边缘场作用下MEMS的猝灭行为
On the quenching behavior of the MEMS with fringing field
摘要：
在具有Dirichlet边界条件的$\mathbb{R}^n$的有界区域$Ω$上的奇异抛物问题$u_t-\triangle u=λ{\frac{1+δ|\nabla u|^2}{(1-u)^2}}$模拟了具有边缘场的微机电系统（MEMS）器件。在本文中，我们专注于该方程的解的猝灭行为。我们首先证明了存在一个临界值$λ_δ^*>0$，使得如果$0<λ<λ_δ^*$，所有的解决方案存在全局;而对于$λ>λ_δ^*$，所有的解决方案将在有限的时间淬火。本文研究了用大电压$λ$估算猝灭时间的方法。进一步地，如果$Ω$是$\mathbb{R}^n$中的凸有界区域，那么它的猝灭集就是$Ω$的紧子集.特别地，如果结构域$Ω$是径向对称的，则原点是唯一的淬灭点。不仅得到了猝灭速率的单侧估计，而且进一步研究了有限猝灭解的精细渐近行为。
英文摘要：
The singular parabolic problem $u_t-\triangle u=λ{\frac{1+δ|\nabla u|^2}{(1-u)^2}}$ on a bounded domain $Ω$ of $\mathbb{R}^n$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $λ_δ^*>0$ such that if $0<λ<λ_δ^*$, all solutions exist globally; while for $λ>λ_δ^*$, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage $λ$ is investigated. Furthermore, the quenching set is a compact subset of $Ω$, provided $Ω$ is a convex bounded domain in $\mathbb{R}^n$. In particular, if the domain $Ω$ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.
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