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nonlinear hyperbolic equation; initial boundary value problem; classical \linebreak global solution; blow up of solutions
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 $$ u_{tt}-\left[ a_0+n a_1(u_x)^{n-1}\right]u_{xx}-a_2 u_{xxtt}=0. $$
[1] Zhuang Wei, Yang Guitong: Propagation of solitary waves in the nonlinear rods. Applied Mathematics and Mechanics 7 (1986), 571-581.
[2] Zhang Shangyuan, Zhuang Wei: Strain solitary waves in the nonlinear elastic rods (in Chinese). Acta Mechanica Sinica 20 (1988), 58-66.
[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. MR 1259814 | Zbl 0822.35094
[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. MR 0399682 | Zbl 0243.35069
[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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