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

Abstract

We consider an Allen–Cahn type equation of the form u t = Δ u + ε −2 f ε ( x , t , u ) , where ε is a small parameter and f ε ( x , t , u ) = f ( u ) − ε g ε ( 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 ε, 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 ε 2 | ln ε | , and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where g ε ≡ 0 . Next we consider systems of reaction–diffusion equations of the form { u t = Δ u + ε −2 f ε ( 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.