[Paper Review] Quenching behaviour of a nonlocal parabolic MEMS equation

Kin Ming Hui|arXiv (Cornell University)|Aug 9, 2009

Numerical methods in inverse problems20 references3 citations

TL;DR

This paper analyzes the quenching behavior of a nonlocal parabolic MEMS equation with a singular nonlinearity driven by a global capacitance effect. Using maximum principle techniques and radial symmetry assumptions, it establishes finite-time quenching for large λ, proves the quenching set is compact and reduces to the origin under radial symmetry, and derives sharp upper bounds for the quenching time, including estimates of the form $1 - u(x,t) \geq C|x|^{2/\beta}$ for $\beta \in (2,3)$.

ABSTRACT

We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=Δu+\lam/(1-u)^2(1+χ\int_Ω1/(1-u) dx)^2$ in $Ω imes (0,\infty)$, $u=0$ on $\1Ω imes (0,\infty)$, $u(x,0)=u_0$ in $Ω$, when $λ$ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When $Ω=B_R$ and $u_0$ is radially symmetric and monotone decreasing in $0\le r\le R$, we prove that the point $x=0$ is the only possible quenching set. When $u_0$ also satisfies some strict concavity assumption, we prove that for any $β\in (2,3)$ the solution satisfies $1-u(x,t)\ge C|x|^{\frac{2}β}$ for some constant $C>0$ and we also obtain the quenching time estimate in this case.

Motivation & Objective

• To understand the quenching behavior of a nonlocal parabolic MEMS equation with global capacitance effects.
• To establish upper bounds for the quenching time when the applied voltage parameter λ is large.
• To prove the compactness of the quenching set under mild initial data conditions.
• To identify the quenching set as the origin when the domain is a ball and the initial data are radially symmetric and monotone decreasing.
• To derive quenching time estimates and pointwise lower bounds on $1 - u(x,t)$ under strict concavity assumptions on the initial data.

Proposed method

• Uses the maximum principle and comparison arguments to analyze the radial solution structure in a ball domain.
• Applies a transformation $v = (1 - u)^{-1}$ to convert the singular equation into a more tractable form.
• Employs a subsolution of the form $w(r,t) = \varepsilon r^2 + w(0,t)$ to control the growth of $1 - u$ near the origin.
• Introduces a function $q = r^{n-1} w_r$ and derives a parabolic PDE for $q$ to analyze the radial derivative behavior.
• Imposes a strict concavity assumption on the initial data to ensure $v(r,0) \geq c_1 r^2 + c_2$ for $r \in [0,R]$, enabling the construction of a lower barrier.
• Uses the maximum principle on the perturbed function $\tilde{q} = q - \varepsilon r^n$ to prove $w_r \geq \varepsilon r$, leading to $1 - u \geq C|x|^{2/\beta}$ for $\beta \in (2,3)$.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of the nonlocal parabolic MEMS equation quench in finite time for large λ?
• RQ2 Where is the quenching set located when the domain is a ball and the initial data are radially symmetric and monotone decreasing?
• RQ3 Can sharp lower bounds of the form $1 - u(x,t) \geq C|x|^{2/\beta}$ be established for $\beta \in (2,3)$ under strict concavity of the initial data?
• RQ4 What upper bounds can be derived for the quenching time in the radial symmetric case with large λ?
• RQ5 Is the quenching set compact under mild regularity and boundedness assumptions on the initial data?

Key findings

• For any $\lambda > \lambda_1$, the solution quenches in finite time $T < 1$, with $T$ bounded above by a constant depending on $\delta_1$, $\lambda$, and $R$.
• When $\Omega = B_R$ and $u_0$ is radially symmetric and monotone decreasing, the quenching set is compact and $x = 0$ is the only possible quenching point.
• For $\beta \in (2,3)$, the solution satisfies $1 - u(x,t) \geq C|x|^{2/\beta}$ in $Q_R^T$ for some $C > 0$, under a strict concavity assumption on $u_0$.
• The quenching time $T$ satisfies $T \leq \frac{2}{\lambda \delta_1}$, where $\delta_1$ is a positive constant depending on $\chi$, $R$, $n$, and $\beta$, with $\delta_1 > [1 + \chi |\omega_{n-1}| (2/\varepsilon)^{1/\beta} (n - 2/\beta)^{-1} R^{n - 2/\beta}]^{-2}$.
• The quenching set is compact under a mild condition on $u_0$, specifically $u_0 \leq a < 1$ a.e. in $\Omega$, and $u_0 \in L^1(\Omega)$.
• The result holds uniformly for $\lambda > \lambda_1$, with all constants independent of the quenching time $T$, ensuring the estimate is robust for large $\lambda$.