On the quenching behaviour of a semilinear wave equation modelling MEMS technology
Nikos I. Kavallaris, Andrew A. Lacey, Christos Nikolopoulos, Dimitrios E. Tzanetis · Discrete and Continuous Dynamical Systems · 2014
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 suitableinitial conditions, which, under appropriate circumstances, serves as amodel of an idealized electrostatically actuatedMEMS device. First we establish local existence of thesolutions 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 profileof $u.$ To this end, the non-existence of a regular similaritysolution near a quenching point is first shownand then a formal asymptotic expansion is used to determinethe local form of the solution. Finally, using a finitedifference scheme, we solve the problem numerically,illustrating the preceding results.