In this paper, we extend the work by Sato devoted to the development of economic growth models within the framework of the Lie group theory. We propose a new growth model based on the assumption of logistic growth in factors and derive the corresponding production functions, as well as a compatible notion of wage share. In the process, it is shown that the new functions compare reasonably well against relevant economic data. The corresponding problem of maximisation of profit under conditions of perfect competition is solved with the aid of one of these functions. In addition, it is explained in reasonably rigorous mathematical terms why Bowley’s law no longer holds true in the post-1960 data.

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centres. For the family studied in this paper, it is enough.

We construct compactly supported self-similar solutions of the modified porous medium equation (MPME)

They have the form

where the similarity exponents α and β depend on ε, m and the dimension N. This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood phenomenon. This paper performs a detailed study of the properties of the anomalous exponents of the MPME.

Fluids in unsaturated porous media are described by the relationship between pressure (p) and saturation (u). Darcy's law and conservation of mass provides an evolution equation for u, and the capillary pressure provides a relation between p and u of the form p∈ pc(u,∂tu). The multi-valued function pc leads to hysteresis effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for oscillatory stochastic coefficients. The effective equations contain a new dependent variable that encodes the history of the wetting process and provide a better description of the physical system.

We wish to understand the macroscopic plastic behaviour of metals by upscaling the micro-mechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete model to be a one dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local. As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of Γ-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.

In this article, we provide a numerical algorithm to reconstruct a convex sound-soft scatterer from phaseless backscattering data, assuming sufficiently high frequency. Certain uniqueness and existence results for the case of circular scatterers are given as well, based on the asymptotic expansion for the normal derivative of the total field.

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.

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.

We study the coupled surface and grain boundary motion in a bi-crystal in the context of the ‘quarter loop’ geometry. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. Three curves meet at points where junction conditions are given. A formulation that describes the coupled normal motion of the curves and preserves arc length parametrisation up to scaling is proposed. The formulation is shown to be well-posed in a simple, linear setting. Equations and junction conditions are approximated by finite difference methods. Numerical convergence to exact travelling wave solutions is shown. The method is applied to other problems of physical interest.

In this paper, Hele-Shaw moving boundary problems with kinetic undercooling regularization are studied. By application of the nonlinear abstract Cauchy–Kovalevskaya theorem the local existence of analytic solutions is shown. Therefore we have to study the behaviour of the gradient of harmonic functions up to the boundary of the domain.

This paper presents a powerful approximate method for modelling the steady single-phase flow into a horizontal well completed with an Inflow Control Device (ICD) in an anisotropic reservoir. Two types of problems are investigated: the forward problem, which allows the user to find the flux distribution along the wellbore for a specified pressure drawdown, and the inverse problem to determine the ICD properties when the flux or reservoir pressure drawdown along the wellbore is given. The method is based on structuring the flow patterns around and, inside the wellbore and across the ICD and on the reduction of the dimensionality of the problem by using boundary integral equations. The resulting one dimensional singular nonlinear integro-differential equation is solved numerically, using the appropriate quadrature formula for singular integrals with Cauchy kernels.

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.

We set up the electromagnetic system and its plane-wave solutions with the associated slowness and wave surfaces. We treat the Cauchy initial-value problem for the electric vector and make explicit the quantities necessary for numerical evaluation. We use the Herglotz-Petrovskii representation as an integral around loops which, for each position and time form the intersection of a plane in the space of slownesses with the slowness surface. The field and especially its singularities are strongly dependent on the varying geometry of these loops; we use a level set numerical technique to compute those real loops which essentially gives us second order accuracy. We give the static term corresponding to the mode with zero wave speed. Numerical evaluation of the solution is presented graphically followed by some concluding remarks.

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

Spontaneous potential well-logging is an important technique in petroleum exploitation. To make the corresponding log interpretation chart, it is supposed that the geometrical structure of the formation, the resistivity in each subdomain, and the spontaneous potential difference on each interface are all known; then in the direct problem, the spontaneous potential u = u(r, z) satisfies an elliptic boundary value problem with jump conditions on interfaces. At the joint points A and B of the interfaces (figure 2), the jumps of the spontaneous potential do not, in general, satisfy the compatibility condition. It turns out that it is impossible to find a piecewise H1 solution to the problem, and the standard finite element method cannot be applied to get an approximate solution. In this paper, by means of a method of removing the singularities at A and B, it is proved that the problem admits a unique weak solution that is piecewise W1,p for any fixed p with 1 ≤ p <2. Moreover, based on this method a numerical scheme is suggested, and some numerical examples and some conclusions of practical interest are given. The techniques used in this paper will find a wider applicability in other problems.

In this article, we consider the spatial homogenisation of a multi-phase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenised partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations, we employ an asymptotic homogenisation of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction–advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug- and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.

This paper analyses a canonical class of one-dimensional superlinear indefinite boundary value problems of great interest in population dynamics under non-homogeneous boundary conditions; the main bifurcation parameter in our analysis is the amplitude of the superlinear term. Essentially, it continues the analysis of López-Gómez et al. (López-Gómez, J., Tellini, A. & Zanolin, F. (2014) High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Comm. Pure Appl. Anal. 13(1), 1–73) with empty overlapping, by computing the bifurcation diagrams of positive steady states of the model and by proving analytically a number of significant features, which have been observed from the numerical experiments carried out here. The numerics of this paper, besides being very challenging from the mathematical point of view, are imperative from the point of view of population dynamics, in order to ascertain the dimensions of the unstable manifolds of the multiple equilibria of the problem, which measure their degree of instability. From that point of view, our results establish that under facilitative effects in competitive media, the harsher the environmental conditions, the richer the dynamics of the species, in the sense discussed in Section 1.

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.

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.

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.