Global solutions of singular parabolic equations arising from electrostatic MEMS

Abstract

We study dynamic solutions of the singular parabolic problem ( P) { u t − Δ u = λ ∗ | x | α ( 1 − u ) 2 , ( x , t ) ∈ B × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) ⩾ 0 , x ∈ B , u ( x , t ) = 0 , x ∈ ∂ B , where α ⩾ 0 and λ ∗ > 0 are two parameters, and B is the unit ball { x ∈ R N : | x | ⩽ 1 } with N ⩾ 2 . Our interest is focussed on ( P) with λ ∗ : = ( 2 + α ) ( 3 N + α − 4 ) 9 , for which ( P) admits a singular stationary solution in the form S ( x ) = 1 − | x | 2 + α 3 . This equation models dynamic deflection of a simple electrostatic Micro-Electro-Mechanical-System (MEMS) device. Under the assumption u 0 ≨︀ S ( x ) , we address the existence, uniqueness, regularity, stability or instability, and asymptotic behavior of weak solutions for ( P) . Given α ∗ ∗ : = 4 − 6 N + 3 6 ( N − 2 ) 4 , in particular we show that if either N ⩾ 8 and α > α ∗ ∗ or 2 ⩽ N ⩽ 7 , then the minimal compact stationary solution u λ ∗ of ( P) is stable and while S ( x ) is unstable. However, for N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , ( P) has no compact minimal solution, and S ( x ) is an attractor from below not from above. Furthermore, the refined asymptotic behavior of global solutions for ( P) is also discussed for the case where N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity.