Existence and dynamic properties of a parabolic nonlocal MEMS equation

Let $Omegasubsetmathbb{R}^n$ be a $C^2$ bounded domain and $chi>0$ be a constant. We will prove the existence of constants $lambda_Ngelambda_N^{ast}gelambda^{ast}(1+chiint_{Omega}frac{dx}{1-w_{ast}})^2$ for the nonlocal MEMS equation $-Delta v=lam/(1-v)^2(1+chiint_{Omega}1/(1-v)dx)^2$ in $Omega$, $v=0$ on $1Omega$, such that a solution exists for any $0lelambda<lambda_N^{ast}$ 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 $1Omega$ as $lambda earrow lambda^{ast}$. 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.