Global existence and blow-up of solutions for logarithmic Klein-Gordon equation

Yaojun Ye; Lanlan Li; Yaojun Ye; Lanlan Li

doi:10.3934/math.2021404

AIMS Mathematics

2021, Volume 6, Issue 7: 6898-6914. doi: 10.3934/math.2021404

Previous Article Next Article

Research article

Global existence and blow-up of solutions for logarithmic Klein-Gordon equation

Yaojun Ye ^,,
Lanlan Li

Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received: 13 December 2020 Accepted: 15 April 2021 Published: 22 April 2021
MSC : 35L05, 35L10, 35B40

This arcitle concerns the initial-boundary value problem for a class of Klein-Gordon equation with logarithmic nonlinearity. By using Galerkin method and compactness criterion, we prove the existence of global solutions to this problem. Meanwhile, the blow-up of solutions in the unstable set is also obtained.
- Klein-Gordon equation,
- logarithmic nonlinearity,
- initial-boundary value problem,
- global solutions,
- blow-up
Citation: Yaojun Ye, Lanlan Li. Global existence and blow-up of solutions for logarithmic Klein-Gordon equation[J]. AIMS Mathematics, 2021, 6(7): 6898-6914. doi: 10.3934/math.2021404

Related Papers:

Abstract

This arcitle concerns the initial-boundary value problem for a class of Klein-Gordon equation with logarithmic nonlinearity. By using Galerkin method and compactness criterion, we prove the existence of global solutions to this problem. Meanwhile, the blow-up of solutions in the unstable set is also obtained.

References

[1]	H. Buljan, A. Siber, M. Soljacic, T. Schwartz, M. Segev, D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E, 68 (2003), 036607. doi: 10.1103/PhysRevE.68.036607
[2]	S. De Martino, M. Falanga, C. Godano, G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Lett., 63 (2003), 472–475. doi: 10.1209/epl/i2003-00547-6
[3]	W. Krolikowski, D. Edmundson, O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122–3126. doi: 10.1103/PhysRevE.61.3122
[4]	P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B, 40 (2009), 59–66.
[5]	I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Serie Sci. Math. Astron. Phys., 23 (1975), 461–466.
[6]	I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62–93. doi: 10.1016/0003-4916(76)90057-9
[7]	K. Bartkowski, P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A: Math. Theor., 41 (2008), 355201. doi: 10.1088/1751-8113/41/35/355201
[8]	T. Cazenave, A. Haraux, Equations devolution avec non linearite logarithmique, Ann. Fac. Sci. Toulouse: Math., 2 (1980), 21–51. doi: 10.5802/afst.543
[9]	P. Brenner, On $L_p$-decay and scattering for nonlinear Klein-Gordon equations, Math. Scand., 51 (1982), 333–360. doi: 10.7146/math.scand.a-11985
[10]	K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201–225. doi: 10.1006/jfan.1999.3503
[11]	K. Nakanishi, Scattering theory for nonlinear Klein-Gordon with Sobolev critical power, Int. Math. Res. Not., 1999 (1999), 31–60. doi: 10.1155/S1073792899000021
[12]	P. Brenner, On space-timr means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186 (1984), 383–391. doi: 10.1007/BF01174891
[13]	P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differ. Equations, 56 (1985), 310–344. doi: 10.1016/0022-0396(85)90083-X
[14]	H. A. Levine, Instablity and nonexistence of global solutions of nonlinear wave equations of the form $Pu_tt = Au+F(u)$, Trans. Am. Math. Soc., 192 (1974), 1–21.
[15]	J. Ball, Remark on blow-up and nonexistence theorems for nonlinear evolution equations, Q. J. Math., 28 (1977), 473–486. doi: 10.1093/qmath/28.4.473
[16]	Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equations, 192 (2003), 155–169. doi: 10.1016/S0022-0396(02)00020-7
[17]	L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273–303. doi: 10.1007/BF02761595
[18]	D. H. Sattinger, On global solutions for nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148–172. doi: 10.1007/BF00250942
[19]	Y. C. Liu, J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665–2687. doi: 10.1016/j.na.2005.09.011
[20]	H. Chen, P. Luo, G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84–98. doi: 10.1016/j.jmaa.2014.08.030
[21]	H. Chen, S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 258 (2015), 4424–4442. doi: 10.1016/j.jde.2015.01.038
[22]	T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127–1140. doi: 10.1016/0362-546X(83)90022-6
[23]	T. Cazenave, A. Haraux, Équation de Schrödinger avec non-linéarité logarithmique, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), 253–256.
[24]	P. Gorka, Logarithmic quantum mechanics: Existence of the ground state, Found. Phys. Lett., 19 (2006), 591–601. doi: 10.1007/s10702-006-1012-7
[25]	P. Gorka, Convergence of logarithmic quantum mechanics to the linear one, Lett. Math. Phys., 81 (2007), 253–264. doi: 10.1007/s11005-007-0183-x
[26]	L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061–1083. doi: 10.2307/2373688
[27]	J. L. Lions, Quelques Méthods de Résolution des Problèm aux Limits nonlinéars, Dunod: Paris, 1969.
[28]	S. M. Zheng, Nonlinear Evolution Equations, New York: Chapman and Hall/CRC, 2004.
[29]	Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Am. Math. Soc., 95 (1960), 101–123. doi: 10.1090/S0002-9947-1960-0111898-8
[30]	M. Willem, Minimax Theorems, Progress Nonlinear Differential Equations Applications, Boston: Birkhäuser, 1996.
[31]	F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincare C, 23 (2006), 185–207.
[32]	S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296–308. doi: 10.1006/jmaa.2001.7697
[33]	K. Agre, M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differ. Integr. Equations, 19 (2006), 1235–1270.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)