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Keywords:
nonlinear hyperbolic equation; initial boundary value problem; classical \linebreak global solution; blow up of solutions
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 $$u_{tt}-\left[ a_0+n a_1(u_x)^{n-1}\right]u_{xx}-a_2 u_{xxtt}=0.$$
References:
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