We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$

$u=1$

$D$

$\Delta _1 \approx 0.00297$

$D \ll 1$

$O(1)$

$u=O(1)$

$u$

$D \to 0^+$

$D \geq \Delta _1$

$2 \sqrt{D}$

$0 \lt D \lt \Delta _1$

$u=0$

$D$

The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.

Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimisation to seek trap arrangements that minimise average escape times. In a recent paper (Iyaniwura (2021) SIAM Rev. 63(3), 525–555), an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps in elliptic domains using global optimisation techniques and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. Further, an asymptotic formula for the average MFPT in elliptic domains with N circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed.

We study the asymptotic behaviour of the periodically mixed Zaremba problem. We cover the part of the boundary by a chess board with a small period (square size) $\varepsilon$ and impose the Dirichlet condition on black and the Neumann condition on white squares. As $\varepsilon \to 0$, we get the effective boundary condition which is always of the Dirichlet type. The Dirichlet data on the boundary, however, depend on the ratio between the magnitudes of the two boundary values.

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns. 21 (1996), 363–394); Grenier (Commun. Pure Appl. Math. 50 (1997), 821–865); Ukai (J. Math. Kyoto Univ. 26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity 24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]

We consider the following inhomogeneous problems

\[ \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \text{ on }\partial \Omega,\\ \end{cases} \]

$\Omega$

$\mathbb {R}^{N}$

$\epsilon >0$

$a$

$L^{1}$

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

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.

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.

We analyse oscillatory instabilities for a coupled partial-ordinary differential equation (PDE-ODE) system modelling the communication between localised spatially segregated dynamically active signalling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2D domain. Each signalling compartment is assumed to secrete a chemical into the extracellular medium (bulk region), and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signalling compartments are circular discs with a small common radius ɛ ≪ 1 and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE–ODE system into a nonlinear ODE system with global coupling. For Sel’kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field are then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronisation of the intracellular dynamics. Quorum sensing behaviour, characterised by the onset of collective behaviour in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

We are concerned with the semi-classical states for the Choquard equation

$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$

In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.

We consider Fokker–Planck equations with tilted periodic potential in the subcritical regime and characterise the spatio-temporal dynamics of the partial masses in the limit of vanishing diffusion. Our convergence proof relies on suitably defined substitute masses and bounds the approximation error using the energy-dissipation relation of the underlying Wasserstein gradient structure. In the appendix, we also discuss the case of an asymmetric double-well potential and derive the corresponding limit dynamics in an elementary way.

We consider two-dimensional mass transport to a finite absorbing strip in a uniform shear flow as a model of surface-based biosensors. The quantity of interest is the Sherwood number Sh, namely the dimensionless net flux onto the strip. Considering early-time absorption, it is a function of the Péclet number Pe and the Damköhler number Da, which, respectively, represent the characteristic magnitude of advection and reaction relative to diffusion. With a view towards modelling nanoscale biosensors, we consider the limit Pe«1. This singular limit is handled using matched asymptotic expansions, with an inner region on the scale of the strip, where mass transport is diffusively dominated, and an outer region at distances that scale as Pe-1/2, where advection enters the dominant balance. At the inner region, the mass concentration possesses a point-sink behaviour at large distances, proportional to Sh. A rescaled concentration, normalised using that number, thus possesses a universal logarithmic divergence; its leading-order correction represents a uniform background concentration. At the outer region, where advection by the shear flow enters the leading-order balance, the strip appears as a point singularity. Asymptotic matching with the concentration field in that region provides the Sherwood number as

$${\rm{Sh}} = {\pi \over {\ln (2/{\rm{P}}{{\rm{e}}^{1/2}}) + 1.0559 + \beta }},$$

In the present paper, we consider the nonlocal Kirchhoff problem

$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$

$\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.

The equation –ε2Δu + F(V(x), u) = 0 is studied over all of ℝn, and solutions are constructed that concentrate at an infinite set as ε → 0. The function V (x) is vector valued. This advances previous studies in which V (x) was scalar valued.

In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundary-value problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli–Kohn–Nirenberg exponent.