Let X be a Banach space or a manifold and G a compact Lie group acting on X. We study G-equivariant (semi)flows on X in the context of forced symmetry breaking. After applying small symmetry breaking perturbations, certain generic invariant manifolds of the above flows persist slightly changed. We obtain necessary and sufficient conditions for the existence of heteroclinic cycles on the perturbed manifolds. Applications are given for the case G = SO(3).

We explore some necessary conditions for quasiconvexity in an attempt to show that rank-one convexity does not imply quasiconvexity when the target space for deformations is two- dimensional. An interesting construction is presented, showing how rank-one directions may fit with each other, making the task harder than in higher dimensions.

In this paper we consider the pseudo-orbit tracing property for dynamical systems generated by iterations of random diffeomorphisms. We first define a type of hyperbolicity by means of a ‘random’ multiplicative ergodic theorem, and then prove our shadowing result by employing the graph transformation methods. That result applies to, for example, the case of small random diffeomorphisms type perturbations of hyperbolic sets of deterministic dynamical systems.

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.

A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].

We consider a class of second-order systems , with q(t) ∊ℝn, for which the potential energy V: ℝn\S→ℝ admits a (possibly unbounded) singular set S ⊂ℝn and has a unique absolute maximum at 0 ∈ℝn. Under some conditions on S and V, we prove the existence of several solutions homoclinic to 0.

Coupled slow and fast motions generated by ordinary differential equations are examined. The qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after. Invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach. In the case of a unique invariant measure for each parameter, the limit of the slow motion is governed by a chattering type equation. Without the uniqueness, the limit of the slow motion solves a differential inclusion. The fast flow, in turn, converges in a statistical sense to the direct integral, respectively the set-valued direct integral, of the invariant measures.

For every k1 0 < k < m ≧ n, there are linear spaces of real n × m matrices which have dimension (m − k)(n − k) and every nonzero element has rank greater than k. Examples of such spaces are constructed and conditions are given under which they have the largest possible dimension.

We give sufficient conditions under which a funclion f: X → Y is an open mapping, where X and y are Banach spaces. This function is not necessarily continuous, but is assumed to have closed graph. We prove our results without requiring that f be Gateaux differentiable; instead, f is assumed to possess a weak type of Gateaux inverse derivative.

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.

We prove partial regularity for minimisers of quasiconvex integrals of the form ∫Ωf(Du(x))dx. More precisely, we consider an integrand f(ξ) having subquadratic growth, i.e. | f(ξ)|≦L(1+|ξ|p) with p < 2. The case of a general integrand depending also on x and u is also considered.

In this paper we discuss C1-linearisations of diffeomorphisms and flows on Banach spaces. Strong foliations of the neighbourhood of the fixed point composed of leaves based on successively larger subspaces (similar to those in [14]) are constructed. Generalised gap conditions which involve the width and separation of vertical bands containing the spectrum of a linear operator are imposed to achieve maximal smoothness. The method of proof generalises that of Hartman and of Mora and Solá-Morales. Our theorems apply to weakly coupled systems of damped wave and beam equations.

We consider the question: is every compact set in a Banach space X contained in the closed unit range of a compact (or even approximable) operator on X? We give large classes of spaces where the question has an affirmative answer, but observe that it has a negative answer, in general, for approximable operators. We further construct a Banach space failing the bounded compact approximation property, though all of its duals have the metric compact approximation property.

A basic question in mathematical ecology is that of deciding whether or not a model for the population dynamics of interacting species predicts their long-term coexistence. A sufficient condition for coexistence is the presence of a globally attracting positive equilibrium, but that condition may be too strong since it excludes other possibilities such as stable periodic solutions. Even if there is such an equilibrium, it may be difficult to establish its existence and stability, especially in the case of models with diffusion. In recent years, there has been considerable interest in the idea of uniform persistence or permanence, where coexistence is inferred from the existence of a globally attracting positive set. The advantage of that approach is that often uniform persistence can be shown much more easily than the existence of a globally attracting equilibrium. The disadvantage is that most techniques for establishing uniform persistence do not provide any information on the size or location of the attracting set. That is a serious drawback from the applied viewpoint, because if the positive attracting set contains points that represent less than one individual of some species, then the practical interpretation that uniform persistence predicts coexistence may not be valid. An alternative approach is to seek asymptotic lower bounds on the populations or densities in the model, via comparison with simpler equations whose dynamics are better known. If such bounds can be obtained and approximately computed, then the prediction ofpersistence can be made practical rather than merely theoretical. This paper describes how practical persistence can be established for some classes of reaction–diffusion models for interacting populations. Somewhat surprisingly, themodels need not be autonomous or have any specific monotonicity properties.

A characteristic wavelength and its parametric dependency are studied for planar interfaces of activator-inhibitor systems as well as their stability in two-dimensional space. When an unstable planar interface is slightly perturbed in a random way, it develops with a characteristic wavelength, that is, the fastest-growing one. A natural question is to ask under what conditions this characteristic wavelength remains finite and approaches a positive definite value as the width of interface, say ε, tends to zero. In this paper, we show that the fastest-growing wavelength has a positive limit value as ε tends to zero for the system:

This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.

We consider the homogenisation of transport kinetic equations with a highly periodic oscillating external field. The external field, acting on the particles, consists of a sum of a field deriving from a periodic potential and a bounded periodic perturbation. For the profile function generated by the oscillating solution of the problem, we derive a kinetic model with transmission boundary conditions in the energy variable. In some cases, for example when the field is not perturbed, we deduce a transport kinetic equation with memory effect satisfied by the weak-* limit of the sequence of solutions.