On the Quenching Behaviour of a Semilinear Wave Equation Modelling MEMS Technology

Andrew Alfred Lacey, Nikos Kavallaris, Christos Nikolopoulos, Dimitrios Tzanetis

Research output: Contribution to journalArticle

4 Citations (Scopus)
64 Downloads (Pure)

Abstract

In this work we study the semilinear wave equation of the form
utt = uxx + lambda/(1 − u)2;
with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any lambda > 0: Then we focus on the singular behaviour of the solution, which occurs through
finite-time quenching, i.e. when ||u(·; t)||1 → 1 as t →t*− < ∞, investigating both conditions for quenching and the quenching profile of u: To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.
Original languageEnglish
Article number13
Pages (from-to)1009-1037
Number of pages29
JournalDiscrete and Continuous Dynamical Systems-Series A
Volume35
Issue number3
Early online date1 Oct 2014
DOIs
Publication statusPublished - Mar 2015

Keywords

  • Electrostatic MEMS
  • Quenching of Solution
  • Hyperbolic Problems

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