Averaging of a high-frequency hyperbolic system of quasi-linear equations with large terms

One of the powerful asymptotic methods of the theory of differential equations is the well-known averaging method, which is associated with the names of famous researchers N. M. Krylov and N. N. Bogolyubov. This method is deeply developed not only for ordinary differential and integral equations, but also for many classes of partial differential equations. However, for hyperbolic systems of differential equations, the averaging method has not been sufficiently studied. For semilinear hyperbolic systems, it is justified in the works of Yu. A. Mitropolsky, G. P. Khoma and some other authors. In addition, a number of authors have previously proposed and justified an algorithm for constructing complete asymptotics of solutions of such systems; the solution of the averaged problem is the main member of the asymptotics. In this paper, we study the Cauchy problem in a multidimensional space-time layer for a hyperbolic system of first-order quasi-linear differential equations with rapidly time-oscillating terms. Among such terms of the right part there may be large — proportional to the square root of the high frequency of oscillations, and the large terms have a zero mean for the fast variable (the product of frequency and time). The specificity of the problem is the fact that the terms of the equations do not explicitly depend on spatial variables. For this problem, a limit (averaged) problem is constructed with the oscillation frequency tending to infinity and a limit transition (averaging method) is justified. The latter means proving the unambiguous solvability of the original (perturbed) problem and substantiating the asymptotic proximity of solutions of the original (perturbed) and averaged problems uniform throughout the layer.