New mathematical objects, called H-measures, are introduced for studying oscillations and concentration effects in partial differential equations. Applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach.

Monotone travelling wave solutions are known to exist for Fisher's equation which models the propagation of an advantageous gene in a single locus, two alleles population genetics model. Fisher's equation assumed that the population size is a constant and that the fitnesses of the individuals in the population depend only on their genotypes. In this paper, we relax these assumptions and allow the fitnesses to depend also on the population size. Under certain assumptions, we prove that in the second heterozygote intermediate case, there exists a constant θ*>0 such that monotone travelling wave solutions for the reaction–diffusion system exist whenever θ > θ*. We also discuss the stability properties of these waves.

Denote by Xt an n-dimensional symmetric Markov process associated with an elliptic operator

where (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:

(i) For almost every and

(ii) Let be a sequence of subdivisions of [0,1] so that

Then

As an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operator

where (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).

The asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β >1), ut, = Δu + uβ(β > l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ > 0, β > 1) are discussed.

If Xt is the diffusion process associated with a second-order uniformly elliptic operator L in divergence form, then without assuming smoothness in L we prove that for each x and y in ℝd,

where p is the fundamental solution to the heat equation associated with L. This allows one to control p when bounded drift terms are added to L; and also allows one to do Stratonovich integration with respect to the process conditioned to start at any point; previous work only dealt with quasi-every starting point.

In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.

We consider the initial boundary value problem for the equations of one-dimensional nonlinear thermoelasticity in ℝ+; and prove a global existence-uniqueness theorem for small smooth data. The asymptotic behaviour is simultaneously obtained.

We study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as t → ∞, and we identify steady states that are stable.

A compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant with

respect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C∞-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.

In this paper we study the existence of a compact attractor for the solutions of the equation utt − Δu + cut + f(u) = h(t, x), x ∊ ℝ3. The phase space is H1 × L2 and periodicity in the x-variables is taken as a boundary condition. Besides the usual coercive condition, we assume f satisfies the growth condition |f′(u)|≦ a + bu2; this growth condition is critical because the embedding H1 → L6 is not compact. In the proof we use an Lp − H1.q estimate for the linear homogeneous wave equation.

The asymptotic decay of L2-solutions of Schrödinger equations (-Δ+V)ψ=0 in ΔR= {x εRn∣∣x∣=r>R} is investigated, where V(x) = V1(r) + V2(x) with V1→ ∞ for r↑∞ and with some ε > 0 for large r. Under additional assumptions on the decay of V1, pointwise upper bounds to |ψ |and lower bounds to the spherical average of ψ are given showing the same asymptotics for r→ ∞. For the case V→ const. > 0 for r→ ℝ (investigated in [8] a simplified treatment is given.

Each singular element α of the full transformation semigroup on a finite set is generated by the idempotents of defect one. The length of the shortest expression of α as a product of such idempotents is given by the gravity function g(α).We use certain consequences of a result by Tatsuhiko Saito to explore connections between the defect and the gravity of α, and then determine the number of elements that have maximum gravity. Finally, we obtain formulae for the number of elements of small gravity. Such elements must have defect 1, and we determine their number within each ℋ-class. Many of the results obtained were suggested, and all have been verified, by programs written in PROLOG, a logic programming language very well suited for algebraic calculations.

By using the concentration-compactness method of Lions [14, 16] and the mountain pass theorem of Ambrosetti and Rabinowitz [3], through a careful inspection of the energy balance for some sequence of approximated solutions, we show that under suitable conditions on f and h, the inhomogeneous problem. −Δu + c2u = λ(f(u) + h(x)) for x ∈ Ω (Ω is an exterior domain in ℝN, N≧ 3) and has at least two positive solutions.

There are some mistakes in [1, Section 4], and since the main result, Theorem 4.4, is central to the theory and has already been applied in various contexts, we felt it advisable to give a complete statement and proof. The applications of vTheorem 4.4 made to date have fortunately been in situations where the results are correct. For convenience, we restate our notation.

In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear initial value problems (x′(t)=A(x(t))x(t) + f(x(t))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.

We consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmèn-Lindelöf and Liouville.

Sufficient conditions are given for a contracted semigroup ring, in which the two-sided ideals have a certain property, to be (a) semiprime, (b) semiprimitive, (c) prime, (d) primitive. The results are applied to the contracted semigroup rings of inverse semigroups, where they provide new proofs of theorems of Domanov.

This paper deals with the number of eigenvalues which appear in the gaps of the spectrum of a Dirac system with real and periodic coefficients when the coefficients are perturbed. The main results provide an upper bound and a condition under which exactly one eigenvalue appears in a given gap.

The aim of this paper is the explicit canonical or standard factorisation of matrix functions with Wiener algebra elements. The present approach covers all regular 2 × 2 matrices where two entries are arbitrary and the remaining two are linear combinations of the former with rational coefficient functions. It is based on the knowledge of how to factorise scalar functions and rational matrix functions. In general, one also needs the approximation of any scalar Wiener algebra function with a rational function. However, this can be easily circumvented in many applications by intuitive manipulations with rational matrix functions.

Functional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.