Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term

Abstract

The main goal of this paper is to study for a fourth-order hyperbolic equation with logarithmic nonlinearity. We obtain several results: Firstly, by using Feado-Galerkin method and a logaritmic Sobolev inequality, we proved local existence of solutions. Later, we proved global existence of solutions by potential well method. Finally, we showed the decay estimates result of the solutions.

Keywords

• Existence,Decay of solution

Kaynakça

1. [1] K. Bartkowski, P. Gorka, One-dimensional Klein–Gordon equation with logarithmic nonlinearities, J. Phys. A., 41(35) (2008), 1-11.
2. [2] I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 23(4) (1975), 461-466.
3. [3] I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100(1–2) (1976), 62-93.
4. [4] 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 3(2003), 68.
5. [5] T. Cazenave, A. Haraux, Equations d’evolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse 2(1) (1980), 21–51.
6. [6] P. Gorka, Logarithmic Klein–Gordon equation, Acta Phys. Pol. B 40(1) (2009), 59–66.
7. [7] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97(4) (1975), 1061–1083.
8. [8] X.S. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc. 50(1) (2013), 275–283.

9. [9] T. Hiramatsu, M. Kawasaki, F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys.6(2010).
10. [10] J. Lions, Quelques methodes de resolution des problems aux limites non lineaires, Dunod Gauthier-Villars, Paris, 1969.
11. [11] S. De Martino, M. Falanga, C. Godano, G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys, 63(3) (2003), 472–475.
12. [12] 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, Journal of Evolution Equations, 18(1) (2018),105-125.
13. [13] M.M. Al-Gharabli, S.A. Messaoudi, The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term, J. Math. Anal. Appl., 454(2017), 1114-1128.
14. [14] H.W. Zhang, G.W. Liu, Q.Y. Hu, Asymptotic Behavior for a Class of Logarithmic Wave Equations with Linear Damping, Appl. Math. Optim., 79(1) (2017), 131-144.