Classical solution of a problem for a system of equations from mechanics with nonsmooth Cauchy conditions

In this article, we study a mixed problem in a quarter-plane for one system of differential equations, which describes vibrations in the string from viscoelastic material, which corresponds to the Maxwell model. At the bottom of the boundary, we pose the Cauchy conditions, and one of them has a discontinuity of the first kind at one point. We set a smooth boundary condition on the lateral boundary. We derive the Klein – Gordon – Fock equation for one function of the studied system. We use the method of characteristics to build the classical solution as a solution of some integral equation. We prove the uniqueness and establish conditions under which a piecewise smooth solution exists. The Cauchy problem is considered the system’s second function. We determine the conditions under which the solution of the system has sufficient smoothness