  The goal in this paper is to demonstrate a new method for constructing
global-in-time approximate (asymptotic) solutions of (pseudodifferential)
parabolic equations with a small parameter. We show that, in the leading term,
such a solution can be constructed by using characteristics, more precisely, by
using solutions of the corresponding Hamiltonian system and without using any
integral representation. For completeness, we also briefly describe the
well-known scheme developed by V.P.Maslov for constructing global-in-time
solutions.
