We study the motion of voids in conductors subject to intense electrical current densities. We use a free-boundary model in which the flow of current around the insulating void is coupled to a law of motion (kinematic condition) for the void boundary. In the first part of the paper, we apply a new complex variable formulation of the model to an infinite domain and use this to (i) consider the stability of circular and flat front travelling waves, (ii) show that, in the unbounded metal domain, the only travelling waves of finite void area are circular, and (iii) consider possible static solutions. In the second part of the paper, we look at a conducting strip (which can be used to model interconnects in electronic devices) and use asymptotic methods to investigate the motion of long wavelength voids on the conductor boundary. In this case we derive a nonlinear parabolic PDE describing the evolution of the free boundary and, using this simpler model, are able to make some predictions about the evolution of the void over long times.

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

In this paper we prove the existence of solutions of the Keller–Segel model in chemotaxis, which blow up in finite or infinite time. This is done without assuming any symmetry properties of the solution.

The often-studied problem known as Kramers' problem, in the general area of rarefied-gas dynamics, is investigated in terms of a linearized, variable collision frequency model of the Boltzmann equation. A convenient change of variables is used to reduce the general case considered to a canonical form that is well suited for analysis by analytical and/or numerical methods. While the general formulation developed is valid for an unspecified collision frequency, a recently developed version of the discrete-ordinates method is used to compute the viscous-slip coefficient and the velocity defect in the Knudsen layer for three specific cases: the classical BGK model, the Williams model (the collision frequency is proportional to the magnitude of the velocity) and the rigid-sphere model.