We study the HN stability of the Vlasov–Poisson Boltzmann system near Maxwellians. Under a suitable smallness assumption on initial data, we show that the global classical solutions constructed by Guo are HN stable. For a stability estimate, we employ the energy methods of Guo.

We study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along an environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparison to that of the random dispersal, we show that the competitor that moves towards more favourable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that, for sufficiently strong advection, the ‘smarter' competitor will move towards more favourable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive, since it can live upon regions with finer quality resources.

We prove that the integral of the product of two functions over a symmetric set in $\mathbb{S}^1\times\mathbb{S}^1$, defined as $E=\{(x,y)\in\mathbb{S}^1\times\mathbb{S}^1:d(\sigma_1(x),\sigma_2(y))\leq\alpha\}$ (where $\sigma_1$, $\sigma_2$ are diffeomorphisms of $\mathbb{S}^1$ with certain properties and $d$ is the geodesic distance on $\mathbb{S}^1$), increases when we pass to their symmetric decreasing rearrangement. We also give a characterization of the diffeomorphisms $\sigma_1$, $\sigma_2$ for which the rearrangement inequality holds. As a consequence, we obtain the result for the integral of the function $\varPsi(f(x),g(y))$ (where $\varPsi$ is a supermodular function) with a kernel given as $k[d(\sigma_1(x),\sigma_2(y))]$, with $k$ decreasing.

This paper deals with singular semilinear elliptic equations in bounded domains with Dirichlet boundary data. The elliptic operator is a second-order operator not necessarily in divergence form. We consider existence, uniqueness and linearized stability of positive solutions for a series of nonlinear eigenvalue problems.

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. We solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular, we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore, we construct examples in which the genericity assumption does not hold and indicate that the situation is different for these.

To prove that certain standard 2-complexes are aspherical we explore a strategy that combines a well-known method based on a graph theoretic lemma of Stallings with a process of reversing the orientation of edges in spherical diagrams. We apply this strategy to labelled-oriented-tree complexes and, more generally, to labelled-oriented-graph (LOG) complexes and obtain classes of aspherical LOG complexes to which, traditionally (without the reversing of edges), Stallings's lemma could not be applied to prove asphericity. These classes contain examples whose asphericity, as far as we know, could not be established by any previous method.

We study smooth mappings with patterns which given by certain divergence diagrams of smooth mappings. The divergent diagrams of smooth mappings can be regard as smooth mappings from manifolds with singular foliations. Our concerns are generic differential topology and generic smooth mappings with patterns. We give a generic semi-local classification of surfaces with singularities and patterns as an application of singularity theory.

A new method for asymptotic summation of linear systems of difference equations is proposed and studied. It is based on the introduction of a certain summation equation that pinpoints sufficient conditions for asymptotic summation. These conditions serve as a framework from which new and old theorems follow. In particular the analogues of the fundamental theorems of Levinson and Hartman and Wintner are shown to follow from one and the same framework. Examples are given that are not amenable to other techniques.

Recent studies of lattice dynamics, with respect to the existence of global compact attractors for certain discrete evolution equations, are based on the derivation of ‘tail estimates of the solution’. We show that, due to the specific nature of the discrete nonlinear Schrödinger (DNLS) system which gives rise to a simple energy equation, the method developed by J. M. Ball is applicable, and provides an alternative proof on the existence of the compactness of the attractor for the weakly damped and driven DNLS equation considered in $\mathbb{Z}^N$, $N\geq1$, lattices, without the usage of tail estimates. The approach covers various DNLS-type equations of physical significance.

We use the method of the moving plane (MMP) to obtain necessary and sufficient conditions for the radial symmetry of positive solutions of the following semi-linear elliptic equation with singular nonlinearity:

$$ \Delta u-\frac{1}{u^\nu}=0\quad\text{in~}\mathbb{R}^n,\quad n\geq2, $$

where $\nu>0$. In order to apply the MMP, it is crucial to obtain the asymptotic expansion of $u$ at $\infty$.

We study the right-definite separated half-linear Sturm–Liouville eigenvalue problems. It is proved that the $n$th real eigenvalue of the problem depends smoothly on the equation, but may have jump discontinuities with respect to the boundary condition. Formulae are found for the derivatives of the $n$th real eigenvalue with respect to all parameters: the endpoints, the boundary condition and the coefficient functions, whenever they exist. Monotone properties and a comparison result for real eigenvalues are deduced as consequences. The generalized Prüfer transformation and the implicit function theorem in Banach spaces play key roles in the proofs.

The action of the symmetric group $S_4$ on the tetrahedron algebra, introduced by Hartwig and Terwilliger, is studied. This action gives a grading of the algebra which is related to its decomposition into a direct sum of three subalgebras isomorphic to the Onsager algebra. The ideals of both the tetrahedron algebra and the Onsager algebra are determined.

Let $D$ be a bounded, finitely connected domain in $\mathbb{C}$ without isolated points in the boundary and let $f$ be a continuous function on $bD$. Let $\tilde{f}$ be a continuous extension of $f$ to $\bar{D}$. We prove that $f$ extends holomorphically through $D$ if and only if the degree of $\tilde{f}+h$ is non-negative for every holomorphic function $h$ on $D$ such that $\tilde{f}+h$ is bounded away from $0$ near $bD$.

We investigate the asymptotic behaviour as $k\to+\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\varOmega)$ of solutions of the equations $\Delta^2u_k=V_k\mathrm{e}^{4u_k}$ on $\varOmega$, where $\varOmega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to+\infty}V_k=1$ in $C^0_{\mathrm{loc}}(\varOmega)$. The corresponding two-dimensional problem was studied by Brézis and Merle and Li and Shafrir, who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Robert and Struwe in 2006, such a quantization does not hold in dimension $4$ for the problem in its full generality. We prove here that, under a natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension $2$.

This paper is devoted to a spectral description of wave propagation phenomena in conservative unbounded media, or, more precisely, the fact that a time-dependent wave can often be represented by a continuous superposition of time-harmonic waves. We are concerned here with the question of the perturbation of such a generalized eigenfunction expansion in the context of scattering problems: if such a property holds for a free situation, i.e. an unperturbed propagative medium, what does it become under perturbation, i.e. in the presence of scatterers? The question has been widely studied in many particular situations. The aim of this paper is to collect some of them in an abstract framework and exhibit sufficient conditions for a perturbation result. We investigate the physical meaning of these conditions which essentially consist in, on the one hand, a stable limiting absorption principle for the free problem, and on the other hand, a compactness (or short-range) property of the perturbed problem.

This approach is illustrated by the scattering of linear water waves by a floating body. The above properties are obtained with the help of integral representations, which allow us to deduce the asymptotic behaviour of time-harmonic waves from that of the Green function of the free problem. The results are not new: the main improvement lies in the structure of the proof, which clearly distinguishes the properties related to the free problem from those which involve the perturbation.

We study the existence of multiple solutions for the problem

and we show that at least one of them is sign changing. Here Ω is a bounded domain in ℝN with N ⩾ 5 whose boundary is of class C2, ∂/∂ν is the outward normal derivative, and f ∈ LN/2(Ω) whose L2N/(N+2)(Ω) norm is sufficiently small.

We investigate the existence and properties of the Jost solution associated with the differential equation $-y''+q(x)y=\lambda y$, $x\geq0$, for a class of real- or complex-valued slowly decaying potentials $q$. In particular, it is shown how the traditional condition $q\in L(\mathbb{R}^{+})$ for the existence of the Jost solution can be replaced by $q'\in L(\mathbb{R}^{+})$ for a class of potentials considered here. We also examine the asymptotics of the Titchmarsh–Weyl function for a class of real- or complex-valued slowly decaying potentials and the form of the spectral density for a class of real-valued slowly decaying potentials.

In a real Hilbert space $H$, we study the bifurcation points of equations of the form $F(\lambda,u)=0$, where $F:\mathbb{R}\times H\rightarrow H$ is a function with $F(\lambda,0)=0$ that is Hadamard differentiable, but not necessarily Fréchet differentiable, with respect to $u$ at $u=0$. In this context, there may be bifurcation at points $\lambda$ where $D_{u} F(\lambda,0):H\rightarrow H$ is an isomorphism. We formulate some additional conditions on $F$ that ensure that bifurcation does not occur at a point where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. Then, in the case where $F(\lambda,\cdot)$ is a gradient, we give conditions that imply that bifurcation occurs at a point $\lambda$. These conditions may be satisfied at points where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. We demonstrate the use of these abstract results in the context of nonlinear elliptic equations of the form

$$ -\Delta u(x)+q(x)u(x)=\lambda\eta(x)^{-1}f(\eta(x)u(x)) $$

and

$$ -\Delta u(x)+q(x)u(x)-\eta(x)^{-1}f(\eta(x)u(x))=\lambda u(x), $$

where $u\in H^{2}(\mathbb{R}^{N})$.

The main goal of this work is to study the existence and uniqueness of a positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.

The steady-state diffusion problem is considered in a thin plate perforated periodically by many cylindrical holes of critical sizes. The plate is scaled to a plate of thickness 1. The asymptotic behaviour of the solution to the resulting rescaled equation is studied when the thickness of the original plate, the holes' size and period converge to 0. The phenomenon of dimension reduction occurs, i.e the limiting equation is posed in the cross-section only. The equation contains a linear term which describes the sink effect of the holes. This term depends on the relationship between the thickness, the period and the size of the perforations.