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

TL;DR

This paper constructs and analyzes a finite-time quenching solution for a nonlocal MEMS model, providing an asymptotic profile and addressing the challenges posed by the nonlocal integral term.

Contribution

It introduces a method to construct a quenching solution with a prescribed profile for a nonlocal MEMS equation, including an asymptotic description and handling the nonlocal term.

Findings

Constructed a solution that quenches at a single interior point.

Provided an asymptotic description of the quenching profile.

Addressed the mathematical challenges of the nonlocal integral term.

Abstract

This work investigates a mathematical model arising in the study of MEMS devices, described by the following parabolic equation on $[0,T)\times\Omega$: $$\partial_t v = \Delta v + \frac{\lambda}{(1-v)^2\left( 1 + \gamma \int_{\Omega} \frac{1}{1-v}\, dx \right)^{2}} , \qquad 0 \leq v \leq 1,$$ where $\Omega \subset \mathbb{R}^N$ is a bounded domain and $\lambda, \gamma > 0$. We construct a solution with a prescribed profile, which quenches in finite time $T$ at exactly one interior point $a \in \Omega$. 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.