We discuss the symmetric solutions of the semilinear elliptic equation Δu + λ(u+ u|u|p−1) = 0, u|∂B = 0 (*), where B is the unit ball in ℝ3. The value of p is taken close to 5, the critical Sobolev exponent for ℝ3. An asymptotic description of the solutions of (*) with large norm is obtained. This predicts a fold bifurcation if p > 5 and the structure of this bifurcation is studied in the limit p – 5→ 0. We find good agreement between the asymptotic description and some numerical calculations. These results are illuminated by recasting the problem (*) in the form of a dynamical system by means of a suitable change of variables. When |p – 5|≪1 and ∥u ≫1, the transformed solutions of (*) are also solutions of a perturbed Hamiltonian system and we study the behaviour of these solutions by using Melnikov methods.

By adapting its well-known proof, the Poincaré–Bendixson theorem, on the existence of periodic orbits of plane autonomous systems, is extended to vector differential equations of the form f(D)x + bφ(g(D)x) = 0. The only restrictions placed on the vector function φ(y) are that its Jacobian matrix should be continuous and lie within a suitably chosen ellintic ball.

The problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) < ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

The use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.

The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

Since the appearance of the article [15] to which this is a sequel, considerable progress has been made in the study of the groups F(r, n) of the title. It is therefore our intention to give a brief account of these developments before proceeding to our main theme, which is to apply the elegant and powerful methods of small cancellation theory to these groups. This has a variety of consequences, perhaps the most important of which is the generalisation to arbitrary r of Lyndon's proof that F(2, n) is infinite for n ≧ 11.

We study entropy travelling wave solutions for first-order hyperbolic balance laws. Results concerning existence, regularity and asymptotic stability of such solutions are proved for convex fluxes and source terms with simple isolated zeros.

This paper is concerned with an iterative functional differential equation x(t) = c1x(t) + c2x[2](t) + … cmχ[m](t) + F(t), where x[i](t) is the i-th iterate of the function x(t). By means of Schauder's Fixed Point Theorem, we establish a local existence theorem for smooth solutions which also depend continuously on the forcing function F(t).

We consider abstract initial boundary value problems in a spirit similar to that of the classical theory of linear semigroups. We assume that the solution u at time t is given by u(t) = S(t) ξ + V(t)g, where ξ and g are respectively the initial and boundary data and S(t) and V(t) are linear operators. We take as a departing point the functional equations satisfied by the propagators S and V. We discuss conditions under which a pair (S, V) describes the solution of an abstract differential initial boundary value problem. Several examples are provided of parabolic and hyperbolic problems that can be accommodated within the abstract theory. We study the backward Euler's method for the time integration of the problems considered.

1. An attempt is made to indicate the quantitative distribution of the Crustaceans inhabiting certain Scottish intertidal areas.

2. Certain areas in the Clyde, notably Cumbrae, show a rich Amphipodan fauna which exhibits zoning by species.

3. Bathyporeia guilliarnsoniana is the dominant amphipod on the Tellina-Nephthys grounds (clean sand) and Corophium volutator is indicative of a muddy to brackish environment.

We consider the boundary-value problem

where Br0 is the ball of radius r0 in ℝN, N ≥ 2, λ > 0 and ν is the outer normal derivative at ∂Br0. This problem is equivalent to the stationary Keller—Segel system from chemotaxis. We show the existence of a solution concentrating at the boundary of the ball as λ goes to 0.

In catalysis theory there is interest in the number of solutions to the equation

with the boundary conditions

the parameters λ,β, γ being all positive and p a non-negative integer. The paper answers this question when γ is large, which is the interesting situation physically. Although the treatment is somewhat different in the cases p = 0 and p ≠ 0, the final answer is the same, that is, given β, there exist two positive functions λ1(γ) and λ2(γ) such that the problem has one solution if λ<λ1(γ), or λ>λ2(γ), three solutions if λ1(γ)<λ <λ2(γ), and two solutions if λ=λ1(γ) or λ=λ2(γ).

After a study made for bounded domains and in the periodic case, we investigate the variational formulation of Schrödinger-Poisson systems set on the whole space ℝd, d ≦ 3. This variational formulation leads to a uniqueness result, while the existence of a solution is proved only for ‘small data’ because of the lack of coerciveness. The end of this paper briefly presents the extension of this formalism to a physically relevant problem where the potential is periodic in one direction.

The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gained prominence as a model for a number of interesting physical situations. At the same time, the modern theory for the initial-value problem for the unforced Korteweg-de Vries equation has taken great strides forward. The mathematical theory pertaining to the forced equation is currently set in narrow function classes and has not kept up with recent advances for the homogeneous equation. This aspect is rectified here with the development of a theory for the initial-value problem for the forced Korteweg-de Vries equation that entails weak assumptions on both the initial wave configuration and the forcing. The results obtained include analytic dependence of solutions on the auxiliary data and allow the external forcing to lie in function classes sufficiently large that a Dirac δ-function or its derivative is included. Analyticity is proved by an infinite-dimensional analogue of Picard iteration. A consequence is that solutions may be approximated arbitrarily well on any bounded time interval by solving a finite number of linear initial-value problems.

Functional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.

This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type

$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$

It is established that under certain restrictions the solution u of the characteristic initial value problem uxy+g(x, y)u = 0, u(x, 0) = p(x) and u(0, y) = q(y), where p(x) > 0 and q(y) > 0, in [0, ∞) x [0, ∞) changes sign along a monotonic decreasing curve which is asymptotic to the axes.

Methods are developed for determining the decomposition matrices for the spin characters of the symmetric groups Sn for an odd prime p. Some general results are obtained which are non-trivial modifications of the corresponding results for ordinary characters. The methods are used to determine the decomposition matrices for 3 ≦ n ≦ ll, and p = 3 but with an interesting ambiguity in the case n = 9. The second author will deal separately with the cases p = 5, 7, 11.

We consider the eigenvalue problem for the one-dimensional (stationary) Dirac operator with some boundary conditions. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, then the potential is actually zero. The analogous statement for the Schrödinger operator is due to Ambarzumian. The proof is based on the fact that the (generalized) moments of a function cannot have alternating signs unless the moments are zero (see §2).

This family includes two of Savigny's genera—Boltenia and Cynthia, and may be defined by their common characters if we omit or modify one point on which considerable stress is usually laid, namely, the number of lobes round the branchial and atrial apertures.

The amended definition of the family is as follows:—

Body attached, sessile or pedunculated; apertures four-lobed or having less than four lobes.

Test coriaceous; rarely cartilaginous or gelatinous.