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.

The bifurcation from a normally conducting to a superconducting state as an external magnetic field is lowered is examined using the Ginzburg–Landau theory. Linear and weakly nonlinear stability analyses are performed near the bifurcation point, and the implications of the results for each of three examples is considered.

The main result of this paper is a complete description of local invertible transformations

relating one equation of the form uxy = F(x, y, u, ux, uy) to another equation of the form υxy = G(x, y, υ, υx, υy).

We describe the slow evolution of the wave speed and reaction temperature in a model of filtration combustion. In the counterflow configuration of the process, a porous solid matrix is converted to a porous solid product by injecting an oxidizing gas at high pressure into one end of a fresh sample of the solid while igniting it at the other end. The solid and gas react exothermically at high activation energy and, under favourable conditions, a self-sustaining combustion wave travels along the sample, converting reactants to product. Since the reaction rate depends on the gas pressure p in the pores, small gradients in p cause variations in the conditions of combustion, which, in turn, cause inhomogeneities in the physical properties of the product. We determine the slow evolution of the wave speed, the reaction temperature, and the mass flux of the gas downstream of the reaction zone. In the absence of a pressure gradient, there is a branch of steadily propagating solutions which has a fold. For planar disturbances on the slow time scale, we show that the middle part of the branch is unstable, with the change of stability occurring at the turning points of the branch. When the pressure gradient is nonzero, there are no steadily propagating solutions and the wave continually evolves. Conditions on the state of the gas at the inlet are described such that the variation in the wave speed and reaction temperature throughout the process can be minimized.

The existence, uniqueness and regularity of the solution to a one-dimensional linear thermoelastic problem with unilateral contact of the Signorini type are established. A finite element approximation is described, and an error bound is derived. It is shown that if the time step is O(h2), then the error in L2 in the temperature and in L∞ in the displacement is O(h). Some numerical experiments are presented.

It is shown that the influence of microstructure in the damage accumulation process leads to a nonlinear diffusion effect, with a strongly stress-dependent diffusion coefficient. A nonlinear parabolic equation with a source term is obtained for the damage parameter. This equation is relevant to blow-up and quenching problems well known to mathematicians with rupture corresponding to blow-up or quenching. However, the damage accumulation equation possesses an additional nonlinearity due to the non-healing of damage. Depending on the value of a dimensionless constant parameter (the ratio of a properly defined microstructural length-size to a characteristic length-size of the initial damage distribution), two essentially different types of damage accumulation process appear to be possible for a given initial damage distribution over the bar length. In processes of the first type, the damage accumulation remains non-homogeneous over the length of the bar, so that the lifetime for the whole specimen is determined by the maximal initial damage within the bar. For processes of the second type the damage distribution over the specimen at first becomes homogeneous (at least in a considerable part of specimen), and then the damage accumulation proceeds uniformly over all or part of the specimen. The lifetime for processes of the second type is essentially longer than the first. Results of a numerical experiment based on the proposed model are presented. In particular, the origin and development of damage propagation waves is demonstrated. Also, it is demonstrated that when there is substantial damage transfer, the ultimate value of the damage parameter in the life-time calculation is of no significance.

Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation ℱ between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that ℱ satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique.

We present a method of solution of a class of fracture problems in the theory of elasticity. The method can be applied to any problem reducible to Poisson's equation, e.g. heat conduction and mass diffusion in solids, theory of consolidation and the like. The novelty of the paper is that we address regions of layered composites with notches, or, in a particular case, with a crack. Within the framework of classical analysis, we apply Fourier and Mellin transforms, 'fit' them together, and reduce the problem to solving a singular integral equation with fixed singularities on a semi-axis. We show the existence and uniqueness of solutions of the equations under consideration, and justify the asymptotics necessary for applications. We show the practical usefulness of the method on the examples of an antiplane problem of fracture mechanics. From our solution, we are able to find the stress intensity factor in the case when a crack tip penetrates a layered composite consisting of 60 layers, and show the limits of applicability of the anisotropic model of such composites.

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.

The self-similar source solution of the Barenblatt equation for elasto-plastic filtration through porous rock is known to be of the second kind. We determine the behaviour of the associated anomalous exponent and the profile of the solution in the limit of large compressibility and small elastic recovery of the rock.

We consider the non-local problem

It is found that for the case of decreasing f then: (i) for

there is a unique steady state which is globally asymptotically stable; (ii) for

then the problem can be scaled so that

in which case: (a) for λ < 8 there is a unique steady state which is globally asymptotically stable; (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, −1 < x, < 1. Some formal asymptotic estimates for the local behaviour of u as it blows up are obtained.

The diffraction of time-harmonic waves in a nonlinear medium with periodic structure is studied in this paper. In particular second harmonic generation – an important phenomenon in nonlinear optics-is modelled. The model, derived from a general nonlinear system of Maxwell's equations, is shown to have a unique solution for all but a discrete number of frequencies. The problem is solved numerically by combining a method of finite elements and a fixed-point iteration scheme. Numerical experiments for some simple grating structures are presented and discussed.

A general method is presented to facilitate the solution of a class of polydisperse spray problems in which a cloud of droplets can be described using a sectional or group model. The procedure involves replacing the original coupled droplet sectional variables conservation equations by a set of uncoupled sectional equations for auxiliary variables. The form of these latter equations is identical to that of the single spray equation for a quasi-monodisperse spray, solutions of which are more readily attainable even for multidimensional spray problems. Thus, these ready-made solutions can be exploited directly for the auxiliary variables, from which solutions can then be constructed in a straightforward manner for the desired original sectional variables. Three illustrative examples for spray diffusion flames with different features of complexity highlight the potential applicability of the proposed method, and indicate the sensitivity of flame characteristics to initial spray conditions and in-spray related phenomena.