2008/08/01 by Kin Ming Hui, Hui, Kin Ming · 2 citations
Engineering · Mathematics · Physics and Astronomy · #35B05 #35B40 #35K20 #35K50 #Advanced MEMS and NEMS Technologies #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Mechanical and Optical Resonators #math.AP #msc:35B05 #msc:35B40 #msc:35K20 #msc:35K50
paper · pdf · doi:10.48550/arxiv.0808.0110
25 pages
arxiv created 2008/08/01 · openalex publication_date 2008/08/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the local and global existence of solutions of the generalized micro-electromechanical system (MEMS) equation ut =Δu+λf(x)/g(u), u<1, in Ω× (0,∞), u(x,t)=0 on ∂Ω× (0,∞), u(x,0)=u0 in Ω, where Ω⊂ℝn is a bounded domain, λ>0 is a constant, 0≤ f∈ Cα(Ω), f\not≡ 0, for some constant 0<α<1, 0<g∈ C2((-∞,1)) such that g'(s)≤ 0 for any s<1 and u0∈ L1(Ω) with u0≤ a<1 for some constant a. We prove that there exists a constant λ∗=λ∗(Ω, f,g)>0 such that the associated stationary problem has a solution for any 0≤λ<λ^* and has no solution for any λ>λ^*. We obtain comparison theorems for the generalized MEMS equation. Under a mild assumption on the initial value we prove the convergence of global solutions to the solution of the corresponding stationary elliptic equation as t→∞ for any 0≤λ<λ^*. We also obtain various conditions for the existence of a touchdown time T>0 for the solution u. That is a time T>0 such that limt\nearrow TsupΩu(⋅,t)=1.