Horizontal patterns from finite speed directional quenching

In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface; in the case presented here the interface moves with speed $c$ in such a way that the bistable region grows, which is the most physically relevant scenario. According to previous results in the literature (Phase separation patterns from directional quenching, Journal of Nonlinear Science, 2017, by Monteiro, R. and Scheel, A.), several patterns exist when $c \gtrsim0$, and here we investigate their persistence for finite $c>0$, clarifying the pattern selection mechanism related to the speed $c$ of the quenching front.