Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Maissâ Boughrara

TL;DR

The work addresses finite-time quenching, i.e., touch-down, in a nonlocal MEMS model governed by $\partial_t v = \Delta v + \frac{\lambda}{(1-v)^2\bigl(1 + \gamma\int_\Omega \frac{1}{1-v}\,dx\bigr)^2}$ with $0\le v\le 1$. By reformulating the problem as a blow-up problem and employing a blow-up–to–parabolic framework, the authors execute a two-tier strategy: a formal regionwise (P1, P2, P3) outer-inner expansion to deduce the quenching profile and a rigorous shrinking-set method to control the dynamics. They achieve a finite-time quenching solution that collapses at a single interior point with explicit intermediate and final profiles, together with a precise asymptotic description of the nonlocal parameter $\theta(t)$ that drives the nonlocal feedback. The analysis combines finite-dimensional reduction of unstable modes with a topological degree argument to guarantee the existence of initial data yielding the desired quenching behavior, even in the critical nonlocal regime. Overall, the paper advances the understanding of critical nonlocal MEMS models by providing a rigorous construction of a single-point quenching solution and detailed asymptotics for the quenching and the nonlocal term.

Abstract

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

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

TL;DR

The work addresses finite-time quenching, i.e., touch-down, in a nonlocal MEMS model governed by with . By reformulating the problem as a blow-up problem and employing a blow-up–to–parabolic framework, the authors execute a two-tier strategy: a formal regionwise (P1, P2, P3) outer-inner expansion to deduce the quenching profile and a rigorous shrinking-set method to control the dynamics. They achieve a finite-time quenching solution that collapses at a single interior point with explicit intermediate and final profiles, together with a precise asymptotic description of the nonlocal parameter that drives the nonlocal feedback. The analysis combines finite-dimensional reduction of unstable modes with a topological degree argument to guarantee the existence of initial data yielding the desired quenching behavior, even in the critical nonlocal regime. Overall, the paper advances the understanding of critical nonlocal MEMS models by providing a rigorous construction of a single-point quenching solution and detailed asymptotics for the quenching and the nonlocal term.

Abstract

This work investigates a mathematical model arising in the study of MEMS devices, described by the following parabolic equation on : where is a bounded domain and . We construct a solution with a prescribed profile, which quenches in finite time at exactly one interior point . 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.

Paper Structure

This paper contains 28 sections, 28 theorems, 361 equations.

Table of Contents

  1. Introduction
  2. Statement of the result
  3. Setting of the problem and strategy
  4. Formal approach
  5. In region $P_1$
  6. In region $P_2$
  7. In Region $P_3$
  8. On the $\theta$'s dynamic
  9. Formulation of the problem
  10. Similarity variables
  11. Localization variable
  12. Shrinking set $S(t)$ to the blow-up profile
  13. The existence proof without technical details
  14. Preparation of the initial data
  15. Dynamics of $\theta$
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\Omega$ be a smooth and bounded domain in $\mathbb{R}^N$ containing the origin and consider equation eq in the critical regime critical condition. Then, there exists $\varepsilon_0$ small such that for all $rq\gamma\in [0,\varepsilon_0]$, there exists an initial data $v_0 \geq 0$ such that the

Theorems & Definitions (61)

  • Theorem 1
  • Remark
  • Remark
  • Definition 3.1: A shrinking set $S(t)$ to the blow-up profile
  • Lemma 3.2: Growth estimates
  • proof
  • Lemma 4.1: existence of $\theta_0$
  • proof
  • Proposition 4.2: Constructing initial data
  • proof
  • ...and 51 more