2025/11/10 by Boughrara, Maissâ
Computer Science · Materials Science · Mathematics · #35B40 #35K55 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Nonlocal and gradient elasticity in micro/nano structures
paper · doi:10.48550/arxiv.2511.06910
openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
This work investigates a mathematical model arising in the study of MEMS devices, described by the following parabolic equation on [0,T)×Ω: ∂t v = Δv + \fracλ(1-v)2( 1 + γ∫Ω (1)/(1-v) dx )2 , 0 ≤ v ≤ 1, where Ω⊂ ℝN 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 ∈ 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.