57
reads

Nikos Kavallaris

2010-07-22
16:30:00 - 17:20:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

In this work we study a non-local hyperbolic equation of the form
\[
u_{tt}=u_{xx} +\lambda\frac{1}{ (1-u)^2\left( 1+ \alpha\int_{0}^{1}\frac{1}{1-u}dx
\right)^{2}},
\]
with homogeneous Dirichlet boundary conditions and
appropriate initial conditions. The problem models an idealised
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 $\lambda$, {\it i.e.} when
$\lambda>\lambda^*_+$ for some constant $\lambda^*_+ \ge \lambda^*_-$,
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.