We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen–Cahn and the Cahn–Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across a trigger line. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses with small speed and increases this bistable region. We find existence and nonexistence results of single interfaces and striped patterns. For zero speed, we find stripes parallel or perpendicular to the trigger line and exclude stripes with an oblique orientation. Single interfaces are always perpendicular to the trigger line. For small positive speed, striped patterns can align perpendicularly. Other orientations are excluded in Allen–Cahn for all nonnegative speeds. Single interfaces for positive trigger speeds are excluded for Cahn–Hilliard and align perpendicularly in Allen–Cahn.
Bibliographical noteFunding Information:
R.M and A.S. are grateful to the University of M?nster, Germany, where part of this work is carried out. R.M. also would like to thank Itsv?n Lagzi and Zolt?n R?cz from ?otvos University, Hungary, for stimulating discussions. R.M. acknowledges financial support through a DAAD Research Grant. A.S. acknowledges partial support through NSF Grants DMS-1612441 and DMS-1311740.
© 2017, Springer Science+Business Media New York.
Copyright 2017 Elsevier B.V., All rights reserved.
- Directional quenching
- Phase separation