Nonradial Quenching Profile for a MEMS Model
Hsuan-Lin Liao, Van Tien Nguyen
TL;DR
The paper constructs a finite-time quenching solution to the parabolic MEMS model $u_t=\Delta u-\dfrac{1}{u^2}$ on the unit disk with $u=1$ on the boundary, whose quenching occurs only at the origin and exhibits a genuinely nonradial final profile. The authors develop a self-similar renormalization, perform a spectral decomposition in the Hermite basis, and implement a bootstrap-energy framework to show the true solution stays near a carefully designed nonradial quenching profile $\mathcal{P}_{\theta}$ across inner, intermediate, and outer scales. The inner expansion yields a cross-shaped leading profile $\Psi_{\theta}$ corrected by $\mathcal{C}$, while outer matching yields a consistent global behavior; the final profile near the origin is $u^*(x)\sim\big(x_1^2x_2^2+\theta(x_1^6+x_2^6)\big)^{1/3}$ as $|x|\to0$, with $\theta\in(0,\theta^*)$. The analysis combines spectral gap estimates, energy methods, and a Brouwer fixed-point argument to establish the existence of such a nonradial quenching solution, marking the first proven nonradial quenching profile in MEMS models and contributing a robust methodological framework for nonradial singularity analysis in parabolic problems.
Abstract
We construct a quenching solution to the parabolic MEMS model \[ u_t = Δu - \frac{1}{u^2} \quad \text{in } \mathcal{B} \times (0,T), \quad u|_{\partial \mathcal{B}} = 1, \] where $\mathcal{B}$ is the unit disc in $\mathbb{R}^2$, and $T > 0$ denotes the quenching time. The constructed solution quenches only at the origin and admits the final profile \[ u(x,T) \sim \left(x_1^2 x_2^2 + θ(x_1^6 + x_2^6)\right)^{\frac{1}{3}} \quad \text{as } |x| \to 0, \] where $θ\in (0, θ^*)$ for some $θ^* > 0$. To our knowledge, this is the first example of a quenching solution with a genuinely non-radial profile. The proof relies on the construction of a good approximate solution, using a perturbative expansion in self-similar variables. We then justify the true solution that remains close to this approximation through a spectral analysis combined with a robust energy method.