The existence and dynamic properties of a parabolic nonlocal MEMS equation

Let Ω⊂RnΩ⊂Rn be a C2C2 bounded domain and χ>0χ>0 be a constant. We will prove the existence of constants λN≥λN∗≥λ∗(1+χ∫Ωdx1−w∗)2 for the nonlocal MEMS equation −Δv=λ/(1−v)2(1+χ∫Ω1/(1−v)dx)2 in ΩΩ, v=0v=0 on ∂Ω∂Ω, such that a solution exists for any 0≤λ<λN∗ and no solution exists for any λ>λNλ>λN where λ∗λ∗ is the pull-in voltage and w∗w∗ is the limit of the minimal solution of −Δv=λ/(1−v)2−Δv=λ/(1−v)2 in ΩΩ with v=0v=0 on ∂Ω∂Ω as λ↗λ∗λ↗λ∗. Moreover λN<∞λN<∞ if ΩΩ is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the solution of the parabolic nonlocal MEMS equation ut=Δu+λ/(1−u)2(1+χ∫Ω1/(1−u)dx)2 in Ω×(0,∞)Ω×(0,∞), u=0u=0 on ∂Ω×(0,∞)∂Ω×(0,∞), u(x,0)=u0u(x,0)=u0 in ΩΩ. We prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λλ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λλ is large.