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http://dx.doi.org/10.4134/BKMS.b190190

GLOBAL SOLUTION AND BLOW-UP OF LOGARITHMIC KLEIN-GORDON EQUATION  

Ye, Yaojun (Department of Mathematics and Information Science Zhejiang University of Science and Technology)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.2, 2020 , pp. 281-294 More about this Journal
Abstract
The initial-boundary value problem for a class of semilinear Klein-Gordon equation with logarithmic nonlinearity in bounded domain is studied. The existence of global solution for this problem is proved by using potential well method, and obtain the exponential decay of global solution through introducing an appropriate Lyapunov function. Meanwhile, the blow-up of solution in the unstable set is also obtained.
Keywords
Klein-Gordon equation; logarithmic nonlinearity; global solution; exponential decay; blow-up;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061-1083. https://doi.org/10.2307/2373688   DOI
2 X. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc. 50 (2013), no. 1, 275-283. https://doi.org/10.4134/BKMS.2013.50.1.275   DOI
3 T. Hiramatsu, M. Kawasaki, and F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys. 2010 (2010), no. 6, 008. https://doi.org/10.1088/1475-7516/2010/06/008   DOI
4 W. Krolikowski, D. Edmundson, and O. Bang, Unfiied model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E 61 (2000), 3122-3126. https://doi.org/10.1103/PhysRevE.61.3122   DOI
5 A. Linde, Strings, textures, inflation and spectrum bending, Phys. Lett. B 284 (1992), no. 3-4, 215-222. https://doi.org/10.1016/0370-2693(92)90423-2   DOI
6 S. De Martino, M. Falanga, C. Godano, and G. Lauro, Logarithmic Schrodinger-like equation as a model for magma transport, Europhys. Lett. 63 (2003), no. 3, 472-475. https://doi.org/10.1209/epl/i2003-00547-6   DOI
7 S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl. 265 (2002), no. 2, 296-308. https://doi.org/10.1006/jmaa.2001.7697   DOI
8 Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101-123. https://doi.org/10.2307/1993333   DOI
9 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. https://doi.org/10.1007/BF02761595   DOI
10 D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148-172. https://doi.org/10.1007/BF00250942   DOI
11 M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhauser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
12 H. Buljan, A. Siber, M. Soljacic, T. Schwartz, M. Segev, and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E (3) 68 (2003), no. 3, 036607, 6 pp. https://doi.org/10.1103/PhysRevE.68.036607
13 K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006), no. 11, 1235-1270.
14 J. D. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D 52 (1995), 5576-5587.   DOI
15 K. Bartkowski and P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A 41 (2008), no. 35, 355201, 11 pp. https://doi.org/10.1088/1751-8113/41/35/355201
16 I. Bia lynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 23 (1975), no. 4, 461-466.
17 I. Bia lynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics 100 (1976), no. 1-2, 62-93. https://doi.org/10.1016/0003-4916(76)90057-9   DOI
18 T. Cazenave, Stable solutions of the logarithmic Schrodinger equation, Nonlinear Anal. 7 (1983), no. 10, 1127-1140.   DOI
19 T. Cazenave and A. Haraux, Equation de Schrodinger avec non-linearite logarithmique, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), no. 4, A253-A256.
20 H. Chen, P. Luo, and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl. 422 (2015), no. 1, 84-98. https://doi.org/10.1016/j.jmaa.2014.08.030   DOI
21 S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations 13 (2008), no. 11-12, 1051-1074.
22 H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudoparabolic equations with logarithmic nonlinearity, J. Dierential Equations 258 (2015), no. 12, 4424-4442. https://doi.org/10.1016/j.jde.2015.01.038   DOI
23 K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B 425 (1998), 309-321.   DOI
24 F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincare Anal. Non Lineaire 23 (2006), no. 2, 185-207. https://doi.org/10.1016/j.anihpc.2005.02.007   DOI
25 P. Gorka, Convergence of logarithmic quantum mechanics to the linear one, Lett. Math. Phys. 81 (2007), no. 3, 253-264. https://doi.org/10.1007/s11005-007-0183-x   DOI
26 S. Gerbi and B. Said-Houari, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Anal. 74 (2011), no. 18, 7137-7150. https://doi.org/10.1016/j.na.2011.07.026   DOI
27 S. Gerbi and B. Said-Houari, Exponential decay for solutions to semilinear damped wave equation, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 3, 559-566. https://doi.org/10.3934/dcdss.2012.5.559
28 P. Gorka, Logarithmic quantum mechanics: existence of the ground state, Found. Phys. Lett. 19 (2006), no. 6, 591-601. https://doi.org/10.1007/s10702-006-1012-7   DOI
29 P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B 40 (2009), no. 1, 59-66.