Blow-Up of the Solution and Global Solvability of the CauchyProblem for the Equation of Vibrations of a Hollow Rod

For a nonlinear partial differential equation of Sobolev type generalizing the equation of vibrations of a hollow flexible rod, the Cauchy problem is studied in the space of continuous functions defined on the entire real line and having limits at infinity. The conditions for the existence of a global classical solution and the blow-up of the solution of the Cauchy problem on a finite time interval are considered.