Asymptotics of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation in several space variables

A formal asymptotic expansion of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation in several space variables with weak nonlinearity is constructed. Under certain conditions imposed on the data of the problem, the asymptotic expansion is constructed in the form of series in powers of a small parameter with coefficients depending on various stretched variables. Problems for determining all terms of the asymptotic expansion are obtained. Specifically, the leading term is determined as the solution of the Cauchy problem for a parabolic equation. The remainder is estimated with respect to the residual.