In this paper, we study the blow-up behaviour of the radially symmetric non-negative solutions u of the semilinear heat equation with supercritical power nonlinearity up (that is, (N – 2)p> N + 2). We prove the existence of non-trivial self-similar blow-up patterns of u around the blow-up point x = 0. This result follows from a convergence theorem for a nonlinear parabolic equation associated to the initial one after rescaling by similarity variables.

We consider the following special problem related to the optimal layout problems of materials: given two linear elastic materials, the elasticity tensors of which are C1 and C2, and a force f, find the strong closure of strains and stresses as the distribution of the materials varies, or, alternatively, find the sets of elasticity tensors which generate these strong closures. In this paper, it is shown that the local incompatibility conditions depending on C1, C2 and the local properties of strains or stresses completely characterize these sets. A connection to multiple-well problems is established.

Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

We show that if the leading coefficient p in a Sturm-Liouville expression is negative on a set E with positive Lebesgue measure, then the minimal operator (and hence any self-adjoint realization of the Sturm-Liouville expression) is not bounded below. The case when E contains an interval follows from standard methods but these methods fail when E contains no interval, e.g. when E is a Cantor-like set. This is the case considered here.

In this paper we establish a non-global Cα estimation on the boundary for the gradient of the solution of a Neumann exterior problem in ℝ3. The boundary data of this problem are with small compact support and with C0 norm tending to infinity when the diameter of support data goes to zero. This problem appears, for instance, in the numerical shape optimization problems when the Newton method is used. It turns out that the gradient decreases when the distance to the boundary data supports increases (Saint Venant's principle), which, when used in applications, leads to a strong reduction of the gradient computational cost.

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problem

where Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equation

for some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

This paper studies geometric properties, in particular extreme points and rays, of various generalizations of Young measures. Applications of the knowledge of extreme points are illustrated on existence results for optimal control problems and on various convergence results for Young measures by using the Choquet theory.

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expression

acting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form

We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.

A vector equation which gives the velocity of a vortex filament embedded in an inviscid incompressible flow is considered. It comprises terms representing effects from the localized self-induction and from the external flow. The initial value problem is proved to have at least a solution for a suitable external flow term.

In this paper, we consider a planar system with two delays:

ẋ1(t) = −a0x1(t) + a1F1 (x1(t − τ1), x2(τ−t2)).

ẋ2(t) = −b0x2(t) + b1F2 (x1(t − τ1), x2(t − τxs2)).

Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.

The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths that lie in a single stratum, except possibly at their initial points, form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

Let X be a Banach space and Y a closed subspace. We introduce an intrinsic geometric property of Y—the k-ball sequence property—which is a weakening of the famous k-ball property due to Alfsen & Effros. We prove that Y satisfies the 2-ball sequence property if and only if Y has the Phelps uniqueness property U (i.e. every continuous linear functional g ∈Y* has a unique norm-preserving extension f ∈X*). We prove that Y is an ideal having property U if and only if Y satisfies the 3-ball sequence property, and in this case, Y satisfies the k-ball sequence property for all k.

Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.

The time-dependent Ginzburg-Landau equations of superconductivity in three spatial dimensions are investigated in this paper. We establish the existence of global weak solutions for this model with any Lp (p ≧ 3) initial data. This work generalizes the results of Wang and Zhan.

The group of self-homotopy equivalences Aut(X ć Y) is represented as a product of two subgroups under the assumption that the self-equivalences of X ć Y can be diagonalized. Moreover, an analogous result holds for the subgroup Aut#(X ć Y) of self-equivalences, which induce identity automorphisms on homotopy groups. Other methods for the computation of Aut(X ć Y) are studied, especially when the spaces involved have an H- or coH-structure, and several examples are considered, among others, some non-simply connected H-spaces of rank 2.

In this note, we establish a variational setting for harmonic morphisms for target spaces of any dimension. We then extend this result to horizontally weakly conformal p-harmonic maps, such maps being p-harmonic morphisms.

We consider a class of inhomogeneous systems of reaction-diffusion equations that includes a model for cavity dynamics in the semiconductor Fabry–Pérot interferometer. By adapting topological and geometrical methods, we prove that a standing pulse solution to this system is stable in a certain parameter regime, under the simplification of homogeneous illumination. Moreover, we explain two bifurcation mechanisms which can cause a loss of stability, yielding travelling and standing pulses, respectively. We compute conditions for these bifurcations to persist when inhomogeneity is restored through a certain general perturbation. Under certain of these conditions, a Hopf bifurcation results, producing periodic solutions called edge oscillations. These inhomogeneous bifurcation mechanisms represent new means for the generation of solutions displaying edge oscillations in a reaction-diffusion system. The oscillations produced by each inhomogeneous bifurcation are expected to depend qualitatively on the properties of the corresponding homogeneous bifurcation.