Nonradial Quenching Profile for a MEMS Model

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 $\hat{I}¸\in (0,\hat{I}{¸}^{\ast })$ for some $\hat{I}{¸}^{\ast }>0$. 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.