We consider the two-phase one-dimensional Stefan problem in a finite interval, with initial and boundary conditions such that the solid phase vanishes at a finite time T and at a single point. We show that the temperature in the solid phase decreases to zero and is bounded by c exp (α/(t − T)) as extinction approaches (C, α > 0) and that phase boundaries at extinction have finite speeds.

We consider a polymer melt in a domain Ω whereby each polymer chain is attached at one endpoint to a fixed surface S contained in ∂Ω. Denote by G(x, t;y) the normalized number density of all subchains from x to y of length t. Then, according to the selfconsistent mean field theory, G satisfies, for each y: Gt - Δ2G + σϕG = 0, where σ is a real parameter, and ϕ is a functional of G(·, ·; ·) both non-local and nonlinear. We establish the existence of G and C∞ regularity of ϕ, as a function of x.

The main result of this paper is a non-uniqueness theorem for the self-similar solutions of a model for phase transitions in binary alloys. The reason for this non-uniqueness is the discontinuity in the coefficients of heat conduction and diffusion at the inter-phase. Also the existence of a self-similar solution and the stability criterion are discussed.

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.

Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.

An analysis is made of the diffraction of sound by a scratch (or groove) in the nominally plane surface of a fluid-loaded elastic solid. The solid is modelled as an isotropic half-space, and an exact representation of the diffracted sound is obtained as an integral over the surface of the scratch involving a Green's function and the acoustic field for the homogeneous solid. The restriction of the range of integration to the surface of the scratch permits the use of an approximate form of Green's function which is valid for source positions confined to a limited region of space. Details are worked out for the case in which the acoustic wavelength is much larger than the diameter of the scratch cross-section (Rayleigh approximation). The results are illustrated by application to sound waves in water inicident on a rectilinear scratch of circular cross-section in steel or rubber. In the former case an important rôle in diffraction is played by leaky Rayleigh waves on the fluid-solid interface.

Uniqueness and continuous dependence on the initial temperature are proved for a onedimensional, quasistatic and frictionless contact problem in linear thermoelasticity. First the problem is reformulated in such a way that it decouples. The resulting problem for the temperature is a nonlinear integro-differential equation. Once the temperature is known the displacement is recovered from an appropriate variational inequality. Uniqueness is proved by considering an integral transform of the temperature. The steady solution is obtained and the asymptotic stability is shown. It turns out that the asymptotic behaviour and the steady state are determined by a relation between the coupling constant a and the initial gap.

We regret that the paper by J. M. Greenberg, ‘Models of elastic–perfectly plastic materials’, vol. 1, part 2, pp. 131–150, was repeated in error as pp. 225–244 in part 3.

The publisher apologizes to the author and readers for this unfortunate error.