For a nonlinear fourth-order partial differential equation modeling the propagation of flexural waves in a Timoshenko beam, the Cauchy problem is studied in the space of continuous functions defined on the entire real axis and having limits at infinity. The time interval of existence and uniqueness of the classical solution to an auxiliary Cauchy problem related to the original one is established, and an estimate for the norm of this local solution is provided. Conditions ensuring the connection between local classical solutions of the original and auxiliary Cauchy problems on a specific time interval are found. Sufficient conditions for extending the local classical solution of the Cauchy problem to a global one and for the blow-up of the solution to the nonlinear Timoshenko equation on a finite time interval are considered.
Kh. G. Umarov (Wed,) studied this question.