On the structure of solutions of a class of hyperbolic systems with several spatial variables in the far field

Regularization of the Singularly Perturbed Cauchy Problem for a Hyperbolic System

“…By choosing the regularizing function as a solution of the Cauchy problem for a first-order partial differential equation, the process of constructing the asymptotic solution of any order of the problem becomes significantly simplified. This approach makes it possible to easily solve the problems studied in works [7][8][9][10][11][12][13][14].…”

“…The papers [10][11][12][13][14] were devoted to the study of scalar and multidimensional systems of hyperbolic equations in the critical case. In [10], a formal asymptotic representation of the solution of an initialboundary value problem for a singularly perturbed system of first-order partial differential equations with small nonlinearity was constructed.…”

“…In the soiltion obtained, there appear splash zones in the vicinity of which the asymptotics is described by a parabolic equation. In [13], the boundary layer method was generalized to systems with an arbitrary number of spatial variables, and the behavior of the solution in the far field was described for large values of x and t. The work [14] was devoted to the construction of an asymptotic expansion of the solution of the Cauchy problem for one class of hyperbolic weakly nonlinear systems with many spatial variables. The parabolic quasilinear equation was obtained, which describes the behavior of the solution for asymptotically large values of independent variables.…”

Asymptotics of the solution of the hyperbolic system with a small parameter

“…Papers [10] - [14] are devoted to the study of scalar and multidimensional systems of hyperbolic equations in the critical case. In [10] a formal asymptotic representation of the solution of an initial-boundary value problem for a MJEN MANAS Journal of Engineering, Volume 9 (Issue 2) © 2021 www.journals.manas.edu.kg singularly perturbed system of first order partial differential equations with small nonlinearity is constructed.…”

“…The work [14] is devoted to the construction of an asymptotic expansion of the solution of the Cauchy problem for one class of hyperbolic weakly nonlinear systems with many spatial variables. A parabolic quasilinear equation is obtained, which describes the behavior of the solution for asymptotically large values of independent variables.…”