A hyperbolic non-local problem modelling MEMS technology

N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos, D. E. Tzanetis

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31 Citations (Scopus)

Abstract

In this work we study a non-local hyperbolic equation of the form

u(tt) = u(xx) + lambda/(1 - u)(2) (1 + alpha integral(1)(0) (1/(1 - u)) dx)(2),

with homogeneous Dirichlet boundary conditions and appropriate initial conditions. The problem models an idealized electrostatically actuated MEMS (Micro-Electro-Mechanical System) device. Initially we present the derivation of the model. Then we prove local existence of solutions for lambda > 0 and global existence for 0 < lambda < lambda_* for some positive lambda_*, with zero initial conditions; similar results are obtained for other initial data. For larger values of the parameter lambda, i.e., when lambda > lambda(+)* for some constant lambda(+)* >= lambda_* and with zero initial conditions, it is proved that the solution of the problem quenches in finite time; again similar results are obtained for other initial data. Finally the problem is solved numerically with a finite difference scheme. Various simulations of the solution of the problem are presented, illustrating the relevant theoretical results.

Original languageEnglish
Pages (from-to)505-534
Number of pages30
JournalRocky Mountain Journal of Mathematics
Volume41
Issue number2
DOIs
Publication statusPublished - 2011

Keywords

  • Electrostatic MEMS
  • quenching of solution
  • hyperbolic non-local problems
  • ELECTROSTATIC MEMS
  • QUENCHING PROBLEM
  • EQUATIONS

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