论文标题
多孔电势和层动态的周期性运动
Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation
论文作者
论文摘要
我们考虑了一种非负势$ W $在有限套件上消失的，并研究了方程式的周期性轨道\ [\ ddot {\ ddot {u} = w_u（u），\; \; \; t \ in \ in \ in \ in \ in \ r，\]，这些物品具有访问$ W $ $ w $ $ w $ sequence sepecente sequence sepece sementite sequence sepece semente sequence sepenite sequence的财产。我们提供了存在这种轨道的条件。引入了新变量$ x =εt$，$ε> 0 $小，这些轨道对应于抛物线方程的固定解决方案\ [u_t = u_t = u_ {xx} -w_u（u），\; \; \; x \; x \ in（0,1），\; t> 0，\; t> 0，\]与周期性边界条件。 在本文的第二篇论文中，我们研究了该方程的解决方案，作为固定溶液具有分层结构。我们得出了一个描述层动力学的ODE系统，并表明它们的运动非常慢。
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 $x=εt$, $ε>0$ 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.
