2008/09/24 by Kin Ming Hui, Hui, Kin Ming · 1 citation
Computer Science · Mathematics · #35B40 (Primary)35B05 #35K20 (Secondary) #35K50 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35B40 #msc:35K20 #msc:35K50
paper · pdf · doi:10.48550/arxiv.0809.4209
29 pages, some typo errors are corrected
openalex publication_date 2008/09/24 · arxiv created 2010/08/17 · arxiv updated 2010/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω⊂ℝn be a C2 bounded domain and χ>0 be a constant. We will prove the existence of constants λN≥λN∗≥λ∗(1+χ∫Ω\fracdx1-w∗)2 for the nonlocal MEMS equation -Δv=\lam/(1-v)2(1+χ∫Ω1/(1-v)dx)2 in Ω, v=0 on \1Ω, such that a solution exists for any 0≤λ<λN∗ and no solution exists for any λ>λN where λ∗ is the pull-in voltage and w∗ is the limit of the minimal solution of -Δv=\lam/(1-v)2 in Ω with v=0 on \1Ω as λ\nearrow λ∗. 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.