Constructing a solution of an initial boundary value problem for a functional-differential equation arising in mechanics of discrete-distributed systems

We consider a three-point initial boundary value problem for a nonlinear functional partial differential equation with an infinite (integral) delay in the argument. The boundary conditions contain a delay in the argument and the highest derivative with respect to time. The initial boundary value problem is a mathematical model of the dynamics of a distributed rotating ideal shaft (rotor) of constant cross section with an ideal rigid circular disk mounted on the shaft. The axes of the shaft and disk coincide, the ends of the shaft rest on bearings. It is assumed that the shaft material obeys a nonlinear rheological model of a hereditarily elastic body. A definition of a solution of the initial boundary value problem is given based on the variational principle. Function spaces for the initial conditions and solutions are introduced, the phase space of the initial boundary value problem is defined. The existence theorem is proved for a solution, as is its uniqueness and continuous dependence on the initial conditions and parameters of the initial boundary value problem in the norm of the phase space. Thus, we demonstrate the well-posedness of the considered initial boundary value problem.