Recently, Sarrión and the authors gave a sufficient condition on invertible Lamperti operators on Lp which guarantees the convergence in the Cesàro-α sense of the ergodic averages and the ergodic Hilbert transform for all f ∈ Lp with p > 1/(1 + α) and −1 < α ≤ 0. The result does not hold for the space L1/(1 + α). In this paper we give a positive result for the smaller Lorentz space L1/(1 + α),1.

Localized travelling waves to reaction-diffusion systems on the real line are investigated. The issue addressed in this work is the transition to instability which arises when the essential spectrum crosses the imaginary-axis. In the first part of this work, it has been shown that large modulated pulses bifurcate near the onset of instability; they are a superposition of the primary pulse with spatially periodic Turing patterns of small amplitude. The bifurcating modulated pulses can be parametrized by the wavelength of the Turing patterns. Furthermore, they are time periodic in a moving frame. In the second part, spectral stability of the bifurcating modulated pulses is addressed. It is shown that the modulated pulses are spectrally stable if and only if the small Turing patterns are spectrally stable, that is, if their continuous spectrum only touches the imaginary-axis at zero. This requires an investigation of the period map associated with the time-periodic modulated pulses.

The asymptotic behaviour of solutions of second-order quasilinear elliptic partial differential equations defined on unbounded domains in Rn contained in strips (when n = 2) or slabs (when n > 2) is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data have appropriate asymptotic behaviour at infinity. We prove Phragmèn–Lindelöf theorems for large classes of elliptic operators, including uniformly elliptic operators and operators with well-defined genre, establish exponential decay estimates for uniformly elliptic operators when the Dirichlet boundary data vanish outside a compact set, establish the uniqueness of solutions, and give examples of solutions for non-uniformly elliptic operators which decay but do not decay exponentially. Our principal theorems are proven using special barrier functions; these barriers are constructed by considering an operator associated to our original operator.

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M defined by a time-dependent heat source and time-dependent boundary conditions.

We consider the universal phantom map out of a non-finite loop space. First we obtain a necessary and sufficient condition for the universal phantom map out of ΩG for a simply connected compact Lie group G to be essential. Next we prove that the universal phantom map out of ΩkX is essential for all k ≥ 2 if X is a simply connected non-contractible finite CW-complex. Ingredients in the proof are the Browder's ∞-implication argument and the Eilenberg–Moore spectral sequence.

The aim of the present paper is to adapt the method of two-scale convergence to the homogenization of a pseudomonotone Dirichlet problem in perforated domains with periodic structure. The limit problem and a corrector result are obtained.

If A and B are C*-algebras and X is an operator A, B-bimodule, then points of X can be separated from closed A, B-absolutely convex subsets of X by completely bounded A, B-bimodule homomorphisms from X into B(K), where K is a Hilbert space and the A, B-bimodule structure on B(K) is induced by a pair of representations π : A → B(K) and σ : B → B(K). If A and B are von Neumann algebras and X is a normal (not necessarily dual) operator A, B-bimodule, those A, B-absolutely convex subsets of X are characterized which can be separated from points of X as above, but with the additional requirement that the two representations π and σ are normal. This requires a new topology on X, which has appeared also in connection with some other questions concerning operator modules.

We conjecture that the mean-field model of superconducting vortices given in [10] is ill-posed wherever the electric current j has some component in the same direction as the vorticity vector ω (which gives the average density and direction of the superconducting vortices). The conjecture is illustrated with a linear stability analysis of a certain solution to the model. A regularised model is then proposed, and this is used to demonstrate the instability of force-free steady states in a certain geometry.

We relate Kaptsov's method of B-determining equations for finding invariant solutions of PDEs to the nonclassical method and to the method of generalised conditional symmetries. An extension of Kaptsov's method is then used to find new solutions of degenerate diffusion equations.

The distortion of a two-dimensional bubble (or drop) in a corner flow of an inviscid incompressible fluid is considered. Numerical solutions are obtained by series truncation. The results confirm and extend previous calculations.

This paper is concerned with a neutral differential equation with four constant coefficients, one delay and one advancement. By means of the theory of envelopes, we consider all possible values of the parameters involved in the equation and obtain a complete set of necessary and sufficient conditions for all solutions to be oscillatory.

The spectral theory for non-self-adjoint elliptic boundary problems involving an indefinite weight function has only been established for the case of higher-order operators under the assumption that the reciprocal of the weight function is essentially bounded. In this paper we are concerned with the spectral theory for a case where the weight function vanishes on a set of positive measure.

An example is given of a quasiconvex f : M2×3 → R such that the transposed function f̃ : M3×2 → R given by f̃(F) = f(FT) is not quasiconvex. For f̃ one can take Sverák's quartic polynomial that is rank-one convex but not quasiconvex. The proof is closely related to the observation that the map v ↦ v1v2v3 is weakly continuous from L3(R3; R3) into distributions provided that A(Dv) = (∂2v1, ∂3v1, ∂1v2, ∂3v2, ∂1v3, ∂2v3) is compact in W−1,3(R3; R6).

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. For simplicity, the fine structure is here taken to be periodic, first in one dimension, and then a lattice of squares in two dimensions. A method of multiple scales is employed, with a classical free-boundary problem being used to model the evolution of the two-phase microstructure. Then a macroscopic model for the mush is obtained by an averaging procedure. The free-boundary temperature is taken to vary according to Gibbs–Thomson and/or kinetic-undercooling effects.

We introduce and analyse a class of quasi-self-similar solutions of the thin film equation to describe the dynamics of expanding liquid films on a solid surface. Using these solutions as intermediate asymptotics profiles, we obtain a quantitative expression for the shape of the film and a relation between the speed of the contact line and the macroscopic and microscopic contact angles.

Let Ω ⊂ RN be a smooth bounded domain. Letbe a second-order strongly elliptic differential operator with smooth symmetric coefficients. Let B denote the Dirichlet or the Neumann boundary operator. We prove the existence of a smooth potential a : Ω → R such that all sufficiently small vector fields on RN + 1 can be realized on the centre manifold of the semilinear parabolic equationby an appropriate nonlinearity f : ( x, s, w ) ∈ Ω x R x RN ↦ f ( x, s, w ) ∈ R.

For N = 2, n, k ∈ N, we prove the existence of a smooth potential a : Ω → R such that all sufficiently small k-jets of vector fields on Rn can be realized on the centre manifold of the semilinear parabolic equationby an appropriate nonlinearity f : ( x, s ) ∈ Ω x R ↦ f (x, s ) ∈ R2 ( here, ‘·’ denotes the scalar product in R2).

The linearized stability problem for fibre spinning of Newtonian and power law fluids, in a one-dimensional, isothermal setting, is reduced to a single Volterra integral equation. A rigorous linear stability criterion is presented, and we compute the corresponding ‘critical stretch ratios’, using methods of complex analysis. The results of this paper prove the correctness of earlier results by Matovich, Pearson and Shah.