We prove the existence of a one-parameter family of solutions of the porous medium equation in which the interface is a half line whose end point advances at a constant speed. Then we prove the stability of the solutions under a suitable class of perturbations. We discuss the relevance of these solutions to gravity-driven flows of thin films, and show that some solutions develop a very thin triangular plateau in the direction of propagation and that the angle of the plateau and its thickness are decreasing functions of the speed.

To study electrostatic actuation, researchers commonly use a setup proposed by G. I. Taylor in [Proc. R. Soc. Lond. Ser. A, 306 (1968), pp. 423–434]. It consists of soap film held at a distance h above a rigid plate so that when a voltage difference is applied between the two components, the top film deflects towards the bottom plate. The most striking feature of this system is when the voltage difference exceeds a critical value V*, the electrostatic forces dominate the surface forces and the soap film gets ‘pulled-into’ or collapses onto the bottom plate. This so-called ‘pull-in’ instability is a ubiquitous feature of electrostatic actuation and as a result, has been the subject of many studies. Recently, Siddique et al. [J. Electrostatics, 69 (2011), pp. 1–6] measured the value of V* as a function of the separation distance and found that the standard prediction breaks down as h increases. Here, we continue the work done in [N. D. Brubaker and J. A. Pelesko, European J. Appl. Math., 22 (2011), pp. 455–470] by investigating the cause of this discrepancy. Specifically, we model the effect of gravity on the generalized version of Taylor's model and study whether it provides the proper correction to the predicted value of V*. In doing so, we derive two nonlinear eigenvalue value problems and investigate their solutions sets.

We consider an anisotropic diffusion equation of the form ut = ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

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.

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.

In this paper a four-dimensional macroeconomic model of a small open economy, describing the development of income, capital stock, interest rate and money stock, which was constructed in [5] (Makovínyiová, K. & Zimka, R. (2009) On stability in generalized Schinasi's macroeconomic model under fixed exchange rates. Tatra Mt. Math. Publ. 43, 115–122), is analysed. Sufficient conditions for the existence of one pair of purely imaginary eigenvalues and two eigenvalues with negative real parts in the linear approximation matrix of the model are found. Formulae for the calculation of the bifurcation coefficients of the model are derived. A statement about the existence of limit cycles is made. A numerical example is given illustrating the results.