Integro-differential equation with a higher-order two-dimensional Whitham operator

We examine the unique solvability of an initial-value problem for a certain higher-order quasilinear partial integro-differential equation with degenerate kernel. Expressing the higher-order partial integro-differential operators as the superposition of first-order partial differential operators, we represent the integro-differential equation considered as an ordinary integro-differential equations that describes the change of the unknown function along characteristics. Using the method of successive approximations, we prove the unique solvability of the initial-value problem and obtain an estimate for the convergence rate of the Picard iterative process.