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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:
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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