The initial value problem for the quasi-linear partial integro-differential equation of higher order with a degenerate kernel

High-order partial differential equations are of great interest when it comes to physical applications. Many problems of gas dynamics, elasticity theory and the theory of plates and shells are reduced to the consideration of high-order partial differential equations. This paper studies the one-valued solvability of the initial value problem for a nonlinear partial integro-differential equation of an arbitrary order with a degenerate kernel. The expression of higher-order partial differential equations as a superposition of first-order partial differential operators has allowed us to apply methods for solving first-order partial differential equations. First-order partial differential equations can be locally solved by the methods of the theory of ordinary differential equations, reducing them to a characteristic system. The existence and uniqueness of the solution to this problem is proved by the method of successive approximation. An estimate of convergence of the iterative Picard process is obtained. The stability of the solution from the second argument of the initial value problem is shown.