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 $\epsilon {\mathrm{\partial }}_t\phi -\epsilon \mathrm{△}\phi +\frac{1}{\epsilon }{\tilde{W}}_{,\phi }(\phi )=0\text{ in }{\mathrm{\Omega }}_T,$ where ${\phi }_{\epsilon }=\phi :{\mathrm{\Omega }}_T:=(0,T)\times \mathrm{\Omega }⟶{\mathbf{R}}^2$ . We assume that the the potential $\tilde{W}$ depends only on the modulus of $\phi$ and vanishes along two concentric circles. We present a priori estimates for the solution $\phi$ , and, in the spatially radially symmetric case, we show rigorously that in the singular limit as $\epsilon \to 0$ , two phases are created. The interface separating the bulk phases evolves by its mean curvature, while $\phi$ evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.