This paper is concerned with the spatial dynamics of a monostable delayed age-structured population model in a 2D lattice strip. When there exists no positive equilibrium, we prove the global attractivity of the zero equilibrium. Otherwise, we give some sufficient conditions to guarantee the global attractivity of the unique positive equilibrium by establishing a series of comparison arguments. Furthermore, when those conditions do not hold, we show that the system is uniformly persistent. Finally, the spreading speed, including the upward convergence, is established for the model without the monotonicity of the growth function. The linear determinacy of the spreading speed and its coincidence with the minimal wave speed are also proved.

Coarsening of solutions of the integro-differential equation

formula here

where Ω ⊂ ℝn, J(·) [ges ] 0, ε > 0 and f(u) = u3 − u (or similar bistable nonlinear term), is examined, and compared with results for the Allen–Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when ε is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen–Cahn equation. The special case J(·) ≡ 1 is explored in detail, giving insight into the behaviour in the more general case J(·) [ges ] 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

Thin-film models for the flow of a low reduced-Reynolds-number poroviscous droplet over a planar substrate are developed. One of the formulations is used to develop a minimal model for active animal cell motion in which the microscopic mechanisms of polymerisation and depolymerisation near the outer cell periphery are modelled by specifying the rate of mass transfer between the phases at the contact-line in terms of the velocity of the latter. An asymptotic analysis in the limit corresponding to strong cell-substrate adhesion is shown to lead to a novel class of multi-valued contact-line laws, a qualitative analysis of which leads in two dimensions to some intriguing behaviour, including (i) periodic contraction and expansion (pulsation), (ii) steady propagation at a contant speed, (iii) an unsteady combination of pulsation and propagation, and (iv) a bistable regime in which both non-motile and motile solutions are admissible, each of them being stable to sufficiently small perturbations, but with sufficiently large perturbations being able to ‘prod’ a stationary cell into motion or halt a moving one; these qualitative predictions are where possible compared with experiment. The contact-line behaviour is likely to be highly sensitive to environmental signals; the formulation may, therefore, provide a useful ‘minimal’ modelling framework for investigation of chemotactic effects at the cell scale. The corresponding extensional flow formulations are also noted.

We consider a family of problems involving two-dimensional Stokes flows with a time dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there to be no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed.

Nonlinear evolution equations including hysteresis functionals are studied. It is the purpose of this paper to investigate the asymptotic stability of the solution as time t → + ∞. In the case when the forcing term of the equation tends to a time-independent function as t → + ∞, we shall show that the solution stabilizes at + ∞ and the system is asymptotically stable (equilibrium stability). In the case when the forcing term is periodic in time, we shall show that there is at least one periodic solution and that in some restricted cases the periodic solution is unique.

In this paper we consider the problem of hedging options in the presence of cost in trading the underlying asset. This work is an asymptotic analysis of a stochastic control problem, as in Hodges & Neuberger and Davis, Panas & Zariphopoulou;. We derive a simple expression for the ‘hedging bandwidth’ around the Black–Scholes delta; this is the region in which it is optimal not to rehedge. The effect of the costs on the value of the option, and on the width of this hedging band is of a significantly greater order of magnitude than the costs themselves. When costs are proportional to volume traded, rehedging should be done to the edge of this band; when there are fixed costs present, trading should be done to an optimal point in the interior of the no-transaction region.

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. For simplicity, the fine structure is here taken to be periodic, first in one dimension, and then a lattice of squares in two dimensions. A method of multiple scales is employed, with a classical free-boundary problem being used to model the evolution of the two-phase microstructure. Then a macroscopic model for the mush is obtained by an averaging procedure. The free-boundary temperature is taken to vary according to Gibbs–Thomson and/or kinetic-undercooling effects.

The incompressible slow viscous flow of a power-law shear-thinning fluid in a wedge-shaped region is considered in the specific instance where the stress is a very small power of the strain rate. Asymptotic analysis is used to determine the structure of similarity solutions. The flow is shown to possess an outer region with boundary layers at the walls. The boundary layers have an intricate structure consisting of a transition layer 0(ɛ) adjoining an inner layer O(ɛlnɛ), which further adjoins an inner-inner layer 0(ɛ) next to the wall. Explicit solutions are found in all the regions and the existence of ‘dead zones’ in the flow is discussed.

Motivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.

This paper proves the existence and uniqueness of a monotone increasing solution of the Painlevé 1 equation y″ = y2+x. The monotonicity of solution is then exploited to show stability of the plasma-sheath transition in a weakly ionizing plasma.

Motivated by the study of fibre dynamics in the carding machine, a continuum model for the motion of a medium composed of fibres is derived under the assumption that the dominant forces are due to fibre-fibre interactions and that the material is in tension. To characterise the material we include the averaged values of density and velocity and introduce variables to describe the mean direction, alignment and entanglement. We assume that the bulk stress of the material depends on the density, entanglement, degree of alignment, average direction and shear-rates. A kinematic equation for average direction and two proposed heuristic laws for the evolution of entanglement and degree of alignment are given to close the system. Extensional and shearing simulations are in good qualitative agreement with experimental results.

In this paper we study the probability density function solutions to a second-order pantograph equation with a linear dispersion term. The functional equation comes from a cell growth model based on the Fokker–Planck equation. We show that the equation has a unique solution for constant positive growth and splitting rates and construct the solution using the Mellin transform.

In a planar smoothly bounded domain $\Omega$ , we consider the model for oncolytic virotherapy given by

$$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$

$ D_w $

$ D_z $

$\beta$

$\beta \lt 1$

$M \gt 0$

$\beta \gt 0$

$$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$

$\beta \gt 1$

$$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$

$\beta$

$\beta = 1$

We study the hole-filling problem for the porous medium equation $u_t= \frac{1}{m} \UDelta u^m$ with $m>1$ in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution $u=t^{2\beta-1}F(|x|t^{-\beta})$, and we establish that the self-similarity exponent $\beta$ is a monotone function of the parameter $m$. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any $m>1$ against perturbations with 2-fold symmetry. In addition, we prove that as $m$ is varied there are bifurcations from the radial solution to self-similar solutions with $k$-fold symmetry for each $k=3,4,5,\dots.$ These bifurcations are simple and occur at values $m_3>m_4>m_5> \cdots\to1$.

The stationary flow of a jet of a Newtonian fluid that is drawn by gravity onto a moving surface is analyzed. It is assumed that the jet has a convex shape and hits the moving surface tangentially. The flow is modelled by a third-order ODE on a domain of unknown length and with an additional integral condition. By solving part of the equation explicitly, the problem is reformulated as a first-order ODE with an integral constraint. The corresponding existence region in the three-dimensional parameter space is characterized in terms of an easily calculable quantity. In a qualitative sense, the results from the model are found to correspond with experimental observations.

In this paper, we use formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau–Lifshitz–Gilbert model for ferromagnets. We also analyse both the harmonic map heatflow and Schrödinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic.

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.

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.

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.