Allen–Cahn equation as a long-time modulation to a reaction-diffusion system

Abstract

We examine a two-component reaction-diffusion system on the real axis with quadratic nonlinearity. Using semigroup estimates, we obtain a solution to our nonlinear system for long-time. For appropriate initial data, we show that a slowly-varying, scaled solution of the Allen–Cahn equation will estimate the solution of our nonlinear system for long-time. We additionally extend this work to $\mathbb{R}^d$.