Abstract
The Cauchy problem is studied for a nonlinear partial differential equation of Sobolev type (i.\,e., an equation not solved for the highest time derivative) that generalizes the equation of small oscillations of an absolutely flexible homogeneous rod. The problem is considered in the space of continuous functions defined on the entire real axis, for which limits exist at infinity. An explicit form of the classical solution to the corresponding linear homogeneous equation is obtained, and norm estimates for the operator-valued functions representing this solution are derived. An estimate for the norm of the solution to the Cauchy problem for the linear homogeneous equation is also given. The time interval of existence and uniqueness of a classical solution to the auxiliary Cauchy problem related to the original one is established, along with an estimate for the norm of this local solution. Conditions are found that ensure a correspondence between the classical solutions of the original and auxiliary Cauchy problems on a certain time interval. Moreover, conditions are examined under which the classical solution to the Cauchy problem blows up on a finite time interval.