We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.

In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn–Hilliard-Reaction model with elasticity in the framework of the Cahn–Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in lithium ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface intersects the free boundary of the Si layer. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.

In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of solutions near numerical approximations. Coexistence of as many as seven nontrivial solutions is proved.

In this paper we establish an existence and uniqueness result for a class of non-local elliptic differential equations with the Dirichlet boundary conditions, which, in general, do not accept a maximum principle. We introduce one monotone sequence of lower–upper pairs of solutions and prove uniform convergence of that sequence to the actual solution of the problem, which is the unique solution for some range of λ (the parameter of the problem). The convergence of the iterative sequence is tested through examples with an order of convergence greater than 1.

We consider a model for the dynamics of growing cell populations with heterogeneous mobility and proliferation rate. The cell phenotypic state is described by a continuous structuring variable and the evolution of the local cell population density function (i.e. the cell phenotypic distribution at each spatial position) is governed by a non-local advection–reaction–diffusion equation. We report on the results of numerical simulations showing that, in the case where the cell mobility is bounded, compactly supported travelling fronts emerge. More mobile phenotypic variants occupy the front edge, whereas more proliferative phenotypic variants are selected at the back of the front. In order to explain such numerical results, we carry out formal asymptotic analysis of the model equation using a Hamilton–Jacobi approach. In summary, we show that the locally dominant phenotypic trait (i.e. the maximum point of the local cell population density function along the phenotypic dimension) satisfies a generalised Burgers’ equation with source term, we construct travelling-front solutions of such transport equation and characterise the corresponding minimal speed. Moreover, we show that, when the cell mobility is unbounded, front edge acceleration and formation of stretching fronts may occur. We briefly discuss the implications of our results in the context of glioma growth.

The present paper is devoted to the study of the existence, the uniqueness and the stability of transition fronts of non-local dispersal equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space non-increasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space non-increasing transition front (if it exists), which implies the uniqueness, up-to-space shifts and monotonicity of transition fronts provided that a space non-increasing transition front exists. Moreover, we show that a transition front must be a periodic travelling front in periodic media and asymptotic speeds of transition fronts exist in uniquely ergodic media. Finally, we prove the existence of space non-increasing transition fronts, whose proof does not need the unbalanced condition.

In this paper we consider a 2π-periodic and two-dimensional Hele-Shaw flow describing the motion of a viscous, incompressible fluid. The free surface is moving under the influence of surface tension and gravity. The motion of the fluid is modelled using a modified version of Darcy's law for Stokesian fluids. The bottom of the cell is assumed to be impermeable. We prove the existence of a unique classical solution for a domain which is a small perturbation of a cylinder. Moreover, we identify the equilibria of the flow and study their stability.

This paper is concerned with the stability issue in determining absorption and diffusion coefficients in photoacoustic imaging. Assuming that the medium is layered and the acoustic wave speed is known, we derive global Hölder stability estimates of the photoacoustic inversion. These results show that the reconstruction is stable in the region close to the optical illumination source, and deteriorate exponentially far away. Several experimental pointed out that the resolution depth of the photoacoustic modality is about tens of millimeters. Our stability estimates confirm these observations and give a rigorous quantification of this depth resolution.

This work studies mathematical issues associated with steady-state modelling of diffusion- reaction-conduction processes in an electrolyte wedge (meniscus corner) of a current-producing porous electrode. The discussion is applicable to various electrodes where the rate-determining reaction occurs at the electrolyte-solid interface; molten carbonate fuel cell cathodes are used as a specific example. New modelling in terms of component potentials (linear combinations of electrochemical potentials) is shown to be consistent with tradition concentration modelling. The current density is proved to be finite, and asymptotic expressions for both current density and total current are derived for suffciently small contact angles. Finally, numerical and asymptotic examples are presented to illustrate the strengths and weaknesses of these expressions.

We prove the uniqueness for the inverse problem of determining a coefficient $q(x)$ in $\partial _t^2 u(x,t) = \uDelta u(x,t) - q(x)u(x,t)$ for $x \in R^n$ and $t > 0$, from observations of $u\vert_{\Gamma\times(0,T)}$ and the normal derivative $\frac{\partial u}{\partial \nu}\vert_{\Gamma\times(0,T)}$ where $\Gamma$ is an arbitrary $C^{\infty}$-hypersurface. Our main result asserts the uniqueness of $q$ over $R^n$ provided that $T > 0$ is sufficiently large and $q$ is analytic near $\Gamma$ and outside a ball. The proof depends on Fritz John's global Holmgren theorem and the uniqueness by a Carleman estimate.

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 study a free boundary problem for the Fisher-KPP equation: ut = uxx + f(u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) − α and g′(t) = −ux(t, g(t)) + β for 0 < β < α. This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We investigate the asymptotic behaviour of bounded solutions. There are two parameters α0 and α* with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β < α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, i.e., h(t) − g(t) → +∞ and u(t, ⋅ + ct) → 1 with c ∈ (cL, cR), where cL and cR are the asymptotic spreading speed of g(t) and h(t), respectively, (cR > 0 > cL when 0 < β < α < α0; cR = 0 >cL when 0 < β < α = α0; 0 > cR > cL when α0 < α < α* and 0 < β < α0); (i-2) vanishing, i.e., limt→Th(t) = limt→Tg(t) and limt→T u(t, x) = 0, where T is some positive constant; (i-3) transition, i.e., g(t) → −∞, h(t) → −∞, 0 < limt→∞[h(t) − g(t)] < +∞ and u(t, x) → V*(x − c*t) with c* < 0, where V*(x − c*t) is a travelling wave with compact support and which satisfies the free boundary conditions. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

In this paper, we study the nonlinear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation of this equation, which is also related to the supercooled Stefan problem, as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

We present an asymptotic theory that includes in a perturbative expansion the coupling effects between the director dynamics and the velocity field in a nematic liquid crystal. Backflow effects are most significant in the presence of defect motion, since in this case the presence of a velocity field may strongly reduce the total energy dissipation and thus increase the defect velocity. As an example, we illustrate how backflow influences the speeds of opposite-charged defects.

An asymptotic reduction of the Gierer–Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity and small activator diffusivity ε leads to a singularly perturbed nonlocal reaction-diffusion equation for the activator concentration. In the limit ε → 0, this nonlocal problem for the activator concentration has localized spike-type solutions. In this limit, we analyze the motion of a spike that is confined to the smooth boundary of a two or three-dimensional domain. By deriving asymptotic differential equations for the spike motion, it is shown that the spike moves towards a local maximum of the curvature in two dimensions and a local maximum of the mean curvature in three dimensions. The motion of a spike on a flat segment of a two-dimensional domain is also analyzed, and this motion is found to be metastable. The critical feature that allows for the slow boundary spike motion is the presence of the nonlocal term in the underlying reaction-diffusion equation.

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

For superconductors of type II the phenomenon of vortex pinning plays an important role in technological applications. Several models have been proposed for this effect (Kim et al., 1963; Bean, 1964; Bossavit, 1994). In Du et al. (1999) and Prigozhin (1996), some of these models are analyzed. In this work we want to contribute to the analysis for the two-dimensional, rate-independent model proposed in Chapman (2000), which has the special feature that vortex movement and creation is an activated process occurring only when a threshold value of the magnetic field is reached. For analytical studies of related rate-dependent models we refer to Chapman et al. (1996), Schätzle & Styles (1999) and Elliott & Styles (2000).