MULTIPLE QUENCHING SOLUTIONS OF A FOURTH ORDER PARABOLIC PDE WITH A SINGULAR NONLINEARITY MODELING A MEMS CAPACITOR

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The soluti...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 2012-01, Vol.72 (3), p.935-958
Main Authors: LINDSAY, A. E., LEGA, J.
Format: Article
Language:English
Subjects:
Applied mathematics
Boundary conditions
Capacitors
Eigenvalues
Electric potential
Equilibrium
Far fields
Geometry
Mathematical analysis
Mathematical models
Mathematics
Microelectromechanical systems
Nonlinearity
Partial differential equations
Quenching
Self-similarity
Singularities
Symmetry
Touchdown
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Summary:Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.
ISSN:0036-1399
1095-712X
DOI:10.1137/110832550