This paper deals with the following fractional elliptic equation with critical exponent

\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]

$\lambda$

$\bar {\nu }\in {{\mathfrak R}}$

$s\in (0,1)$

$2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$

$(-\Delta )^{s}$

$\Omega \subset {{\mathfrak R}}^{N}$

$\varphi _{1}$

$u=0$

${{\mathfrak R}}^{N}\setminus \Omega$

$\lambda$

$\bar {\nu }$

$|\bar \nu | \to \infty$

We consider a system of reaction–diffusion equations including chemotaxis terms and coming out of the modelling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.

In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on $L^{2}$-energy method and some time-decay estimates in $L^{p}$-norm for the smoothed rarefaction wave.

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for $u(r,t)$ as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series.

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.

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small \[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in \[{\mathbb{R}^2}\]. The expansion of this principal eigenvalue proceeds in powers of \[\nu\equiv- 1/\log (\varepsilon {d_c})\], where dc is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

We consider the boundary Hardy–Hénon equation

\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]

$B_1(0)\subset \mathbb {R}^{N}$

$(N\geq 3)$

$1$

$0$

$p>0$

$\alpha \in \mathbb {R}$

$0< p<({N+2})/({N-2})$

$\alpha \leq -2$

$p>1$

$\alpha >-2$

$1< p<({N+2})/({N-2})$

$0< p\leq 1$

$\alpha \leq -2$

$0< p<1$

$\alpha >-2$

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.

This work studies functional difference equations of the second order with a potential belonging to a special class of meromorphic functions. The equations depend on a spectral parameter. Consideration of this type of equations is motivated by applications in diffraction theory and by construction of eigenfunctions for the Laplace operator in angular domains. In particular, such eigenfunctions describe eigenoscillations of acoustic waves in angular domains with ‘semitransparent’ boundary conditions. For negative values of the spectral parameter, we study essential and discrete spectrum of the equations and describe properties of the corresponding solutions. The study is based on the reduction of the functional difference equations to integral equations with a symmetric kernel. A sufficient condition is formulated for the potential that ensures existence of the discrete spectrum. The obtained results are applied for studying the behaviour of eigenfunctions for the Laplace operator in adjacent angular domains with the Robin-type boundary conditions on their common boundary. At infinity, the eigenfunctions vanish exponentially as was expected. However, the rate of such decay depends on the observation direction. In particular, in a vicinity of some directions, the regime of decay is switched from one to another and such asymptotic behaviour is described by a Fresnel-type integral.

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

We study the fractional parabolic Lichnerowicz equation

$$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$

$p>0$

$ 0<s<1 $

$p>1$

$0<p\leq 1$

$$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$

$p>0$

$0<s<1$

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝn. We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$, $d\geq 2$, equipped with surface measure $\sigma _{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation

$$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$

$a\in C^\infty (\mathbb {S}^{d-1})$

$q\geq 2(d+1)/(d-1)$

$f\in L^2(\mathbb {S}^{d-1})$

$C^\infty (\mathbb {S}^{d-1})$

$\mathbb {S}^{d-1}$

$C^\infty $

A hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as “traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps.

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

An analysis is undertaken of the formation and stability of localised patterns in a 1D Schanckenberg model, with source terms in both the activator and inhibitor fields. The aim is to illustrate the connection between semi-strong asymptotic analysis and the theory of localised pattern formation within a pinning region created by a subcritical Turing bifurcation. A two-parameter bifurcation diagram of homogeneous, periodic and localised patterns is obtained numerically. A natural asymptotic scaling for semi-strong interaction theory is found where an activator source term \[a = O(\varepsilon )\] and the inhibitor source \[b = O({\varepsilon ^2})\], with ε2 being the diffusion ratio. The theory predicts a fold of spike solutions leading to onset of localised patterns upon increase of b from zero. Non-local eigenvalue arguments show that both branches emanating from the fold are unstable, with the higher intensity branch becoming stable through a Hopf bifurcation as b increases beyond the \[O(\varepsilon )\] regime. All analytical results are found to agree with numerics. In particular, the asymptotic expression for the fold is found to be accurate beyond its region of validity, and its extension into the pinning region is found to form the low b boundary of the so-called homoclinic snaking region. Further numerical results point to both sub and supercritical Hopf bifurcation and novel spikeinsertion dynamics.

Poroelastic effects have been of great interest in the seismic literature as they have been identified as a major cause of wave attenuation in heterogeneous porous media. The observed attenuation in the seismic wave can be explained in part by energy loss to fluid motion in the pores. On the other hand, it is known that the attenuation is particularly pronounced in stratified structures where the scale of spatial heterogeneity is much smaller than the seismic wavelength. Understanding of poroelastic effects on seismic wave attenuation in heterogeneous porous media has largely relied on numerical experiments. In this work, we present a homogenisation technique to obtain an upscaled viscoelastic model that captures seismic wave attenuation when the sub-seismic scale heterogeneity is periodic. The upscaled viscoelastic model directly relates seismic wave attenuation to the material properties of the heterogeneous structure. We verify our upscaled viscoelastic model against a full poroelastic model in numerical experiments. Our homogenisation technique suggests a new approach for solving coupled equations of motion.

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.