Opuscula Math. 40, no. 1 (2020), 111-130
https://doi.org/10.7494/OpMath.2020.40.1.111

Opuscula Mathematica

# Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity

Wei Lian
Md Salik Ahmed
Runzhang Xu

Abstract. In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels ($$E(0)\lt d$$, $$E(0)=d$$ and $$E(0)\gt 0$$) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.

Keywords: global existence, blow-up, logarithmic and polynomial nonlinearity, potential well.

Mathematics Subject Classification: 35L71, 35L20, 35L05.

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1. M.M. Al-Gharabli, S.A. Messaoudi, Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ. 18 (2018), 105-125.
2. J.M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. 28 (1977), 473-486.
3. J.D. Barrow, P. Persons, Inflationary models with logarithmic potentials, Phys. Rev. D 52 (1970), 5576.
4. K. Bartkowski, P. Gorka, One dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A 41 355201 (2008), 1-11.
5. I. Bialynicki-Birula, J. Mycielski, Gaussons: Solitons of the logarithmic Schrödinger equation, Phys. Scr. 20 (1979), 539-544.
6. 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), 1-5.
7. T. Cazenave, A. Haraux, Équations d'évolution avec non-linéarité logrithmique, Ann. Fac. Sci. Toulouse Math. 2 (1980), 21-51.
8. J.A. Esquivel-Avila, The dynamics of a nonlinear wave equation, J. Math. Anal. Appl. 279 (2003), 135-150.
9. F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré Anal. Non Linéaire 23 (2006), 185-207.
10. P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B 40 (2009), 59-66.
11. X.S. Han, Global exitence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc. 50 (2013), 275-283.
12. T. Hiramatsu, M. Kawasaki, F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cos. Astr. Phys. 6 (2010).
13. Q. Hu, H. Zhang, G. Liu, Global exitence and exponential growth of the solution for the logarithmic Boussinesq-type equation, J. Math. Anal. Appl. 436 (2016), 990-1001.
14. W. Krolikowski, D. Edmundson, O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E 61 (2000), 3122-3126.
15. H.A. Levine, Instability and non-existence of global solutions to nonlinear wave equations of the form $$Pu_{tt}=-Au+F(u)$$, Trans. Amer. Math. Soc. 192 (1974), 1-21.
16. H.A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138-146.
17. W. Lian, M.S. Ahmed, R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal. 184 (2019), 239-257.
18. A.D. Linde, Strings, textures, inflation and spectrum bending, Phys. Lett. B284 (1992), 215-222.
19. Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations 192 (2003), 155-169.
20. Y. Liu, R. Xu, Nonlinear wave equations and reaction-diffusion equations with several nonlinear source terms of different signs, Discrete Contin. Dyn. Syst. Ser. B 7 (2007), 171-189.
21. Y. Liu, J. Zhao, On potential wells and applications to semolinear hyperbolic equations and parabolic equations, Nonlinear Anal. 64 (2006), 2665-2687.
22. S. De Martino, M. Falanga, C. Godon, G. Lauro, Logarithmic Schröinger-like equation as a model for magma transport, Europhys. Lett. 63 (2003), 472-475.
23. E.M. Maslov, Pulsons, bubbles and the corresponding nonlinear wave equations in $$n+1$$ dimensions, Phys. Lett. A 151 (1990), 47-51.
24. L.E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273-303.
25. P. Pucci, J. Serrin, Some new results on global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (1998), 203-214.
26. D.H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968), 148-172.
27. B. Straughan, Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc. 48 (1975), 381-390.
28. Y. Wang, A suffiecient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc. 136 (2008), 3477-3482.
29. R. Xu, Initial boundary value problem of a semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. J. Math. 68 (2010), 459-468.
30. R. Xu, Y. Ding, Global solution and finite time blow up for damped Klein-Gordon equation, Acta Math. Sci. Ser. A (Chin. Ed.) 33 (2013), 643-652.
31. R. Xu, X. Wang, H. Xu, M. Zhang, Arbitrary energy global existence for wave equation with combined power-type nonlinearities of different signs, Bound Value Probl. (2016), Article number: 214.
32. R. Xu, Y. Yang, B. Liu, J. Shen, S. Huang, Global existence and blow up of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys. 66 (2015), 955-976.
33. H. Zhang, G. Liu, Q. Hu, Exponential decay of energy for a logarithmic wave equation, J. Partial Differential Equation 28 (2015), 269-277.
• Runzhang Xu (corresponding author)
• https://orcid.org/0000-0003-4703-9319
• College of Automation, College of Mathematical Sciences, Harbin Engineering University, 150001, People's Republic of China
• Communicated by Vicentiu D. Radulescu.
• Accepted: 2019-08-01.
• Published online: 2020-02-17.