The steady states of the Cahn–Hilliard equation are studied as a function of interval length, L, and average mass, m. We count the number of nontrivial monotone increasing steady state solutions and demonstrate that if m lies within the spinodal region then for a.e. there is an even number of such solutions and for a.e. there is an odd number of such solutions. If m lies within the metastable region, then for a.e. L > 0 there is an even number of solutions. Furthermore, we prove that for all values of m, there are no secondary bifurcations.

We consider some properties of completely integrable first-order differential equations for real-valued functions. In order to study this subject, we introduce the theory of Legendrian unfoldings. We give a characterisation of equations with classical complete solutions in terms of Legendrian unfoldings, and also assert that the set of equations with singular solutions is an open set in the space of completely integrable equations even though such a set is thin in the space of all equations.

The singularly perturbed differential-delay equation

is studied for a class of step-function nonlinearities f. We show that in general the discrete system

does not mirror the dynamics of (*), even for small ε, but that rather a different system

does. Here F is related to, but different from, f, and describes the evolution of transition layers. In this context, we also study the effects of smoothing out the discontinuities of f.

We prove that every operator preserving orthogonality in a real Banach space is an isometry multiplied by a constant.

A one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.

This work is devoted to the study of subharmonic solutions of a second-order Hamiltonian system

near an equilibrium point, say q = 0. The problem of existence of periodic solutions from the global point of view is also considered.

This problem has been studied for the case where the potential is positive and superquadratic. In this work a potential V that has change in sign is considered. The potential is decomposed as

where P is homogeneous, superquadratic and nondegenerate, and is of higher order near 0. In this paper the existence is shown of a sequence of subharmonic solutions of the equation above that converges to the equilibrium, such that their minimal periods converge to infinity.

This problem is approached from a variational point of view. Existence of subharmonic and periodic solutions is obtained via minimax techniques.

A systematic approach is proposed for the generalised factorisation of certain non-rational n × n matrix functions. The first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means. It closes a gap between the classical Wiener-Hopf procedure and the operator theoretic method of generalised factorisation. Secondly, as examples we consider certain matrix functions of Jones form or of N-part form, which are equivalent to each other, in a sense. The factorisation procedure is complete and explicit, based only on the factorisation of scalar functions, of rational matrix functions and upon linear algebra. Applications in elastodynamic diffraction theory are treated in detail and in a most effective way.

It is known that the parametric equation u'∊+ a∊u∊ = f, u∊ (0)= 0,with α ≦ a∊ ≦ β, for all ∊ > 0 and almost everywhere in a bounded domain Ω of ℝN, and f in L∞((0, T) × Ω), shows, at the limit, a memory effect. In this work the associated minimisation problem is considered and we describe how the memory effect appears in the Γ-limit, for the weak topology H1:(0, T; L2(Ω)) of the corresponding functional. The sequence a∊ has no dependence in time.

Let Cf, Pf, Qf and Rf be respectively the convex, polyconvex, quasi-convex and rank-one-convex envelopes of a given function f. If fp: RNxM→R and fq(ξ) behaves as |ξ|q at infinity q ∈ (1, ∞), we show that . This is the case for (Pfp)p provided that q ≠1,…, min (N, M), otherwise . In the last part of this work, we show that f(ξ) = g(|ξ|) does not imply in general Pf = Qf.

Let s be a natural number, s ≥ 2. We seek a positive number λ(s) with the following property:

Let ε > 0. Let Q1(x1, …, xs), Q2(x1, …, xs) be real quadratic forms, then for N > C1(s, ε) we have

for some integers n1, …, ns,

The aim of this paper is to formulate and study phase transition problems in materials with memory, based on the Gurtin–Pipkin constitutive assumption on the heat flux. As different phases are involved, the internal energy is allowed to depend on the phase variable (besides the temperature) and to take its past history into account. By considering the standard equilibrium condition at the interface between two phases, we deal with a hyperbolic Stefan problem reckoning with memory effects. Then, substituting this equilibrium condition with a relaxation dynamics, we represent some dissipation phenomena including supercooling or superheating. The application of a fixed point argument helps us to show the existence and uniqueness of the solution to the latter problem (still of hyperbolic type). Hence, by introducing a suitable regularisation and taking the limit as a kinetic parameter goes to zero, we prove an existence result for the former Stefan problem. Moreover, its uniqueness is deduced by contradiction.

We consider three models of multiple-step combustion processes on bounded spatial domains. Previously, steady-state convergence results have been established for these models with zero Neumann boundary conditions imposed on the temperature as well as the mass fractions. We retain here throughout the same boundary conditions on the mass fractions, but in our first set of results we establish steady-state convergence results with fixed Dirichlet boundary conditions on the temperature. Next, under certain physically reasonable assumptions, we develop, for two of the models, estimates on the decay rates of both mass fractions to zero, while for the remaining model we develop estimates on the decay rate of one concentration to zero and establish a positive lower bound on the other mass fraction. These results hold under either set of boundary conditions, but when the Dirichlet conditions are imposed on the temperature, we are able to obtain estimates on the rate of convergence of the temperature to its (generally nonconstant) steady-state. Finally, we improve the results of a previous paper by adding a temperature convergence result.

We study the initial growth of the interfaces of non-negative local solutions of the equation ut = (um)xx−λuq when m ≧ 1 and 0<q <1. We show that if with C < C0, for some explicit C0 = C0(λ, m, q), then the free boundary Ϛ(t) = sup {x:u(x, t) > 0} is a ‘heating front’. More precisely Ϛ(t) ≧at(m−q)/2(1−q) for any t small enough and for some a>0. If on the contrary, with C<C0, then Ϛ(t) is a ‘cooling front’ and in fact Ϛ(t) ≧ −atm−q)/2(1−q) for any t small enough and for some a > 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.

In this paper we characterise left orders in strongly regular rings, both for classical left orders and left orders in the sense of Fountain and Gould [2] where the ring of quotients need not have an identity.

We establish a necessary and sufficient condition for the existence of a positive solution of the integrodifferential equation

where nis an increasing real-valued function on the interval [0, α); that is, if and only if the characteristic equation

admits a positive root.

Consider the difference equation , where is a sequence of non-negative numbers. We prove this has positive solution if and only if the characteristic equation admits a root in (0, 1). For general results on integrodifferential equations we refer to the book by Burton [1] and the survey article by Corduneanu and Lakshmikantham [2]. Existence of a positive solution and oscillations in integrodifferential equations or in systems of integrodifferential equations recently have been investigated by Ladas, Philos and Sficas [5], Györi and Ladas [4], Philos and Sficas [12], Philos [9], [10], [11].

Recently, there has been some interest in the existence or the non-existence of positive solutions or the oscillation behavior of some difference equations. See Ladas, Philos and Sficas [6], [7].

The purpose of this paper is to investigate the positive solutions of integrodifferential equations (Section 1) and difference equations (Section 2) with unbounded delay. We obtain also some results for integrodifferential and difference inequalities.

The confluent hypergeometric function Φ2(β,β′, γ, x, y) satisfies a system of partial differential equations which possesses the singular loci x = 0, y = 0, x − y = 0 of regular type and x = ∞, y = ∞ of irregular type. Near x = ∞ (|y| is bounded) and near y = ∞ (|x| is bounded), asymptotic expansions and Stokes multipliers of linearly independent solutions of the system are obtained. By a connection formula, the asymptotic behaviour of Φ2(β,β′, γ, x, y) itself is also clarified near these singular loci.