Multiple-interface solutions of one dimensional generalized parabolic Cahn-Hilliard equation

Abstract

We consider one dimensional generalized parabolic Cahn-Hilliard equation $$u_t=-{\partial }_{xx}[{\partial }_{xx}u-W^{\prime }(u)]+W^{\prime \prime }(u)[{\partial }_{xx}u-W^{\prime }(u)],\forall (t,x)\in [0,+\infty )\times \mathbb{R},$$ where the function $W(\cdot )$ is the standard double-well potential. For any given positive integer $k\ge 2$, we construct a solution $u(t,x)$ with $k$ interfaces, which has the form $$u(t,x)\approx \sum _{j=1}^k(-1{)}^{j+1}\ddot{I}‰(x-{\hat{I}}_j^3(t))-\frac{1+(-1{)}^k}{2}\text{as}t\to +\infty ,$$ where $\ddot{I}‰$ is the solution to the Allen-Cahn equation $$ ω''-W'(ω)=0,\quadω'>0\quad\mbox{in }{\mathbb R}, \quad ω(0)=0, \quad ω(\pm\infty)=\pm 1. $$ The interfaces are described by the functions ${\hat{I}}_j^3(t)$ with $j=1,\cdots ,k$, which are determined by a Toda system and have the forms $${\hat{I}}_j^3(t)=\frac{1}{2\sqrt{2}}\left(j-\frac{k+1}{2}\right)\ln t+O(1).$$ The Toda system is different from the one that determine the dynamics of the multiple interfaces of solutions to one dimensional parabolic Allen-Cahn equation established by M. del Pino and K. Gkikas in {\em Proc. R. Soc. Edinb. Sect. A}, 148 (2018), 6: 1165-1199.