A model describing the evolving shape of a growing pile is considered, and is shown to be equivalent to an evolutionary quasi-variational inequality. If the support surface has no steep slopes, the inequality becomes a variational one. For this case existence and uniqueness of the solution are proved.

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

This paper deals with the spherically symmetric Stefan problem in three space dimensions. The melting temperature satisfies the Gibbs–Thomson law. The solution is obtained as a limit of solutions of similar problems containing a small additional kinetic term in the melting temperature. Under some structural assumptions we show that the phase-change boundary has at most one discontinuity point t = T0 (see the corresponding result for the planar Stefan problem in Götz & Zaltzman (1995)). In the one-phase problem the discontinuity point always exists. At the time T0 the whole solid phase melts instantaneously. We study also the asymptotical stability (t → ∞) of stationary solutions satisfying boundary conditions of thermostat type.

In the focussing problem for the porous medium equation, one considers an initial distribution of material outside some compact set K. As time progresses material flows into K, and at some finite time T first covers all of K. For radially symmetric flows, with K a ball centred at the origin, it is known that the intermediate asymptotics of this focussing process is described by a family of self-similar solutions to the porous medium equation. Here we study the postfocussing regime and show that its onset is also described by self-similar solutions, even for nonsymmetric flows.

We show by using formal asymptotics that the zero level set of the solution to the Cahn–Hilliard equation with a concentration dependent mobility approximates to lowest order in ɛ. an interface evolving according to the geometric motion,

(where V is the normal velocity, Δ8 is the surface Laplacian and κ is the mean curvature of the interface), both in the deep quench limit and when the temperature θ is where є2 is the coefficient of gradient energy. Equation (0.1) may be viewed as motion by surface diffusion, and as a higher-order analogue of motion by mean curvature predicted by the bistable reaction-diffusion equation.

We consider the propagation of an exothermic reaction wave which converts liquid reactants into a solid product. Such reaction waves are observed, for example in addition polymerization, where there is a propagating localized polymerization zone, in which monomer is converted to polymer. We study uniformly propagating travelling waves and their linear stability. We show that, though this problem is similar to the problem of gaseous combustion, which exhibits a hydrodynamic instability, here there is no such hydrodynamic instability. This theoretical result is consistent with experimental observations in the case when the polymerization process occurs under high external pressure conditions.