This work treats the injection of certain thermoplastics into a planar mould cavity. The problem is to determine the filling pattern. It is assumed that the thermoplastic can be modelled as a non-Newtonian fluid of power-law type whose power-law exponent is relatively small (the pseudo-plastic case). The dependence of the viscosity on thermal variations is neglected. The mathematical description leads to a moving boundary problem, for which an asymptotic solution is found. According to this solution, the expansion of the polymer melt follows the level sets of an interior distance function, which is determined by the geometry of the mould, and the position of the injection point. The solution is easily computed and results of numerical experiments are given.

We consider phase transition processes in which the thermodynamic variables are the temperature and an order parameter. Various classes are identified and many specific examples are illustrated. In this framework the question of the range of applicability of the so-called ‘additivity rules’ is investigated, showing that they apply only to a very special type of processes.

We construct a theory for maximal viscosity solutions of the Cauchy problem for the modified porous medium equation ut + γ|ut| = Δ(um) with γ∈(−1, 1) and m > 1. We investigate the existence, uniqueness, finite propagation speed and optimal regularity of these solutions. As a second main theme, we prove that the asymptotic behaviour is given by a certain family of self-similar solutions of the so-called second kind with anomalous similarity exponents.

We treat the mathematical properties of the one-parameter version of the Mróz model for plastic flow. We present continuity results and an energy inequality for the hardening rule, and discuss different versions of the flow rule regarding their relation to the basic laws of thermodynamics.

We consider an equilibrium problem for a thin inclusion in a shell. The faces of the inclusion are assumed to satisfy a non-penetration condition, which is an inequality imposed on the tangential shell displacements. The properties of the solution are studied, in particular, the smoothness of the stress field in the vicinity of the inclusion. The tangential displacements are proved to belong to the space H2 near the internal points of the inclusion. The character of the contact between the inclusion faces is described in terms of a suitable non-negative measure. The stability of the solution is investigated for small perturbations to the inclusion geometry.

A sharp estimate of the growth of solutions of the initial value problem for systems of the form

where Cj(t) are matrices with elements of power growth, is found. As a corollary of this result, it follows, for instance, that each solution of the initial value problem satisfies the estimate ‖u(t)‖ ≤ Cexp{γln2(1+|t|)} for some C > 0 and γ > 0.

We consider the GI/M/1 – K queue which has a capacity of K customers. Using singular perturbation methods, we construct asymptotic approximations to the stationary queue length distribution. We assume that K is large and treat several different parameter regimes. Extensive numerical comparisons are used to show the quality of the proposed approximations.