For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.

In this article, we study the higher-order Moreau's sweeping process introduced in [1], in the case where an exogenous time-varying function u(·) is present in both the linear dynamics and in the unilateral constraints. First, we show that the well-posedness results (existence and uniqueness of solutions) obtained in [1] for the autonomous case, extend to the non-autonomous case when u(·) is smooth and piece-wise analytic, after a suitable state transformation is done. Stability issues are discussed. The complexity of such non-smooth non-autonomous dynamical systems is illustrated in a particular case named the higher-order bouncing ball, where trajectories with accumulations of jumps are exhibited. Examples from mechanics and circuits illustrate some of the results. The link with complementarity dynamical systems and with switching differential-algebraic equations is made.

We study the analyticity properties of solutions for a class of non-linear evolutionary pseudo-differential equations possessing global attractors. In order to do this we utilise an analyticity criterion for spatially periodic functions, which involves the rate of growth of a suitable norm of the nth derivative of the solution, with respect to the spatial variable, as n tends to infinity. This criterion can be used to a wide class of dissipative-dispersive partial differential equations, provided they possess global attractors. Using this criterion and the spectral method developed in Akrivis et al. [1] we have improved previous results.

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

The onset of shear-banding in a deforming elastoplastic solid has been linked to change of type of the governing partial differential equations. If uniform material properties are assumed, then (i) deformations prior to shear-banding are uniform, and (ii) the onset of shear-banding occurs simultaneously at all points in the sample. In this paper we study, in the context of a model for anti-plane shearing of a granular material, the effect of a small variation in material properties (e.g. in yield strength) within the sample. Using matched asymptotic expansions, we find that (i) the deformation is extremely non-uniform in a short time period immediately preceding the formation of shear-bands; and (ii) generically, a shear-band forms at a single location in the sample.

We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn–Hilliard functional coupled with a small deformation energy. We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen–Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long-time behaviour raft-like structures can emerge and stabilise, and their parameter dependence is further explored.

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An asymptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae have been obtained for the first six eigen-frequencies.

Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealth but benefit from it, have always existed, and are likely to be present in all societies in the foreseeable future. Their existence brings about serious problems since they act as sinks for the community wealth and deplete resources which are always limited and often scarce.

To fight cheaters, a society can invest additional resources to pursue one or several aims. For instance, an improvement in social solidarity (e.g. by fostering education) may be sought. Alternatively, deterrence (e.g. by increasing police budget) may be enhanced. Then the following questions naturally arise: (i) how much to spend and (ii) how to allocate the expenditure between both strategies above. This paper addresses this general issue in a simplified setting, which however we believe of some interest. More precisely, we consider a society constituted by two productive classes and an unproductive one, the cheaters, and proposes a dynamical system that describes their evolution in time. We find it convenient to formulate our model as a three-dimensional ordinary differential equation (ODE) system whose variables are the cheater population, the total wealth and one of the productive social classes. The stationary values of the cheater population and the total wealth are studied in terms of the two parameters: φ (how much to invest) and s (how to distribute such expenditure). We show that it is not possible to simultaneously minimize the cheater population and maximize the total wealth with respect to φ and s. We then discuss the possibility of defining a compromise function to find suitable values of φ and s that optimize the response to cheating. In our opinion, this qualitative approach may be of some help to plan and implement social strategies against cheating.

In the paper, Bifurcation analysis of the twist-Fréedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions (2009 Eur. J. Appl. Math.20, 269–287) by da Costa et al. the twist-Fréedericksz transition in a nematic liquid-crystal one-dimensional cell of lenght L was studied, imposing an antisymmetric net twist Dirichlet condition at the cell boundaries. In the present paper, we extend that study to the more general case of net twist Dirichlet conditions without any kind of symmetry restrictions. We use phase-plane analysis tools and appropriately defined time maps to obtain the bifurcation diagrams of the model when L is the bifurcation parameter, and related these diagrams with the one in the antisymmetric situation. The stability of the bifurcating solutions is investigated by applying the method of Maginu (1978 J. Math. Anal. Appl.63, 224–243).

We model larviciding of mosquitoes taking into account the evolution of resistance to the larvicides, the evolutionary costs of resistance and the implications for malaria control. There is evidence that resistance comes with various costs, one of which is reduced adult longevity for resistant mutants. The mosquito adult lifespan is one of the most crucial parameters in malaria transmission due to a long developmental time for the malaria parasite in the insect. A possible malaria control strategy is therefore to shorten this adult lifespan by larviciding with a potent larvicide to which mosquitoes become resistant. This novel strategy is studied using a mathematical model for the wild type and resistant mutants and by incorporating the malaria disease dynamics using an SEI type model with standard incidence that incorporates the latency period of the parasite in wild-type and resistant mosquitoes. We consider the linear stability of the malaria-free equilibrium in which the resistant strain is dominant and derive a condition for the global eradication of malaria. Numerical simulations are presented, which offer further insights. The parameter to which the analysis is most sensitive is the per-capita death rate of resistant adult mosquitoes. Increasing this parameter dramatically reduces the basic reproduction number R0. However, increasing it too much causes the wild type to outcompete the resistant mutants, and the control strategy fails. Exploitation of costs of resistance to larvicides thus offers a possible malaria control measure if the larvicide is sufficiently potent and costs of resistance are neither too great nor too small.

A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is not allowed to revisit a node in one step. Second-order random walks can model physical diffusion phenomena in a more realistic way than traditional random walks and have been very successfully used in various network mining and machine learning settings. However, numerous questions are still open for this type of stochastic processes. In this work, we extend well-known results concerning mean hitting and return times of standard random walks to the second-order case. In particular, we provide simple formulas that allow us to compute these numbers by solving suitable systems of linear equations. Moreover, by introducing the ‘pullback’ first-order stochastic process of a second-order random walk, we provide second-order versions of the renowned Kac’s and Random Target Lemmas.

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

We consider the two-phase one-dimensional Stefan problem in a finite interval, with initial and boundary conditions such that the solid phase vanishes at a finite time T and at a single point. We show that the temperature in the solid phase decreases to zero and is bounded by c exp (α/(t − T)) as extinction approaches (C, α > 0) and that phase boundaries at extinction have finite speeds.

When ceramics are heated inside a microwave cavity, a well-known phenomenon is the occurrence of hot spots – localised regions of high temperature. This phenomenon was modelled by Kriegsmann ((1997), IMA J. Appl. Math. 59(2), pp. 123–146; (2001), IMA J. Appl. Math. 66(1), pp. 1–32) using a non-local evolution PDE. We investigate profile and the stability of hot spots in one and two dimensions by using Kriegsmann's model with exponential non-linearity. The linearised problem associated with hot-spot-type solutions possesses two classes of eigenvalues. The first type is the large eigenvalues associated with the stability of the hot-spot profile and in this particular model there cannot be instability associated with these eigenvalues. The second type is the small eigenvalues associated with translation invariance. We show that the hot spots can become unstable due to the presence of small eigenvalues, and we characterise the instability thresholds. In particular, we show that for the material with low heat conductivity (such as ceramics), and in the presence of a variable electric field, the hot spots are typically stable inside a plate (in two dimensions) but can become unstable for a slab (in one dimension) provided that the microwave power is sufficiently large. On the other hand, for materials with high heat conductivity, the interior hot spots are unstable and move to the boundary of the domain in either one or two dimensions. For materials with moderate heat conductivity, the stability of hot spots is determined by both the geometry and the electric field inside the microwave cavity.

In this paper, we study a quasi-static frictional contact problem for a viscoelastic body with damage effect inside the body as well as normal compliance condition and multi-valued friction law on the contact boundary. The considered friction law generalizes Coulomb friction condition into multi-valued setting. The variational–hemi-variational formulation of the problem is derived and arguments of fixed point theory and surjectivity results for pseudo-monotone operators are applied, in order to prove the existence and uniqueness of solution.

An unforced oscillator which obeys the equation x¨+α (x2+x˙2−1)x˙+x=0 has a pure sine-wave limit cycle which is attained rapidly for a range of values of α. In this paper, free and forced oscillation of this equation are examined experimentally and compared with those for the Van der Pol oscillator. Asymptotic solutions for small α confirm the experimental results.

We consider a state-dependent GI/G/1 queueing system characterized by the unfinished work U(t) in the system at time t. We introduce state-dependence by allowing (i) the arrival process to depend on the instantaneous value of U(t), (ii) the service rate, that is, the rate at which U(t) decreases in the absence of arrivals, to depend on U(t), and (iii) the customer's service requirement to depend on U(t*) where t* denotes the instant in which that customer entered the system. We consider the limit of short inter-arrival times and small service requests and compute asymptotic approximations to the stationary density of the unfinished work, including the stationary probability of finding the system empty, using the WKB method and the method of matched asymptotic expansions.

The static compression between two smooth plates of an axisymmetric capsule or vesicle is investigated by means of asymptotic analysis. The governing equations of the vesicle are derived from thin-shell theory and involve a bending stiffness B, a shear modulus H, the unstressed vesicle radius a and a constant surface-area constraint. The sixth-order free-boundary problem obtained by a balance-of-forces approach is addressed in the limit when the dimensionless parameter C = Ha2/B is large and the plate displacements are small. When the plate displacement is of order aC−1/2, the vesicle undergoes a sub-critical buckling instability which is captured by leading-order asymptotics. Asymptotic linear and quadratic force–displacement relations for the pre- and post-buckled solutions are determined. The leading-order post-buckled solution is described by a simple fourth-order problem, exhibiting stress-focusing with stretching and bending confined to a narrow boundary layer. In contrast, in the pre-buckled state, stretching occurs over a larger length scale than bending. The results are in good qualitative agreement with numerical simulations for finite values of C.

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.