In this paper, we study the following Ginzburg–Landau functional:

where (G, C), and p is a smooth bounded and non-negative map, having minima on the boundary of Ḡ. We give the location of the singularities in the case where the degree around each singularity is equal to 1.

A new formula is given for the spectral function in singular Sturm–Liouville problems where the potential is like – (const.)xc (0 < c ≦ 2). The formula is used to answer an open question concerning the asymptotic form of the spectral function.

Sharp weak type (1,1) and Lp estimates in dimension one are obtained for uncentred maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions, uncentred maximal functions fail to satisfy such estimates. Analogous results for centred maximal functions are given in all dimensions.

We study the behaviour of the non-negative blowing up solutions to the quasilinear parabolic equation with a typical reaction–diffusion right-hand side and with a singularity in the space variable which takes the form

where m ≧ 1, p > 1 are arbitrary constants, in the critical exponent case q = (p–1)/m > 0. We impose zero Dirichlet boundary conditions at the singular point x = 0 and at x = 1, and take large initial data. For a class of ‘concave’ initial functions, we prove focusing at the origin of the solutions as t approaches the blow-up time T in the sense that x = 0 belongs to the blow-up set. The proof is based on an application of the intersection comparison method with an explicit ‘separable’ solution which has the same blow-up time as u. The method has a natural generalisation to the case of more general nonlinearities in the equation. A description of different fine structures of blow-up patterns in the semilinear case m = 1 and in the quasilinear one m > 1 is also presented. A numerical study of the semilinear equation is also made using an adaptive collocation method. This is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour.

We discuss the local Cauchy problem for the generalised Kadomtsev–Petviashvili equation, namely , in the periodic and nonperiodic settings.

The maximum and anti-maximum principles are extended to the case of eigenvalue Sturm–Liouville problems

with boundary conditions of Dirichlet type (if possible) on a bounded interval [a, b]. The function r is assumed to be continuous and > 0 on ]a, b[, but the function 1/r is not necessarily integrable on [a, b]. The conditions on the functions p, m and h depend on the integrability or nonintegrability of 1/r on [a, c] and/or [c, b], for some c ∈ ]a, b[. The weight function m is not necessarily of constant sign.

In this paper we prove nonexistence of positive C2 solutions for systems of semilinear elliptic inequalities, for polyharmonic semilinear inequalities in cones and, under better conditions on the nonlinearity, for bounded positive solutions of elliptic semilinear equations in half spaces. Using a blow-up argument, these results allow us to prove a-priori bounds for a class of semilinear elliptic systems of equations in bounded domains.

An initial-boundary-value problem is considered for the heat equation in an infinite angle dθr ⊆ R2 × [0, ∞) with the oblique derivative boundary conditions on the faces λi of the angle:

with either h0 + h1 > 0, or h0 + h1 ≦ 0. The unique solvability of such a problem is proved in appropriate weighted Sobolev spaces according to the sign of h0 + h1. Estimates of the solution are obtained under ‘natural’ restrictions on the opening of the angle.

The norm of an integral operator occurring in the partial wave decomposition of an operator B introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined. The result implies that B is non-negative and has no eigenvalue at 0 when the nuclear charge does not exceed a specified critical value.

The structure of a categorical, E*-dense, E*-unitary E-semigroup S is elucidated in terms of a ‘B-quiver’, where B is a primitive inverse semigroup. In the case where S is strongly categorical, B is a Brandt semigroup. A covering theorem is also proved, to the effect that every categorical E*-dense E-semigroup has a cover which is a categorical, E*-dense, E*-unitary E-semigroup.

The Hartman–Wintner problem on asymptotic equivalence of fundamental systems of solutions (FSSs) for two Sturm–Liouville equations is studied. The following results are obtained: a criterion of asymptotic equivalence of FSSs, and sufficient conditions of asymptotic equivalence of FSSs which are expressed in terms of the coefficients of the considered equations only.

The Riemann problem for a resonant nonlinear system of conservation laws is considered here. The Riemann solution was constructed by employing the viscosity approximation approach. One kind of new discontinuity, which is called the Dirac-contact wave, appeared in the Riemann solution. Because the strict hyperbolicity as well as the genuine nonlinearity of the system considered failed, the solution we obtained in this paper is not unique for some initial data. An additional condition was explored to guarantee the uniqueness of the Riemann problem.

The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the form

is investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.

For a class of nonlinear Schrodinger equations, we prove the existence of semiclassical stationary states with possibly infinitely many concentration points. As h → 0, these states concentrate near critical points of the potential. Furthermore, for periodic potential, these states can be constructed to satisfy periodic boundary conditions.

We first prove existence and uniqueness of non-negative solutions of the equation

in in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form

where g = ga satisfies

After uniqueness is proved, the asymptotic behaviour of solutions of

is studied. In particular, we show that

The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.

An infinite system of reaction–diffusion equations that represents a particular case of the discrete coagulation–fragmentation model with diffusion is studied. The reaction part of the model describes the rate of clusters break-up into smaller particles. Diffusion constants are assumed to be different in each equation and concentration-dependent fragmentation coefficients are considered. Existence of solutions is studied under fairly general assumptions on fragmentation coefficients and initial data. Uniqueness in the class of mass-preserving solutions is proved. Convergence of solutions to spatially homogeneous equilibrium state is obtained.

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves (N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.

The monoids considered are the free monoid Mx and the free monoid-with-involution MIx on a nonempty set X. In each case, relative to a simply-defined involution, an explicit construction is given for a separating family of continuous star matrix representations of the l1-algebra of the monoid and it is shown that this algebra admits a faithful trace. The results are based on earlier work by M. J. Crabb et al. concerning the complex semigroup algebras of Mx and MIx.

We give a representation formula for the convex, polyconvex and rank one convex hulls of a set of n × n matrices with prescribed singular values.