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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 e∂tφ− e4φ + 1 e W,φ(φ) = 0 in ΩT , where φe = φ : ΩT := (0, T )×Ω −→ R2. We assume that the the potential W depends only on the modulus of φ and vanishes along two concentric circles. We present a priori estimates for the solution φ, and, in the spatially radially symmetric case, we show rigorously that in the singular limit as e → 0, two phases are created. The interface separating the bulk phases evolves by its mean curvature, while φ evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.