In this paper, we develop a mathematical model to describe interactions between tumour cells and a compliant blood vessel that supplies oxygen to the region. We assume that, in addition to proliferating, the tumour cells die through apoptosis and necrosis. We also assume that pressure differences within the tumour mass, caused by spatial variations in proliferation and degradation, cause cell motion. We couple the behaviour of the blood vessel into the model for the oxygen tension. The model equations track the evolution of the densities of live and dead cells, the oxygen tension within the tumour, the live and dead cell speeds, the pressure and the width of the blood vessel. We present explicit solutions to the model for certain parameter regimes, and then solve the model numerically for more general parameter regimes. We show how the resulting steady-state behaviour varies as the key model parameters are changed. Finally, we discuss the biological implications of our work.

The shape stability of the reaction interface for reactive flow in a porous medium is investigated. Previous work showed that the Reaction-Infiltration Instability could cause the reaction zone to lose stability when the Peclet number exceeded a critical value. The new feature of this study is to include a velocity-dependent hydrodynamic dispersion. A mathematical model for this phenomenon is given in the form of a moving free-boundary problem. The spectrum of the linearized problem is obtained, and the related analysis and numerical calculations show that the onset of the instability is not eliminated by the new dispersive terms. The details of analysis show that the instability is reduced especially by the transverse dispersion.

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation

[formula here]

with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution

[formula here]

of the complex Hamilton–Jacobi equation

[formula here].