The singular limit of a vector-valued reaction-diffusion process

We study the asymptotic behaviour of the solution to the vector–valued reaction–diffusion equation ε∂tφ−ε△φ+1εW~,φ(φ)=0inΩT,\begin{equation*}\varepsilon {\partial _{t}}\varphi -\varepsilon \triangle \varphi + {\frac {1}{\varepsilon }} \tilde W_{,\varphi } (\varphi ) = 0 \quad \text { in } \Omega _{T}, \end{equation*} where φε=φ:ΩT:=(0,T)×Ω⟶R2\varphi _{\varepsilon }=\varphi :\Omega _{T}:=(0,T)\times \Omega \longrightarrow \mathbf {R}^{2}. We assume that the the potential W~\tilde W depends only on the modulus of φ\varphi and vanishes along two concentric circles. We present a priori estimates for the solution φ\varphi, and, in the spatially radially symmetric case, we show rigorously that in the singular limit as ε→0\varepsilon \to 0, two phases are created. The interface separating the bulk phases evolves by its mean curvature, while φ\varphi evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.