Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation

We consider singularly perturbed operator differential transport equations of a special form in the case where the transport operator acts on space–time variables; a linear operator acting on an additional variable describes the interaction that “scrambles” the solution with respect to that variable. We construct a formal asymptotic expansion of the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearity and weak diffusion in the case of several spatial variables. Under some conditions assumed for these problems, the leading term of the asymptotics is described by a quasilinear parabolic equation. The remainder term is estimated with respect to the residual under certain conditions.