We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

In this paper, we consider the bi-stable equation proposed by Rosenau to replace the Allen–Cahn equation in the case of large gradients. We discuss the bifurcation problem for stationary solutions of this equation on an interval as the diffusion coefficient and the length of the interval are varied, concentrating on classical solutions.

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of type Mu = −N∇p + F with suitable matrices M and N with F depending on the right-hand side in the bulk term and in the boundary condition.

Continuous dependence upon the elastic moduli of the total energy, and consequently in appropriate measure of the stress, strain and displacement, is established in the initial boundary value problem defined on a semi-infinite cylinder occupied by a linear anisotropic compressible elastic material. Free, constrained and mixed boundary conditions on the lateral surface of the cylinder are discussed. The conclusions are obtained using differential inequalities and partly depend upon previously published results.

Typical problems with solutions characterised by first-kind discontinuities occurring at interfaces of layered inhomogeneous media are considered with respect to second-order differential equations in partial derivatives. Direct, inverse and mixed types of solution discontinuities are considered. Presented are generalised formulations of problems under consideration, having discontinuous solutions and allowing a uniform description of the processes of heat, mass and charge transfer in multilayer media. Homogeneous difference schemes built on the basis of generalised solutions, which are illustrated by test problems with analytical solutions, are given.

We go back to the results of Poincaré [Poincare, H (1891) Sur lintegration des equations differentielles du premier ordre et du premier degre I and II, Rendiconti del circolo matematico di Palermo5, 161–191] on the multipliers of a periodic orbit for proving the C1 non-integrability of differential systems. We apply these results to Lorenz, Rossler and Michelson systems, among others.