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Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order. (English) Zbl 0839.35085
The authors prove the existence and uniqueness of a classical local solution and classical global solution and blow up of the solution for the equation \(u_{tt}- au_{xx}- bu_{xxtt}= f(u_x)_x\) satisfying Dirichlet or Neumann boundary conditions. The results are applied to the equation of nonlinear waves in elastic rod, where \(f(s)= sn\).

MSC:
35L75 Higher-order nonlinear hyperbolic equations
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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