Solutions with multiple interfaces to the generalized parabolic Cahn–Hilliard equation in one and three space dimensions

Abstract

AbstractWe consider the generalized parabolic Cahn–Hilliard equation where , the function is the standard double‐well potential and the time region For and any given integer , we construct a solution with interfaces, which has the form where is the solution to the Allen–Cahn equation The dynamics of the interfaces are determined by a Toda system and the functions with have the forms For and any integer , we construct an ancient radial solution of the form where all satisfy another Toda system and have the forms In particular, these Toda systems are totally different from those driving the multiple interfaces of solutions to the parabolic Allen–Cahn equation established by M. del Pino and K. Gkikas [Proc. R. Soc. Edinb. Sect. A, 148 (2018), 6: 1165–1199; and Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), 1: 187–215].