具有临界参数的非局部MEMS模型触地解的性质
Profile of a Touch-Down solution to a nonlocal MEMS model with critical parameters
AI总结：
本文研究具有临界参数的非局部MEMS抛物方程，构造了在有限时间于单个内点淬灭的解，并通过约化为有限维动力系统和拓扑指标论证给出淬灭形态的渐近描述。
摘要：
本文研究MEMS器件研究中出现的一个数学模型，该模型由$[0,T)\timesΩ$上的如下抛物方程描述：$$\partial_t v = Δv + \fracλ{(1-v)^2\left( 1 + γ\int_Ω \frac{1}{1-v}\\, dx \right)^{2}} ,\qquad 0 \leq v \leq 1,$$ 其中$Ω\subset \mathbb{R}^N$是有界区域，$λ, γ> 0$。我们构造一个具有指定形态的解，该解在有限时间$T$于恰好一个内点$a \in Ω$处发生淬灭。此外，我们能够给出淬灭形态的渐近描述。我们将该问题重新表述为一个爆破问题，以利用Merle、Zaag在1997年，Duong、Zaag在2019年以及Duong、Ghoul、Kavallaris、Zaag在2022年所采用的技术。证明通过两个主要步骤进行：约化为有限维动力系统，以及运用指标理论的经典拓扑论证。主要挑战在于处理非局部积分项，当问题变换到爆破框架时，该项会产生一个额外的梯度项。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
