On the quenching behaviour of a semilinear wave equation modelling  MEMS technology

Nikos I. Kavallaris; Andrew A. Lacey; Christos V. Nikolopoulos; Dimitrios E. Tzanetis

doi:10.3934/dcds.2015.35.1009

Article Contents

2015, Volume 35, Issue 3: 1009-1037. Doi: 10.3934/dcds.2015.35.1009

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On the quenching behaviour of a semilinear wave equation modelling MEMS technology

1.
Department of Mathematics, School of Science and Engineering, University of Chester, Thornton Science Park, Pool Lane, Ince Chester CH2 4NU
2.
Maxwell Institute for Mathematical Sciences & Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS
3.
Department of Mathematics, University of the Aegean, GR-832 00 Karlovassi, Samos
4.
Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, 157 80 Athens

Received: February 28, 2014

Revised: June 30, 2014

Published: February 2015

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Abstract

In this work we study the semilinear wave equation of the form \[ u_{tt}=u_{xx} + {\lambda}/{ (1-u)^2}, \] 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 $||u(\cdot,t)||_{\infty}\to 1$ as $t\to t^*- < \infty$, 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.

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Mathematics Subject Classification: Primary: 35L81, 35J60; Secondary: 74H35, 74G55, 74K15.

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