Profile of a Touch-Down solution to a nonlocal MEMS model with critical parameters

Abstract

This work investigates a mathematical model arising in the study of MEMS devices, described by the following parabolic equation on $[0,T)\times \hat{I}\text{c◯}$: $${\partial }_{t}v=\hat{I}”v+\frac{\hat{I}}{»}(1-v{)}^2{\left(1+{\hat{I}}^3{\int }_{\hat{I}}\text{c◯}\frac{1}{1-v}dx\right)}^2,0\le v\le 1,$$ where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is a bounded domain and $\hat{I}»,{\hat{I}}^3>0$. We construct a solution with a prescribed profile, which quenches in finite time $T$ at exactly one interior point $a\in \hat{I}\text{c◯}$. Moreover, we are able to provide an asymptotic description of the quenching profile. We reformulate the problem as a blow-up problem to utilize the techniques employed in Merle, Zaag in 1997, Duong, Zaag in 2019 and Duong, Ghoul, Kavallaris, Zaag 2022. The proof proceeds through two principal steps: a reduction to a finite-dimensional dynamical system and a classical topological argument employing index theory. The main challenge lies in managing the nonlocal integral term, which generates an additional gradient term when the problem is transformed into the blow-up framework.

61 pages