2014/02/01 by Xue Luo, Luo, Xue, Stephen S. -T. Yau +1
Mathematics · #35B40 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B40 #msc:35J60
paper · pdf · doi:10.48550/arxiv.1402.0066
27 pages, 3 figures, 3 tables; accepted by Quart. Appl. Math. arXiv admin note: text overlap with arXiv:0712.3071 by other authors
arxiv created 2014/02/01 · arxiv updated 2014/02/04
The singular parabolic problem ut-\triangle u=λ(1+δ|∇ u|2)/((1-u)2) on a bounded domain Ω of ℝn with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value λδ^*>0 such that if 0<λ<λδ^*, all solutions exist globally; while for λ>λδ^*, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage λ is investigated. Furthermore, the quenching set is a compact subset of Ω, provided Ω is a convex bounded domain in ℝn. In particular, if the domain Ω is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.