Fringing field can prevent infinite time quenching

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.