Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.

The coupling of the Stefan equation for the heat flow with the Gibbs–Thomson law relating the melting temperature to the mean curvature of the phase interface is considered. Solutions, global in time, are constructed which satisfy the natural a priori estimates. Mathematically the main difficulty is to prove a certain regularity in time for the temperature and the indicator function of the phase separately. A capacity type estimate is used to give an L1 bound for fractional time derivatives.

In this paper we consider the numerical analysis of a parabolic variational inequality arising from a deep quench limit of a model for phase separation in a binary mixture due to Cahn and Hilliard. Stability, convergence and error bounds for a finite element approximation are proven. Numerical simulations in one and two space dimensions are presented.

An optimization problem for the fundamental eigenvalue $\lam_0$ of the Laplacian in a planar simply-connected domain that contains $N$ small identically-shaped holes, each of radius $\eps\ll 1$, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue $\lam_0$ is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains $N$ small traps. For small hole radii $\eps$, a two-term asymptotic expansion for $\lam_0$ is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations $x_{i}$, for $i=1,\ldots,N$, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize $\lam_0$. For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes $\lam_0$. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize $\lam_0$. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer–Meinhardt reaction-diffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg–Landau theory of superconductivity.

For a large class of vorticities we prove that a steady periodic deep-water wave must be symmetric if its profile is monotone between crests and troughs.

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

Decades of research demonstrate that crime is concentrated at a range of spatial scales. Such findings have clear implications for crime forecasting and police resource allocation models. More recent work has also shown that crime clusters in space and time with a regularity that might improve methods of crime prediction. In this paper I review some of the available evidence and provide illustrations of the types of analysis – spatial and spatio-temporal – conducted hitherto. With a few exceptions, the application of formal Mathematics in the study of space–time patterns of crime has been rather limited, and so a central aim of the paper is to stimulate interest in this area of research.

Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst–Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current–voltage (I–V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numerical I–V curve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).

We present several algorithms, executable in a finite number of steps, which have been implemented in the symbolic language maple. The standard form algorithm reduces a system of PDEs to a simplified standard form which has all of its integrability conditions satisfied (i.e. is involutive). The initial data algorithm uses a system's standard form to calculate a set of initial data that uniquely determines a local solution to the system without needing to solve the system. The number of arbitrary constants and arbitrary functions in the general solution to the system is directly calculable from this set. The taylor algorithm uses a system's standard form and initial data set to determine the Taylor series expansion of its solution about any point to any given finite degree. All systems of linear PDEs and many systems of nonlinear PDEs can be reduced to standard form in a finite number of steps. Our algorithms have simple geometric interpretations which are illustrated through the use of diagrams. The standard form algorithm is generally more efficient than the classical methods due to Janet and Cartan for reducing systems of PDEs to involutive form.

We consider the non-local problem

which models the temperature when an electric current flows through a material with temperature dependent electrical resistivity f(u) > 0, subject to a fixed potential difference. It is found that for some special cases where f is decreasing and

so the problem can be scaled to make

then:(a) for λ < 8 there is a unique steady state which is globally asymptotically stable: (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, – 1 < x < 1.

In this paper we consider a non-local anisotropic model for phase separation in two-phase fluids at equilibrium, and show that when the thickness of the interface tends to zero in a suitable way, the classical surface tension model is recovered. Relevant examples are given by continuum limits of ferromagnetic Ising systems in equilibrium statistical mechanics.

We prove short-time well-posedness of a Hele–Shaw system with two fluids and no surface tension (this is also known as the Muskat problem). We restrict our attention here to the stable case of the problem. That is, in order for the motion to be well-posed, the initial data must satisfy a sign condition which is a generalization of a condition of Saffman and Taylor. This sign condition essentially means that the more viscous fluid must displace the less viscous fluid. The proof uses the formulation introduced in the numerical work of Hou, Lowengrub, and Shelley, and relies on energy methods.

For the Keller–Segel model, it was conjectured by Childress and Percus (1984, Chemotactic collapse in two dimensions. In Lecture Notes in Biomath. Vol. 55, Springer, Berlin-Heidelberg-New York, 1984, pp. 61–66) that in a two-dimensional domain there exists a critical number C such that if the initial mass is strictly less than C, then the solution exists globally in time and if it is strictly larger than C blowup happens. For different versions of the Keller–Segel model, the conjecture has essentially been proved. The case of several chemotactic species introduces an additional question: What is the analogue for the critical mass obtained for the single species system? In this paper, we investigate for a two-species model for chemotaxis in 2 the conditions on the initial data, which determine blowup or global existence in time. Specifically, we find a curve in the plane of masses such that outside of it there is blowup and inside of it global existence in time is proved when the initial masses satisfy a threshold condition. Optimality of this condition is discussed through an analysis in the radial case. Finally, we show in the case of blowup for general data how it is possible to obtain a balance between entropies and prove what species should aggregates first.

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

A model describing the evolving shape of a growing pile is considered, and is shown to be equivalent to an evolutionary quasi-variational inequality. If the support surface has no steep slopes, the inequality becomes a variational one. For this case existence and uniqueness of the solution are proved.

We consider a competition–diffusion system and study its singular limit as the interspecific competition rate tend to infinity. We prove the convergence to a Stefan problem with zero latent heat.

We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol.67 1457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

Leading-order equations governing the dynamics of a two-dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill-posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length-and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalized to give new models for fully three-dimensional sheets.

We consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach, together with a convective term due to gravity, results in a pseudo-parabolic Burgers type equation. We give a rigorous study of global travelling-wave solutions, with emphasis on the role played by the dynamic term and the appearance of fronts.