Cauchy problem for a loaded integro-differential equation

PDEs and integro-differential equations of the convolution type arise in mathematical models of physical, biological, and technical systems and in other areas where it is necessary to take into account the history of processes. Constitutive relations in a linear processes of inhomogeneous diffusion and propagation of waves with memory contain a time- and space-dependent memory kernel. Problems of memory kernels identification in parabolic and hyperbolic integro-differential equations have been intensively studied.

In many cases, the equations describing the propagation electrodynamic and elastic waves reduced to hyperbolic equations with integral convolution.  Based on the foregoing, iwe study an analog of the Cauchy problem for a generalized loaded wave equation involving convolution-type operators. The study aims to construct optimal representations of the solution of the hyperbolic type equation and to study questions of the existence and uniqueness of the solution to the Cauchy problem for the loaded differential equation.