# zbMATH — the first resource for mathematics

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.)
##### Keywords:
global solution; blow up
Full Text: