Abstract
This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.
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References
Bogolubsky J L. Some examples of inelastic soliton interaction[J]. Computer Physics Communications, 1977, 13(1):149–155.
Clarkson P A, Leveque R J, Saxton R A. Solitary wave interaction in elastic rods[J]. Studies in Applied Mathematics, 1986, 75:95–122.
Zhuang Wei, Yang Guitong. Propagation of solitary waves in the nonlinear elastic rods[J]. Appl Math Mech-Engl Ed, 1986, 7(7):571–582.
Zhang Shanyuan, Zhuang Wei. Strain solitary waves in the nonlinear elastic rods[J]. Acta Mech Sinica, 1988, 20(1):58–66 (in Chinese).
Seyler C E, Fanstermacher D L. A symmetric regularized long wave equation[J]. Phys Fluids, 1984, 27(1):4–7.
Chen Guowang, Yang Zhijian, Zhao Zhancai. Initial value problem and first boundary problem for a class of quasilinear wave equations[J]. Acta Math Appl Sinica, 1993, 9(4):289–301.
Yang Zhijian. Existence and non-existence of global solutions to a generalized modification of the improved Boussinesq equation[J]. Math Mech Appl Sci, 1998, 21(16):1467–1477.
Guo Boling. The spectral method for symmetric regularized wave equations[J]. J Comp Math, 1987, 5(4):297–306.
Yang Zhijian, Song Changming. On the problem of the existence of global solutions for a class of nonlinear evolution equations[J]. Acta Math Appl Sinica, 1997, 20(3):321–331 (in Chinese).
Guo Boling. The vanishing viscosity method and the viscosity of the difference scheme[M]. Beijing: Science Press, 1993 (in Chinese).
Zhu Weiqiu. Nonlinear waves in elastic rods[J]. Acta Solid Mechanica Sinica, 1980, 1(2):247–253.
Shang Yadong. Initial boundary value problem of equation u tt − Δu − Δu tt = f(u)[J]. Acta Mathematicae Applicatae Sinica, 2000, 23(3):385–393 (in Chinese).
Liu Yacheng, Li Xiaoyuan. Some remarks on the equation u tt − Δu − Δtt = f(u)[J]. Journal of Natural Science of Heilongjiang University, 2004, 21(3):1–6 (in Chinese).
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Communicated by GUO Xing-ming
Project supported by the National Natural Science Foundation of China (No. 10271034) and the Natural Science Foundation of Heilongjiang Province of China (No. A2007-02).
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Xu, Rz., Zhao, Xr. & Shen, Jh. Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms. Appl. Math. Mech.-Engl. Ed. 29, 259–262 (2008). https://doi.org/10.1007/s10483-008-0213-y
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DOI: https://doi.org/10.1007/s10483-008-0213-y