Existence and dynamic properties of a parabolic nonlocal MEMS equation

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a $C^2$ bounded domain and $\chi >0$ be a constant. We will prove the existence of constants ${\lambda }_N\ge {\lambda }_N^{\ast }\ge {\lambda }^{\ast }(1+\chi {\int }_{\mathrm{\Omega }}\frac{dx}{1-w_{\ast }}{)}^2$ 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$\1\Omega$,suchthatasolutionexistsforany$0\le\lambda<\lambda_N^{\ast}$andnosolutionexistsforany$\lambda>\lambda_N$where$\lambda^{\ast}$ is the pull-in voltage and $w_{\ast }$ is the limit of the minimal solution of $-\mathrm{\Delta }v=\text{lam}\mathrm{/}(1-v{)}^2$ in $\mathrm{\Omega }$ with $v=0$ on $\text{1}\mathrm{\Omega }$ 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.