The Cauchy problem for the equation of bending vibrations of a nonlinear-elastic rod of infinite length

For the differential equation mentioned in the title of the article, the solvability of the Cauchy problem in the space of continuous functions on the whole real axis by reducing to an abstract Cauchy problem in a Banach space is studied. An explicit form of the solution of the corresponding linear equation is found. The time interval for the existence of the classical solution of the Cauchy problem for a nonlinear equation is established and an estimate of the norm of this local solution is obtained. The conditions for the existence of a global solution and the destruction of the solution on a finite interval are considered.