Nikos I. Kavallaris

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.

Touchdown and related problems in electrostatic MEMS device equation

Abstract.We study the electrostatic MEMS-device equation,

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

u_{t} -\Delta u = \frac{\lambda |x|^{\beta}}{(1-u)^{p}}

A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

, with Dirichlet boundary condition. First, we describe the touchdown of non-stationary solution in accordance with the total set of stationary solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the Morse-Smale property for radially symmetric non-stationary solutions.

The dual integral equation method in hydromechanical systems

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.

Gierer–Meinhardt System

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.

Linear Friction Welding

The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitors response to the activators growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.

A Non-local Model Illustrating Replicator Dynamics

This article has been accepted for publication and will appear in a revised form, subsequent to peer review and/or editorial input by Cambridge University Press, in European Journal of Applied Mathematics published by Cambridge University Press. Copyright Cambridge University Press 2017.