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- W2951144246 abstract "We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+ u(1-u^2)quad (x,t)in mathbb{R}times (-infty, 0].$$ The steady state $w(x) =tanh (x/sqrt{2})$, connects, as a layer the stable phases $-1$ and $+1$. We construct a solution $u$ with any given number $k$ of transition layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling copies of $w$ with interfaces diverging one to each other as $tto -infty$. More precisely, we find $$ u(x,t) approx sum_{j=1}^k (-1)^{j-1}w(x-xi_j(t)) + frac 12 ((-1)^{k-1}- 1)quad hbox{as} tto -infty, $$ where the functions $xi_j(t)$ satisfy a first order Toda-type system. They are given by $$xi_j(t)=frac{1}{sqrt{2}}left(j-frac{k+1}{2}right)log(-t)+gamma_{jk},quad j=1,...,k,$$ for certain explicit constants $gamma_{jk}.$" @default.
- W2951144246 created "2019-06-27" @default.
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- W2951144246 date "2017-03-26" @default.
- W2951144246 modified "2023-09-27" @default.
- W2951144246 title "Ancient multiple-layer solutions to the Allen-Cahn equation" @default.
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