2023/03/30 by Liu, Chao, yang, Jun
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2303.17288
We consider one dimensional generalized parabolic Cahn-Hilliard equation ut=-∂xx[∂xxu-W'(u)]+W''(u)[∂xx u -W'(u)], ∀ (t,x)∈ [0,+∞)× \mathbb R, where the function W(⋅) is the standard double-well potential. For any given positive integer k≥2, we construct a solution u(t,x) with k interfaces, which has the form u(t,x)≈∑j=1k(-1)j+1ω(x-γj(t))-(1+(-1)k)/(2) as t→ +∞, where ω is the solution to the Allen-Cahn equation ω''-W'(ω)=0, ω'gt;0\quadin \mathbb R, ω(0)=0, ω(±∞)=± 1. The interfaces are described by the functions γj(t) with j=1,⋯,k, which are determined by a Toda system and have the forms γj(t)=(1)/(2√(2))(j-(k+1)/(2))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.