The singular limit of a vector-valued reaction-diffusion process
Lia Bronsard, Barbara E. E. Stoth · Transactions of the American Mathematical Society · 1998
We study the asymptotic behaviour of the solution to the vector–valued reaction–diffusion equation ε ∂ t φ − ε △ φ + 1 ε W ~ , φ ( φ ) = 0 in Ω T , \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 φ ε = φ : Ω T := ( 0 , T ) × Ω ⟶ R 2 \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