A hyperbolic non-local problem modelling MEMS technology

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.