[1] K. Bartkowski, P. Gorka, One-dimensional Klein–Gordon equation with logarithmic nonlinearities, J. Phys. A., 41(35) (2008), 1-11.
[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] I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100(1–2) (1976), 62-93.
[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] T. Cazenave, A. Haraux, Equations d’evolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse 2(1) (1980), 21–51.
[6] P. Gorka, Logarithmic Klein–Gordon equation, Acta Phys. Pol. B 40(1) (2009), 59–66.
[7] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97(4) (1975), 1061–1083.
[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] T. Hiramatsu, M. Kawasaki, F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys.6(2010).
[10] J. Lions, Quelques methodes de resolution des problems aux limites non lineaires, Dunod Gauthier-Villars, Paris, 1969.
[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] 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] 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] 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.
Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term
Year 2019,
Volume: 2 Issue: 1, 27 - 36, 30.10.2019
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.
[1] K. Bartkowski, P. Gorka, One-dimensional Klein–Gordon equation with logarithmic nonlinearities, J. Phys. A., 41(35) (2008), 1-11.
[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] I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100(1–2) (1976), 62-93.
[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] T. Cazenave, A. Haraux, Equations d’evolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse 2(1) (1980), 21–51.
[6] P. Gorka, Logarithmic Klein–Gordon equation, Acta Phys. Pol. B 40(1) (2009), 59–66.
[7] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97(4) (1975), 1061–1083.
[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] T. Hiramatsu, M. Kawasaki, F. Takahashi, Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys.6(2010).
[10] J. Lions, Quelques methodes de resolution des problems aux limites non lineaires, Dunod Gauthier-Villars, Paris, 1969.
[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] 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] 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] 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.
Pişkin, E., & Irkıl, N. (2019). Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term. Conference Proceedings of Science and Technology, 2(1), 27-36.
AMA
Pişkin E, Irkıl N. Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term. Conference Proceedings of Science and Technology. October 2019;2(1):27-36.
Chicago
Pişkin, Erhan, and Nazlı Irkıl. “Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation With Logarithmic Source Term”. Conference Proceedings of Science and Technology 2, no. 1 (October 2019): 27-36.
EndNote
Pişkin E, Irkıl N (October 1, 2019) Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term. Conference Proceedings of Science and Technology 2 1 27–36.
IEEE
E. Pişkin and N. Irkıl, “Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term”, Conference Proceedings of Science and Technology, vol. 2, no. 1, pp. 27–36, 2019.
ISNAD
Pişkin, Erhan - Irkıl, Nazlı. “Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation With Logarithmic Source Term”. Conference Proceedings of Science and Technology 2/1 (October 2019), 27-36.
JAMA
Pişkin E, Irkıl N. Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term. Conference Proceedings of Science and Technology. 2019;2:27–36.
MLA
Pişkin, Erhan and Nazlı Irkıl. “Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation With Logarithmic Source Term”. Conference Proceedings of Science and Technology, vol. 2, no. 1, 2019, pp. 27-36.
Vancouver
Pişkin E, Irkıl N. Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term. Conference Proceedings of Science and Technology. 2019;2(1):27-36.