.35240 · doi:10.1142/S0219199717500651 |
| [17] | Xu, RZ; Lian, W.; Niu, Y., Global well-posedness of coupled parabolic systems, Sci. China Math., 63, 321-356 (2020) · Zbl 1431.35063 · doi:10.1007/s11425-017-9280-x |
| [18] | Lian, W.; Wang, J.; Xu, RZ, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equ., 269, 4914-4959 (2020) · Zbl 1448.35322 · doi:10.1016/j.jde.2020.03.047 |
| [19] | Zhang, MY; Ahmed, MS, Sharp conditions of global existence for nonlinear Schröinger equation with a harmonic potential, Adv. Nonlinear Anal., 9, 882-894 (2020) · Zbl 1434.35199 · doi:10.1515/anona-2020-0031 |
| [20] | Liao, ML; Liu, Q.; Ye, HL, Global existence and blow-up of weak solutions for a class of fractional \(p\)-Laplacian evolution equations, Adv. Nonlinear Anal., 9, 1569-1591 (2020) · Zbl 1436.35319 · doi:10.1515/anona-2020-0066 |
| [21] | Papageorgiou, NS; Rǎdulescu, VD; Repovš, DD, Nonlinear Analysis-Theory and Methods. Springer Monographs in Mathematics (2019), Cham: Springer, Cham · Zbl 1414.46003 |
| [22] | Xu, RZ, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Q. Appl. Math., 3, 459-468 (2010) · Zbl 1200.35035 |
| [23] | Lian, W.; Xu, RZ, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9, 613-632 (2020) · Zbl 1421.35222 · doi:10.1515/anona-2020-0016 |
| [24] | Wang, XC; Xu, RZ, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10, 261-288 (2021) · Zbl 1447.35078 · doi:10.1515/anona-2020-0141 |
| [25] | Xu, RZ; Su, J., Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264, 2732-2763 (2013) · Zbl 1279.35065 · doi:10.1016/j.jfa.2013.03.010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
