2009/08/09 by Kin Ming Hui, Hui, Kin Ming
Engineering · Mathematics · #35B05 (Secondary) #35B40 (Primary) #35K20 #35K50 #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems #math.AP #msc:35B05 #msc:35B40 #msc:35K20 #msc:35K50
paper · pdf · doi:10.48550/arxiv.0908.1227
13 pages
openalex publication_date 2009/08/09 · arxiv created 2010/03/16 · arxiv updated 2010/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation ut=Δu+\lam/(1-u)2(1+χ∫Ω1/(1-u) dx)2 in Ω× (0,∞), u=0 on \1Ω× (0,∞), u(x,0)=u0 in Ω, when λ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When Ω=BR and u0 is radially symmetric and monotone decreasing in 0≤ r≤ R, we prove that the point x=0 is the only possible quenching set. When u0 also satisfies some strict concavity assumption, we prove that for any β∈ (2,3) the solution satisfies 1-u(x,t)≥ C|x|^\frac2β for some constant C>0 and we also obtain the quenching time estimate in this case.