Geodesic
Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order
Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 475-487.

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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 $$ u_{tt}-\left[ a_0+n a_1(u_x)^{n-1}\right]u_{xx}-a_2 u_{xxtt}=0. $$
Classification : 35L35, 35L75, 35Q72, 74H45, 74K10
Keywords: nonlinear hyperbolic equation; initial boundary value problem; classical \linebreak global solution; blow up of solutions 
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     author = {Chen, Guowang and Wang, Shubin},
     title = {Existence and non-existence of global solutions for nonlinear hyperbolic equations  of higher order},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {475--487},
     publisher = {mathdoc},
     volume = {36},
     number = {3},
     year = {1995},
     mrnumber = {1364488},
     zbl = {0839.35085},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1995__36_3_a8/}
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Chen, Guowang; Wang, Shubin. Existence and non-existence of global solutions for nonlinear hyperbolic equations  of higher order. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 475-487. http://geodesic.mathdoc.fr/item/CMUC_1995__36_3_a8/