Averaging a High-Frequency Hyperbolic System of Quasilinear Equations with Large Summands

One of the powerful asymptotic methods of the theory of differential equations is the well-known averaging methodassociated with the famous researchers Krylov and Bogolyubov. Thismethod is well developed for ordinary differential and integral equations as well asfor many classes of partial differential equations, but not studied sufficiently for hyperbolic systems.The method was justified for semilinear hyperbolic systems in the articles by Mitropolsky,Khoma, and some other authors. Moreover, some authorsproposed and justified an algorithm for constructing complete asymptotics of solutions tothese systems; the solution of the averaged problem is the main term of the asymptotics.We study the Cauchy problem in a multidimensional space-time layer fora hyperbolic system of first-order quasilinear differential equations with rapidlytime-oscillating terms. Such terms on the right-hand side may be large andproportional to the square root of the high-frequency oscillations; the large termshave zero mean in the fast variable (the product of frequency and time). The peculiarityof the problem is the fact that the summands of the equations do not dependexplicitly on spatial variables.We construct some limit (averaged) problem as theoscillation frequency tends to infinity and justify passage to the averaging method.This proves the unique solvability of the original (perturbed)problem and substantiates the asymptotic proximity of solutions to the original (perturbed)and averaged problems uniformly through the layer.