Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
837007 | Nonlinear Analysis: Real World Applications | 2016 | 17 Pages |
Abstract
Consider the following problem arising in Micro-Electro-Mechanical Systems (MEMS) {ut−Δu=λ(1+δ∣∇u∣2)(1−u)p,(x,t)∈Ω×(0,T),u=0,(x,t)∈∂Ω×(0,T),u(x,0)=u0(x),0⩽u0(x)<1,x∈Ω, where δ>0δ>0, p>1p>1 and ΩΩ is a bounded smooth domain in RN(N⩾1)RN(N⩾1). We prove that infinite time quenching is impossible for any λ>0λ>0 in this problem. It provides a remarkable contrast to the case of δ=0δ=0, in which infinite time quenching must happen for some λλ when ΩΩ is a ball in RN(N⩾8)RN(N⩾8). This means that the presence of the fringing field δ|∇u|2δ|∇u|2 dramatically changes the quenching behavior of the solution. We also obtain some new results about global convergence and quenching in finite time.
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Authors
Hongjing Pan, Ruixiang Xing,