Cauchy Problem for the Torsional Vibration Equation of a Nonlinear-Elastic Rod of Infinite Length

for the differential equation of torsional vibrations of an infinite nonlinear-elastic rod, the solvability of the Cauchy problem in the space of continuous functions on the real axis is studied. An explicit form of the solution of the corresponding linear partial differential equation is obtained. The time interval for the existence of the classical solution to the Cauchy problem for a nonlinear equation is found and an estimate of this local solution is obtained. Conditions for the existence of a global solution and blow-up of the solution on a finite interval are considered.