Global solutions of singular parabolic equations arising from electrostatic MEMS

We study dynamic solutions of the singular parabolic problemequation(P){ut−Δu=λ∗|x|α(1−u)2,(x,t)∈B×(0,∞),u(x,0)=u0(x)⩾0,x∈B,u(x,t)=0,x∈∂B, where α⩾0α⩾0 and λ∗>0λ∗>0 are two parameters, and B   is the unit ball {x∈RN:|x|⩽1} with N⩾2N⩾2. Our interest is focussed on (P) with λ∗:=(2+α)(3N+α−4)9, for which (P) admits a singular stationary solution in the form S(x)=1−|x|2+α3. This equation models dynamic deflection of a simple electrostatic Micro-Electro-Mechanical-System (MEMS) device. Under the assumption u0≨︀S(x)u0≨︀S(x), we address the existence, uniqueness, regularity, stability or instability, and asymptotic behavior of weak solutions for (P). Given α∗∗:=4−6N+36(N−2)4, in particular we show that if either N⩾8N⩾8 and α>α∗∗α>α∗∗ or 2⩽N⩽72⩽N⩽7, then the minimal compact stationary solution uλ∗uλ∗ of (P) is stable and while S(x)S(x) is unstable. However, for N⩾8N⩾8 and 0⩽α⩽α∗∗0⩽α⩽α∗∗, (P) has no compact minimal solution, and S(x)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⩾8N⩾8 and 0⩽α⩽α∗∗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.