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Higher-Order One-Point Touchdown in a Nonlocal MEMS Model
Abstract
Micro-electro-mechanical systems, commonly called MEMS, can be modeled by singular parabolic equations describing an elastic membrane attracted toward a fixed conducting plate. Touchdown occurs when the gap between the membrane and the plate vanishes in finite time.
In this talk, we consider a radially symmetric nonlocal MEMS model. In this model, the electrostatic force depends not only on the gap at each point, but also on a singular integral over the whole device. This global interaction changes both the touchdown rate and the relevant spatial scale near the touchdown point.
We construct smooth initial data for which the solution touches down at exactly one point and has a prescribed higher-order contact profile. In particular, the resulting contact can be much flatter than the usual quadratic profile. We also determine the asymptotic behavior of the solution near the touchdown point and describe how the nonlocal term influences the leading-order profile.
The proof combines adapted similarity variables, a finite-dimensional reduction that incorporates the nonlocal feedback, estimates for the infinite-dimensional remainder, outer barrier arguments, and a topological selection method. This is joint work with Saoussen Latrach, Tetsuji Tokihiro, and Hatem Zaag.
