Front Motion in the Parabolic Reaction-Diffusion Problem

A singularly perturbed initial-boundary value problem is considered for a parabolic equation known in applications as the reaction-diffusion equation. An asymptotic expansion of solutions with a moving front is constructed, and an existence theorem for such solutions is proved. The asymptotic expansion is substantiated using the asymptotic method of differential inequalities, which is extended to the class of problems under study. The method is based on well-known comparison theorems and is a development of the idea of using formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.