# Article

 Title: Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order (English) Author: Chen, Guowang Author: Wang, Shubin Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 36 Issue: 3 Year: 1995 Pages: 475-487 . Category: math . Summary: The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation $$u_{tt}-\alpha u_{xx}-\beta u_{xxtt}=\varphi (u_x)_x$$ are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods $$u_{tt}-\left[ a_0+n a_1(u_x)^{n-1}\right]u_{xx}-a_2 u_{xxtt}=0.$$ (English) Keyword: nonlinear hyperbolic equation Keyword: initial boundary value problem Keyword: classical \linebreak global solution Keyword: blow up of solutions MSC: 35L35 MSC: 35L75 MSC: 35Q72 MSC: 74H45 MSC: 74K10 idZBL: Zbl 0839.35085 idMR: MR1364488 . Date available: 2009-01-08T18:19:29Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118776 . Reference: [1] Zhuang Wei, Yang Guitong: Propagation of solitary waves in the nonlinear rods.Applied Mathematics and Mechanics 7 (1986), 571-581. Reference: [2] Zhang Shangyuan, Zhuang Wei: Strain solitary waves in the nonlinear elastic rods (in Chinese).Acta Mechanica Sinica 20 (1988), 58-66. Reference: [3] Chen Guowang, Yang Zhijian, Zhao Zhancai: Initial value problems and first boundary problems for a class of quasilinear wave equations.Acta Mathematicae Applicate Sinica 9 (1993), 289-301. Zbl 0822.35094, MR 1259814 Reference: [4] Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.SIAM J. Math. Anal. 5 (1974), 138-146. Zbl 0243.35069, MR 0399682 Reference: [5] Levine H.A.: Instability & nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+F(u)$.Trans. of AMS 192 (1974), 1-21. MR 0344697 .

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