Existence and dynamic properties of a parabolic nonlocal MEMS equation

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a $C^2$ bounded domain and $\chi >0$ be a constant. We will prove the existence of constants ${\lambda }_N\ge {\lambda }_N^{\ast }\ge {\lambda }^{\ast }(1+\chi {\int }_{\Omega }\frac{dx}{1-w_{\ast }}{)}^2$ for the nonlocal MEMS equation $-Δv=\lam/(1-v)^2(1+χ\int_Ω1/(1-v)dx)^2$ in $\Omega$, $v=0$ on $\1Ω$, such that a solution exists for any $0\le \lambda <{\lambda }_N^{\ast }$ and no solution exists for any $\lambda >{\lambda }_N$ where ${\lambda }^{\ast }$ is the pull-in voltage and $w_{\ast }$ is the limit of the minimal solution of $-Δv=\lam/(1-v)^2$ in $\Omega$ with $v=0$ on $\1Ω$ as $\lambda \nearrow {\lambda }^{\ast }$. We will prove the existence, uniqueness and asymptotic behaviour of the global solution of the corresponding parabolic nonlocal MEMS equation under various boundedness conditions on $\lambda$. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when $\lambda$ is large.

29 pages, some typo errors are corrected

References in corpus (1)

• Global and touchdown behaviour of the generalized MEMS device equation

Cited by in corpus (1)

• Quenching behaviour of a nonlocal parabolic MEMS equation