Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

Giorgio Fusco

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arXiv · 2010.05628

Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

Giorgio Fusco

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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 \[\ddot{u}=W_u(u),\;\;t\in\R,\] 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 \[u_t=u_{xx}-W_u(u),\;\;x\in(0,1),\;t>0,\] with periodic boundary conditions. In the second paper 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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Giorgio Fusco. 2020-10-12. Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation. https://arxiv.org/abs/2010.05628

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