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

An asymptotic expansion of the solution to the Cauchy problem for a class of hyperbolic weakly nonlinear systems with many spatial variables is constructed. A parabolic quasilinear equation describing the behavior of the solution at asymptotically large values of the independent variables is obtained. The pseudo-diffusion processes that depend on the relationship between the number of equations and the number of spatial variables are analyzed. The structure of the subspace in which there are pseudo-diffusion evolution processes of the solution in the far field is described.