Nonradial Quenching Profile for a MEMS Model

We construct a quenching solution to the parabolic MEMS model $$u_t=\mathrm{\Delta }u-\frac{1}{u^2}\text{in }\mathcal{B}\times (0,T),u{\mathrm{\mid }}_{\mathrm{\partial }\mathcal{B}}=1,$$ 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 $$u(x,T)\sim {(x_1^2{x}_2^2+\theta (x_1^6+x_2^6))}^{\frac{1}{3}}\text{as }\mathrm{\mid }x\mathrm{\mid }\to 0,$$ where $\theta \in (0,{\theta }^{\ast })$ for some ${\theta }^{\ast }>0$. 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.