We discuss a nonlinear diffusion equation in which the diffusivity has a non-local dependence on the surface concentration. Exact results are obtained for some special cases by means of similarity methods and non-local transformations. We indicate some of the implications the analysis has for more general cases.

Integral characteristics, such as elastic polarization matrices of elastic inclusions and cavities, are described. The matrix of elastic polarization of a finite cavity is constructed in the case of the two-dimensional Lamé operator under the assumption that the geometry of the domain occupied by the cavity is defined by a conformal mapping from the unit disk. Examples and applications of these integral characteristics in the theory of cracks are considered.

It is shown that for a class of pairs of energy wells the only Young measures having these wells as support must reduce to spatially constant Dirac masses. This implies the prediction that fine structures will be absent in certain crystal phase transitions.

We consider a well-known thermo-diffusive model for the propagation of a premixed, adiabatic flame front in the large-activation-energy limit. That model depends only on one nondimensional parameter β, the reduced Lewis number. Near the pulsating instability limit, as β↓β0= 32/3, we obtain an asymptotic model for the evolution of a quasi-planar flame front, via a multi-scale analysis. The asymptotic model consists of two complex Ginzburg–Landau equations and a real Burgers equation, coupled by non-local terms. The model is used to analyse the nonlinear stability of the flame front.

Bennett functions , (h, k), defined below and elsewhere, have been used extensively in Fourier analysing the output of certain devices such as rectifiers, limiters and other instantaneous modulators responding to a two-frequency input for which the output at time t depends only on the device input–output characteristic Y = Y(X) and the input at time t. Here, with the aid of a new set of modified Bennett functions , (h, k), these methods are extended to solve the same problems for non-instantaneous bang-bang controls with multivalued input–output characteristics Y = Y(X) and hence having memory.

A simple computational procedure is described for the evaluation of the modified Bennett functions, which arise in the multiple Fourier analysis for bang-bang controls with memory.