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 $-\Delta v=\lam/(1-v)^2(1+\chi\int_{\Omega}1/(1-v)dx)^2$ in $\Omega$, $v=0$ on $\1\Omega$, 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 $-\Delta v=\lam/(1-v)^2$ in $\Omega$ with $v=0$ on $\1\Omega$ 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.