Dimitrios E. Tzanetis

A hyperbolic non-local problem modelling MEMS technology

In this work we study a non-local hyperbolic equation of the form utt = uxx + λ 1 (1− u)2 ( 1 + α ∫ 1 0 1 1−udx )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 λ > 0 and global existence for 0 λ+ for some constant λ+ ≥ λ−, 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.

On the blow-up of a non-local parabolic problem

Abstract We investigate the conditions under which the solution of the initial-boundary value problem of the non-local equation u t = Δ u + λ f ( u ) / ( ∫ Ω f ( u ) d x ) p , where Ω is a bounded domain of R N and f ( u ) is a positive, increasing, convex function, performs blow-up.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

We consider an initial boundary value problem for the non-local equation, u t = u xx +λ f ( u )/(∫ 1 -1 f ( u ) dx ) 2 , with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u ( x , t ) blows up globally in finite time t *, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t *, by using comparison methods. For f ( u ) = e − u , we give an asymptotic estimate: t * ∼ t u (λ−λ*) −1/2 for 0 < (λ−λ*) [Lt ] 1, where t u is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.