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Periodic Motions for Multi-wells Potentials and Layers Dynamic for the Vector Allen–Cahn Equation

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Abstract

We consider a nonnegative potential W that vanishes on a finite set and study the existence of periodic orbits of the equation

$$\begin{aligned} \ddot{u}=W_u(u),\;\;t\in {\mathbb {R}}, \end{aligned}$$

that have the property of visiting neighborhoods of zeros of W in a given finite sequence. We give conditions for the existence of such orbits. After introducing the new variable \(x=\epsilon t\), \(\epsilon >0\) small, these orbits correspond to stationary solutions of the parabolic equation

$$\begin{aligned} u_t=u_{xx}-W_u(u),\;\;x\in (0,1),\;t>0, \end{aligned}$$

with periodic boundary conditions. In the second part of the paper we study solutions of this equation that, as the stationary solutions, have a layered structure. We derive a system of ODE that describes the dynamics of the layers and show that their motion is extremely slow.

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  1. Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio, 67010, Coppito, L’Aquila, Italy

    Giorgio Fusco

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Fusco, G. Periodic Motions for Multi-wells Potentials and Layers Dynamic for the Vector Allen–Cahn Equation. J Dyn Diff Equat 34, 3165–3215 (2022). https://doi.org/10.1007/s10884-021-09949-5

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  • DOI: https://doi.org/10.1007/s10884-021-09949-5

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