The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen–Cahn type equation of the form u t = Δ u + e −2 f e ( x , t , u ) , where e is a small parameter and f e ( x , t , u ) = f ( u ) − e g e ( x , t , u ) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u 0 that is independent of e , we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order e 2 | ln e | , and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e . This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where g e ≡ 0 . Next we consider systems of reaction–diffusion equations of the form { u t = Δ u + e −2 f e ( u , v ) , v t = D Δ v + h ( u , v ) , which include the FitzHugh–Nagumo system as a special case. Given a rather general initial data ( u 0 , v 0 ) , we show that the component u develops a steep transition layer and that all the above-mentioned results remain true for the u -component of these systems.