We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:

with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:

uniformly on any compact set [δ, 1], δ ∈ (0, 1).

For a smooth, simple closed curve α in the plane, the perpendicular bisector map P associates to each pair of distinct points (p, q) on α the perpendicular bisector of the chord joining p and q. To a pair (p, p), the map P associates the normal to α at p. The set of critical values of this map is the union of the dual of the symmetry set of α and the dual of the evolute. (The symmetry set is the locus of the centres of circles bitangent to α.) We study the mapP and use it to give a complete list of the transitions which take place on the dual of the symmetry set and the dual of the evolute, as α varies in a generic one-parameter family of plane curves.

In this note, we first consider the monotonicity of the Maslov-type index theory. More precisely, for any two 1-periodic symmetric continuous matrix functions B0(t) and B1(t) with B0(t) < B1(t), we consider the relations between the Maslov-type indices (i(B0), ν (B0)) and (i(B1), ν (B1)). We then apply this theory to study the existence and multiplicity of some kinds of asymptotically linear Hamiltonian systems

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

(1) Homarus vulgáris, Eupagurus bernhardus, and Carcinus mœnas all form limb-buds or papillæ in the process of limb regeneration. These are covered by a chitinous envelope, and the observations here recorded show that their outer form and size are adaptations to the requirements of the animal. That of the lobster is straight, that of the hermit crab curved, while the shore crab has a papilla which may be folded on itself three times inside the envelope.

(2) Valvular action of the diaphragm at the breaking plane plays a greater part in the stopping of haemorrhage after self-amputation than clotting, and the dilatation of small vessels which pass beneath the epidermis detaches a layer of cells. This layer of epidermis proliferates from its free edges to form the new limb.

(3) A new diaphragm is the first structure laid down, and differentiation takes place from the base outwards. Muscle arises at the growing tip from cells proliferated from the old epidermis (an ectodermal structure), and the nerve grows outwards from the torn end by cell proliferation.

(4) Muscle-fibres are anatomically complete immediately before moulting. The fibrillæ are cross-striated and enclosed in a sarcolemma, but full functional activity does not come till several days after moulting, beginning with slow rhythmic movements. Sarcoplasm seems to be less plentiful than in the normal fibre.

(5) When moulting occurs the papilla is at once expanded to several times its previous size by valvular action, and the epidermis, previously composed of several layers of cells, now thins to a single layer, as is seen in the normal limb.

Two non-trivial solutions for semilinear elliptic resonant problems are obtained via the Lyapunov—Schmidt reduction and the three-critical-points theorem. The difficulty that the variational functional does not satisfy the Palais—Smale condition is overcome by taking advantage of the reduction and a careful analysis of the reduced functional.

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

In a previous paper (Jack 1970, § 5) the pre-zonal polynomials were defined by the surface integral (1) below. In that paper they were evaluated using a series of differential type operators acting on the sums of powers of the roots of a matrix.

The present paper, although it is a continuation of Jack (1970), is self-contained and consists of three parts. The first part, §§ 3–7, shows how the pre-zonal polynomials may be calculated, certainly for low degree, using only the first of the differential operators. The next part, §§ 8–11, is devoted to establishing various recurrence relations between the pre-zonal polynomials and to conjecturing similar recurrence relations between the zonal polynomials (James 1964, p. 478). In the final part, §§ 12–14 the pre-zonal polynomials are evaluated directly from the integral (1) below, in terms of the augmented monomial symmetric functions.

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds Gk(ℝn) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝn. In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of Gk(ℝn) as a function of n.

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the ‘charge’, that is, the weight of the Riesz interaction energy.

In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

We consider a phase-field model for diffusion-induced grain boundary motion. The model couples a parabolic variational inequality to a degenerate diffusion equation. Using a regularization technique, we prove an existence theorem for the resulting system. We also obtain a uniqueness result, provided the solution has some additional regularity.

Explicit formulae and numerical values for upper and lower bounds for the best constant in Landau/s inequality on the real line are given. For p > 3, the value of the upper bound is less than the value of the best constant conjectured by Gindler and Goldstein (J. Analyse Math. 28 (1975), 213–238).

We study the homogenization of ferromagnetic equations with periodic coefficients in space dimension 2. The obtained nonlinear homogenized law can only be written using a two-scale framework, which couples microscopic and macroscopic scales. It also involves corrector terms, at the microscopic scale, in the form of pseudo-differential operators. We prove the L2 two-scale strong convergence in the laminar case (one-dimensional periodicity).

The existence of periodic solutions is studied for certain singularly perturbed differential inclusions. Applications are given to dry friction problems.

We study the question of whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (non-commutative) algebra, is attained. This question was considered by Anick in his 1983 paper, ‘Generic algebras and CW-complexes’ (Princeton University Press), where he proved that the estimate is attained for the number of quadratic relations d ≤ ¼n2 and d ≥ ½n2, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to ½n(n – 1) was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.

We prove that over any infinite field, the Anick conjecture holds for d ≥ (n2 + n) and an arbitrary number of generators n, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

If G is a discontinuous group of homeomorphisms of a connected, locally path connected space X, which acts freely on X, then the projection π: X → X/G is a covering map and has the homotopy lifting property. Here we allow the elements of G to have fixed points and use work of Rhodes to investigate how two loops in X are related if their projections are homotopic in X/G. This enables us to establish a formula for the fundamental group of the orbit space of a discontinuous group under very general conditions. Finally we show by means of an example that some restriction on the action near fixed points is needed for the formula to be valid.

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators Ap, Aac and Asc such that there exists an orthonormal basis of eigenvectors for the operator Ap, the operator Aac has purely absolutely continuous spectrum and the operator Asc has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component Asc into a direct sum of two self-adjoint operators and . The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Let Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.

We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .