Nonautonomous parabolic equations of the form ut − Δu = f(u, t) on a symmetric domain are considered. Using the moving-hyperplane method, it is proved that any bounded nonnegative solution symmetrises as t → ∞. This is then used to show that for nonlinearities periodic in t, any non-negative bounded solution approaches a periodic solution.

A characterisation is provided for the weak closure of the set of rearrangements of a function on an unbounded domain. The extreme points of this convex, weakly compact set are classified. This result is used to study the maximising sequences of a variational problem for steady vortices.

In this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete, noncompact Riemannian manifold (M, g) with Sobolev constant S2(M) > 0 and Ricci curvature Ric (M) ≧ 0 outside some compact subset, into a complete manifold of nonpositive curvature or a regular ball. In particular, we prove the uniqueness and existence of bounded harmonic functions on (M, g).

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

Let (X, ∑, μ) denote a σ-finite measure space. We show that the kernel condition on a weighted composition operator acting on L2(X, ∑, μ), which is necessary for hyponormality of the adjoint, implies that a certain subset of X has the localising property defined by Lambert. For operators satisfying this condition, we find a reducing subspace whose orthocomplement in L2 is annihilated by both the operator and its adjoint, allowing us to obtain characterisations of seminormality for the operator by looking only at the restriction to the reducing subspace. This simplifies the analysis significantly, giving transparent characterisations for the hyponormality and quasinormality of the adjoint, as well as a characterisation of normality for the operator which does not require the computation of any conditional expectations. Several examples are given. We then characterise the semi-hyponormal class for both the operator and its adjoint.

For a 2 × 2 periodic system with a perturbation P whose first moment is finite, Jy′ = [λI + R(x) + P(x)]y, we study the behaviour of the Titchmarsh–Weyl m(λ)-coefficient at the spectral gap endpoints. Assuming gap nondegeneracy, our main result is that as λ → λ0, (λ − λ0)½(m(λ) → c ≠ 0 if and only if λ0 is a φ-half-bound state, which follows from an analysis of Jost-type functions.

We consider the Cauchy problem for the quasilinear heat equation

where σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].

In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivity

with σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

The paper studies orbits in a function space described by solutions of a system of reaction–diffusion equations in a bounded domain with a boundary condition of homogeneous Robin type. The omega-limit set of a bounded semi-orbit is shown to have a simple structure, provided that certain hypotheses are satisfied. For a two-dimensional system of Fitz-Hugh Nagumo type, these hypotheses yield explicit sufficient conditions for the existence of at least one periodic trajectory.

In this paper we characterise the linear transformations of an infinite-dimensional vector space that can be written as the product of nilpotent transformations. This and a linear version of Malcev's congruence on transformation semigroups are then used to construct a new class of congruence-free semigroups.

We consider an integrable case of the Henon-Heiles system and use an isomorphism with the two-gap KdV-flow to construct families of real elliptic trajectories which are associated with two-gap elliptic solitons of the KdV equation. Some of these solutions exhibit blow-up in finite time.

We consider an emulsion of two Stokes fluids, one of which is periodically distributed in the form of small spherical bubbles. The effects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles, we consider the two-scale convergence, and in the fine phase limit we find that the bulk flow is described by an anisotropic Stokes fluid. The effective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2].

Classes of exact solutions of the Navier–Stokes equations for incompressible fluid flow are explored. These have spatially-uniform velocity gradients at each instant, but often display complex temporal behaviour. Particular illustrative cases are described and related to previously-known solutions.

Margulis has given conditions under which a lattice in a semisimple Lie group admits the structure of an arithmetic subgroup. We show that these arithmetic structures are unique. The result is not subject to the condition “rkR(G) ≧ 2” required by the Margulis result. In the lowest dimensions, the result has previously been observed by Takeuchi, Maclachlan and Reid.

Although all the coefficients in the equation of a plane algebraic curve may be real numbers, it by no means follows that the equations of all its bitangents are real. But Plücker perceived that this could happen for the 28 bitangents of a nonsingular plane quartic. Where can this be observed in a body of 28 explicit linear equations? This modest note affords an example.

We consider the problem

where u ∊ Rn, n ≧ 2, and V ∊ C2(Rn∖e, R) is a potential having an absolute maximum at 0 and such that V(x) → − ∞ as x → e. We prove that, under some conditions on V, this problem has at least n − 1 geometrically distinct solutions.

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku) dx on the space BVk(Ω:ℝm) of the integrable functions, whose the f-th derivative in the sense of distributions is a Radon measure with bounded total variation.

Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.