Existence and dynamic properties of a parabolic nonlocal MEMS equation

Let $Ω\subset\mathbb{R}^n$ be a $C^2$ bounded domain and $χ>0$ be a constant. We will prove the existence of constants $λ_N\geλ_N^{\ast}\geλ^{\ast}(1+χ\int_Ω\frac{dx}{1-w_{\ast}})^2$ for the nonlocal MEMS equation $-Δv=\lam/(1-v)^2(1+χ\int_Ω1/(1-v)dx)^2$ in $Ω$, $v=0$ on $\1Ω$, such that a solution exists for any $0\leλ<λ_N^{\ast}$ and no solution exists for any $λ>λ_N$ where $λ^{\ast}$ is the pull-in voltage and $w_{\ast}$ is the limit of the minimal solution of $-Δv=\lam/(1-v)^2$ in $Ω$ with $v=0$ on $\1Ω$ as $λ\nearrow λ^{\ast}$. 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 $λ$. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when $λ$ is large.

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