Parabolic singular perturbations: formal matched asymptotics

Abstract

For each ∈ ∈ (0, 1] we denote by u∈ the solution to the singularly perturbed parabolic problem (15.2), which for convenience of the reader we rewrite in the following form:

$$\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial u}}{{\partial t}} - \Delta u + { \in ^{ - 2}}W'\left( u \right) = 0}&{in\;\left( {0,T} \right) \times \Omega ,} \\ {\frac{{\partial u}}{{\partial {n_\Omega }}} = 0}&{on\;\left( {0,T} \right) \times \partial \Omega ,} \\ {u = u_ \in ^0}&{in\;\left\{ {t = 0} \right\} \times \Omega .} \end{array}} \right.$$

(16.1)

In this chapter we perform two asymptotic expansions of u∈, which will be suitably matched one each other. In spite of the fact that the argument is formal, it eventually leads to a rigorous proof of convergence of {u∈(t, …) = 0} to a mean curvature flow as ∈ ↓ 0, valid for short times (see Chapter 17). Asymptotic expansions for reaction-diffusion equations of the type in (16.1) have been performed, among other places, in [145, 118, 120] (see also [59, 226] and [81, 6]). Here we will closely follow the arguments of [229].