This paper is concerned with some non-linear propagation phenomena for reaction–advection–diffusion equations with Kolmogrov–Petrovsky–Piskunov (KPP)-type non-linearities in general periodic domains or in infinite cylinders with oscillating boundaries. Having a variational formula for the minimal speed of propagation involving eigenvalue problems (proved in Berestycki, H., Hamel, F. & Nadirashvili, N. (2005) The speed of propagation for KPP type problems (periodic framework). J. Eur. Math. Soc. 7, 173–213), we consider the minimal speed of propagation as a function of diffusion factors, reaction factors and periodicity parameters. There we study the limits, the asymptotic behaviours and the variations of the considered functions with respect to these parameters. One of the sections deals with homogenization problem as an application of the results in the previous sections in order to find the limit of the minimal speed when the periodicity cell is very small.

We consider the Cauchy problem for the system

where . Let e ∈ ℝ2 with |e| = 1. If u(x, 0) is smooth, bounded and

we prove u → e uniformly in x as t → ∞. Of particular interest is the motion of the zeros (vortices) of u. In this case, all zeros disappear after a finite time.

The mathematical model of the flow of a viscous lubricant between elastic bearings leads to the study of a highly non-linear and non-local elliptic variational inequality. We discuss the existence of a solution by using an a prioriL∞-estimate. This method allows us to solve a large class of problems, including those arising from the linear Hertzian theory, and yields new existence results for the cases of a pressure-dependent viscosity or the inclusion of a load constraint. For small data the uniqueness of the solution holds, and we show that in the cylindrical journal bearing problem with small eccentricity ratio, the free boundary is given by two disjoint differentiable arcs close to the free boundary of the first-order approximate solution.

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

We introduce and analyse a class of quasi-self-similar solutions of the thin film equation to describe the dynamics of expanding liquid films on a solid surface. Using these solutions as intermediate asymptotics profiles, we obtain a quantitative expression for the shape of the film and a relation between the speed of the contact line and the macroscopic and microscopic contact angles.

We consider the eigenvalue problem associated with the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulae for the eigenvalues and eigenfunctions when the size of the microstructure ε is shorter than the wavelength of the eigenfunctions 1/√λε. On the other hand, it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions (ε ∼ 1/√λε) singular phenomena may occur. In this paper we study the behaviour of the eigenvalues and eigenfunctions when 1/√λε is larger than the critical size ε. We use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to ε. Our analysis provides all order correction formulae for the limit eigenvalues and eigenfunctions above the critical size. Each term of the asymptotic expansion requires one more derivative of the density. Thus, a full description requires the density to be C∞ smooth.

An initial value problem for the functional differential equation

y′(t) =Ay(t) +By(qt) +Cy′(qt) +f(t), t ≥ t0 > 0

where A, B, C are complex matrices, q∈(0, 1), and f is a vector of continuous functions, is considered in this paper. Its solution is represented in terms of the fundamental solution via the variation-of-constants formula. For some special cases, the fundamental solutions are formulated as piecewise Dirichlet series. The variation-of-constants formula is used to analysis the asymptotic behaviour of the solutions of some scalar equations, including one with variable coefficients related to coherent states of the q-oscillator algebra in quantum mechanics.

This article is concerned with the structure and stability properties of a combustion front that propagates in the axial direction along the surface of a cylindrical solid fuel element. The fuel consists of a mixture of two finely ground metallic powders, which combine upon ignition in a one-step chemical reaction. The reaction is accompanied by a melting process, which in turn enhances the reaction rate. The combustion products are in the solid state. The reaction zone, inside which the melting occurs, is modelled as a front that propagates along the surface of the cylinder. The different modes of propagation that have been observed experimentally (such as single- and multiheaded spin combustion and multiple-point combustion) are explained as the results of bifurcations from a uniformly propagating plane circular front. The stability properties of the various modes are discussed.

The purpose of this paper is to study a mathematical model of lubricating flow between elastic surfaces obeying the linear Hertzian theory when cavitation takes place. Cavitation is a free boundary phenomenon that is described in this paper by the New Elrod–Adams model. This model introduces the concentration of fluid as well as the pressure as unknown functions and is suggested in preference to the classical variational inequality due to its ability to describe inflow and outflow. This leads to a nonlinear variational and nonlocal equation. Herein, an existence theorem is proved by means of two different techniques.

Solutions to a boundary-value problem involving a second-order linear functional differential equation with an advanced argument are investigated in this paper. The boundary conditions imposed on the differential equation are analogous to conditions defining various singular Sturm-Liouville problems, and if an eigenvalue parameter is introduced certain properties of the spectrum can be deduced having analogues with the classical problem. Dirichlet series solutions are constructed for the problem and it is established that the spectrum contains an infinite number of real positive eigenvalues. A Laplace transform analysis of the problem then reveals that the spectrum does not generically consist of isolated points and that there may be an infinite number of eigenfunctions corresponding to a given eigenvalue. In contrast, it is also shown that there is a subset of eigenvalues that correspond to the zeros of an entire function for which the corresponding eigenfunctions are unique.

We construct an asymptotic representation for the solution $u(x,t)$ of a singularly-perturbed linear fifth-order evolution equation which accounts for the relevant exponentially-small terms in all regions of the complex $x$ plane. The particular equation that we study is chosen in part to highlight the complexities that arise in high-order examples, resulting in particular from the non-existence of a suitable (steady-state) heteroclinic connection. Key points of this calculation are the identification, location and evolution of the active (in the sense that non-zero, though exponentially-small, terms are switched on across them) Stokes lines, and of the higher-order Stokes lines across which these can be activated or inactivated. In doing so, we need in particular to analyse two ‘levels’ of higher-order Stokes lines and to present the associated mechanisms by which they can themselves be activated or inactivated. By piecing together the information concerning which Stokes lines (both ordinary and higher-order) are active, we are able to deduce systematically which of the competing exponentials that can potentially arise within the asymptotic solution are actually present in each region of the complex plane.

A one-dimensional model of clarifier-thickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. This model also includes a singular feed source. In this paper, the clarifier-thickener model studied in a previous paper (Numer. Math.97 (2004) 25–65) is extended by a singular sink through which material is extracted from the unit. A difficulty is that in contrast to the singular source, the sink term cannot be incorporated into the flux function; rather, the sink is represented by a new non-conservative transport term. To focus on the new analytical difficulties arising due to this non-conservative term, a reduced problem is formulated, which contains the new sink term of the extended clarifier-thickener model, but not the source term and flux discontinuities. The paper is concerned with numerical methods for both models (extended and reduced) and with the well-posedness analysis for the reduced problem. For the reduced problem, a definition of entropy solutions, based on Kružkov-type entropy functions and fluxes, is provided. Jump conditions are derived and uniqueness of the entropy solution is shown. Existence of an entropy solution is shown by proving convergence of a monotone difference scheme. Two variants of the numerical scheme are introduced. Numerical examples illustrate that all three variants converge to the entropy solution, but introduce different amounts of numerical diffusion.

We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven extensional flow for the Upper Convected Maxwell and Oldroyd-B constitutive models. Assuming the axial stress in the filament has a spatial dependence provides the simplest coupling of viscoelastic effects to the motion of the filament, and yields a closed system of ODEs with an exact solution for the stretch rate and filament thickness satisfied by both constitutive models. This viscoelastic solution, which is a generalization of the exact solution for Newtonian filaments, converges to the Newtonian power-law scaling as $t \rightarrow \infty$. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly- and strongly-viscoelastic limits. We compare the viscoelastic solution to measurements of the thinning filament that forms behind a falling drop for several semi-dilute (strongly-viscoelastic) polymer solutions. We find the exact solution correctly predicts the time-dependence of the filament diameter in all of the experiments. As $t \rightarrow \infty$, observations of the filament thickness follow the Newtonian scaling $1/\sqrt{t}$. The transition from viscoelastic to Newtonian scaling in the filament thickness is coupled to a stretch-to-coil transition of the polymer molecules.

Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

This paper is devoted to an analysis of the formation of oscillatory viscous fingers in a Hele-Shaw cell on the basis of the interfacial wave theory, previously established for the pattern formation dynamics in dendrite growth, as well as in the classic Saffman–Taylor flow. In particular, we study the problem of selection and persistence of oscillatory fingers with a tiny bubble at the finger tip. We obtain uniformly valid asymptotic solutions for this problem, and derive the linear, global wave instability mechanism for this more complicated system. The global, neutrally stable modes are computed in a large region of parameters, which select the form of oscillatory fingers in the later stage of evolution. We have compared the theoretical predictions with the experimental data by Couder et al. (1986) and by Kopf-Sill & Homsy (1987), and found excellent quantitative agreement.

Nonstationary two-dimensional filtration in a porous medium is considered, whereby part of the medium is saturated, another part is unsaturated but wet, and the remaining part is dry. The saturated/unsaturated and unsaturated/dry interfaces are free boundaries. It is shown that there exists a unique solution, and that the saturation function is continuous in the wet portion of the medium; this implies that the two interfaces are separated. Under some monotonicity-type conditions on the initial and boundary data it is shown that the free boundaries are continuous.

We prove the existence of a family of Travelling Wave (TW) solutions for a large class of scalar reaction-diffusion equations with degenerate, nonlinear diffusion coefficients and monostable nonlinear reaction terms. We also investigate stability. Specifically, we show that, as in the linear diffusion case [6], the slowest TW in the family yields the asymptotic rate of the propagation of disturbances from the unstable rest state in these systems. In addition, we give conditions on the reaction term and diffusion coefficient ensuring the existence of interfaces.

In this paper, three iterative procedures (Landweber-Fridman, conjugate gradient and minimal error methods) for obtaining a stable solution to the Cauchy problem in slow viscous flows are presented and compared. A section is devoted to the numerical investigations of these algorithms. There, we use the boundary element method together with efficient stopping criteria for ceasing the iteration process in order to obtain stable solutions.

In this paper we investigate the model for the motion of a contact line over a smooth solid surface developed by Shikhmurzaev, [24]. We show that the formulation is incomplete as it stands, since the mathematical structure of the model indicates that an additional condition is required at the contact line. Recent work by Bedeaux, [4], provides this missing condition, and we examine the consequences of this for the relationship between the contact angle and contact line speed for Stokes flow, using asymptotic methods to investigate the case of small capillary number, and a boundary integral method to find the solution for general capillary number, which allows us to include the effect of viscous bending. We compare the theory with experimental data from a plunging tape experiment with water/glycerol mixtures of varying viscosities [11]. We find that we are able to obtain a reasonable fit using Shikhmurzaev's model, but that it remains unclear whether the linearized surface thermodynamics that underlies the theory provide an adequate description for the motion of a contact line.

In this paper, we consider the penalty method to solve the unilateral contact with friction between an electro-elastic body and a conductive foundation. Mathematical properties, such as the existence of a solution to the penalty problem and its convergence to the solution of the original problem, are reported. Then, we present a finite elements approximation for the penalised problem and prove its convergence. Finally, we propose an iterative method to solve the resulting finite element system and establish its convergence.