We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to 0 of solutions as time approaches ∞. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.

Weighted inequalities for certain Hardy-type averaging operators in are shown to be equivalent to weighted inequalities for one-dimensional operators. Known results for the one-dimensional operators are applied to give weight characterisations, with best constants in some cases, in the higher-dimensional setting. Operators considered include averages over all dilations of very general starshaped regions as well as averages over all balls touching the origin. As a consequence, simple weight conditions are given which imply weighted norm inequalities for a class of integral operators with monotone kernels.

If E is the set of idempotents and G the group of units within a full transformation semigroup ℐx, then EG = GE = ℐx if X is finite. The question of identifying the subsemigroup EG = GE = 〈G∪E〉 in the case where X is infinite leads to an investigation of interrelations among various naturally occurring subsemigroups of ℐX. In the final section it is shown that precisely two additional elements µ, v are needed in order that G∪E∪{µ, v} should generate ℐx.

We consider the following problem:

where

for x ∈ RN, f(x, t), ft (x, t) ∈ C(RN × R), and f (x, t) ≧ 0 for all x ∈ RN and t ∈ R+, f(x, t) is an odd function of t. We show that if the maximum of Q(x) is achieved at k different points of RN, then for μ large enough the above problem has at least k positive solutions and k nodal solutions.

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.

A system of partial differential equations modelling a van der Waals fluid or an elastic medium with nonmonotone pressure-density relation is studied. As the system changes type, regularisations are considered. The existence of one-dimensional periodic travelling waves, with prescribed average density in a certain range, average velocity and wavelength, is proved. They exhibit layer structure when the regularisation parameter is small. Similarities with the Cahn–Hilliard equation are explored.

In this paper, we study global weak solutions for elastic equations with damping using the compensated compactness method. When the two end states at ± ∞ are not equal, the selfsimilar solutions for the corresponding parabolic equation are used to get the entropic estimates for both the L∞ and L2 cases.

For some singular perturbed Dirichlet and Neumann problems in a bounded smooth domain, we study solutions which have a spike in the interior. We obtain both necessary and sufficient conditions for the existence of interior spike solutions. We use, among others, the methods of projections and viscosity solutions, weak convergence of measures and Liapunov–Schmidt reduction.

We show that some free discontinuity problems can be obtained as a limit of nonconvex local functionals with a singular perturbation of higher order.

We prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower-order perturbations of the critical term. The results highlight the phenomenon of bifurcation of the critical dimensions discovered by Pucci and Serrin; moreover, we show that another bifurcation seems to appear for ‘nonresonant’ dimensions.

Long-time behaviour of solutions to porous medium equations with convection is investigated when the initial datum is a non-negative and integrable function on the real line. The long-time profile of the solutions is determined, and depends on whether the convective or the diffusive effect dominates for large times. Sharp temporal decay estimates are also provided.

We discuss divergence- and volume-preserving geodesic transformations with respect to submanifolds and in particular, with respect to hypersurfaces. We use these transformations to derive characterisations of special classes of hypersurfaces such as isoparametric hypersurfaces and Hopf hypersurfaces with constant principal curvatures. Furthermore, we consider divergence-preserving geodesic transformations with respect to geodesic spheres.

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 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 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.

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

We answer the question, raised in 1989, whether the ℐ2-radical of a structural matrix near-ring can be written as the sum of two nontrivial ideals, one of which is nilpotent, in the affirmative.

We prove that locally minimising solutions of − Δu = u(1 − |u|2) in R2, i.e. solutions that minimise the action in any bounded domain of R2, are such that ∫R2(1 − |u|2)2(x) dx < + ∞. We prove a similar property for locally minimising solutions in a half-plane.