Fringing field can prevent infinite time quenching

Abstract Consider the following problem arising in Micro-Electro-Mechanical Systems (MEMS) { u t − Δ u = λ ( 1 + δ ∣ ∇ u ∣ 2 ) ( 1 − u ) p , ( x , t ) ∈ Ω × ( 0 , T ) , u = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , 0 ⩽ u 0 ( x ) 1 , x ∈ Ω , where δ > 0 , p > 1 and Ω is a bounded smooth domain in R N ( N ⩾ 1 ) . We prove that infinite time quenching is impossible for any λ > 0 in this problem. It provides a remarkable contrast to the case of δ = 0 , in which infinite time quenching must happen for some λ when Ω is a ball in R N ( N ⩾ 8 ) . This means that the presence of the fringing field δ | ∇ u | 2 dramatically changes the quenching behavior of the solution. We also obtain some new results about global convergence and quenching in finite time.