The linearized Ginzburg–Landau equations in both semi-infinite strips and rectangles are transformed into equivalent one-dimensional integral equations. Then, the properties of the integral equations are utilized to prove that the onset field for a semi-infinite strip is isolated. We solve the integral equations numerically to obtain the onset field for both rectangles and semi-infinite strips. A formal asymptotic expansion of the onset field in the long rectangle limit is also obtained. Using this formal expansion, we show that the onset field converges in this limit faster than any finite exponential rate, and as a byproduct, that the onset mode in a semi-infinite strip must be asymptotically symmetric.

We prove the existence of a solution to a free boundary value problem arising in elastohydrodynamic lubrication. Previously, the existence was proved only under severe restrictions on the range of the physical parameters. We remove those restrictions here.

When individuals move together in large groups, as seen in schools of fish, they adapt their speed and direction to that of their neighbours. We present and analyse a model for the speed adaptation process in the case in which all individuals move in the same or in two opposite directions. The model consists of a hyperbolic conservation law for the density of individuals coupled to a parabolic or elliptic equation for speed. A detailed linear analysis reveals several mechanisms for the appearance of instabilities of the homogeneous steady state, which trigger the formation of schools, herds, flocks, etc. Long-term existence of weak solutions is shown using the vanishing viscosity approach.

An eigenvalue problem is considered for a multi-structure consisting of a three-dimensional finite solid connected to an arbitrary smooth elastic thin shell of revolution. Two-sided estimates are obtained for the first six eigenfrequencies of the multi-structure. Explicit asymptotic formulae are given.

We prove the existence of a family of Travelling Wave (TW) solutions for a large class of scalar reaction-diffusion equations with degenerate, nonlinear diffusion coefficients and monostable nonlinear reaction terms. We also investigate stability. Specifically, we show that, as in the linear diffusion case [6], the slowest TW in the family yields the asymptotic rate of the propagation of disturbances from the unstable rest state in these systems. In addition, we give conditions on the reaction term and diffusion coefficient ensuring the existence of interfaces.