We assume that the Stefan problem with underheating has a classical solution until the moment of contact of two distinct free boundaries and the free boundaries have continuous velocities until the moment of contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with underheating, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution that is the solution of the heat equation.

Layered nanocrystals consist of a core of one material surrounded by a shell of a second material. We present computation of the atomistic strain energy density in a layered nanocrystal, using an idealised model with a simple cubic lattice and harmonic interatomic potentials. These computations show that there is a critical size r*s for the shell thickness rs at which the energy density has a maximum. This critical size is roughly independent of the geometry and material parameters of the system. Interestingly, this critical size agrees with the shell thickness at which the quantum yield has a maximum, as observed in several systems and thus leads one to support the hypothesis that maximal quantum yield is strongly correlated with maximal elastic energy density.

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, non-linear selection of the minimal speed occurs.

The rich dynamics of quantized vortices governed by the Ginzburg-Landau-Schrödinger equation (GLSE) is an interesting problem studied in many application fields. Although recent mathematical analysis and numerical simulations have led to a much better understanding of such dynamics, many important questions remain open. In this article, we consider numerical simulations of the GLSE in two dimensions with non-zero far-field conditions. Using two-dimensional polar coordinates, transversely highly oscillating far-field conditions can be efficiently resolved in the phase space, thus giving rise to an unconditionally stable, efficient and accurate time-splitting method for the problem under consideration. This method is also time reversible for the case of the non-linear Schrödinger equation. By applying this numerical method to the GLSE, we obtain some conclusive experimental findings on issues such as the stability of quantized vortex, interaction of two vortices, dynamics of the quantized vortex lattice and the motion of vortex with an inhomogeneous external potential. Discussions on these simulation results and the recent theoretical studies are made to provide further understanding of the vortex stability and vortex dynamics described by the GLSE.