In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using the n-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

Linked equations of the form

are studied on a bounded smooth domain in RN for λ ∈ R2. Existence and uniqueness of solutions are discussed for fi homogeneous of order p – 1 in ui, generalising the ‘Klein Oscillation Theorem’ when p = 2, N = 1. Bifurcation from the principal eigenvalue is also considered for nonhomogeneous perturbations fi of order greater than p – 1.

We study the dynamical behaviour, as t → ∞, of admissible weak solutions of the scalar balance law

with x ∊ ≡ ℝ/Lℤ, L > 0, 0 < t < ∞, and f(·) ∊ C2, g(·) ∊ C1. We assume that f(·) is strictly convex, while g(·) is of at most linear growth, has finitely many zeros and changes sign across them. We show that, if u(·,t) stays bounded in L∞(S1), as t → ∞, then it either converges to a constant state or approaches asymptotically a rotating wave, i.e. an admissible weak solution of (1.1) of the form ũ(x − ct), c ∈ ℝ. Hence, the asymptotic state of every bounded solution of (1.1) consists precisely of either an equilibrium or one time-periodic solution. Furthermore, each one of these two alternatives is characterised by the Conley indices of the critical points of the ordinary differential equation .

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.

We obtain the Γ(L1(Ώ))-limit of the sequence

where Eε is the family of anisotropic perturbations

of the nonconvex functional of vector-valued functions

The proof relies on the blow-up argument introduced by Fonseca and Müller.

A Dirac system is considered which has a matrix-valued long-range, short-range and oscillatory potentials. The system has one singular endpoint at infinity. Additional conditions on the potential are given which guarantee particular asymptotic behaviour of an energy functional associated with a certain set of solutions. This asymptotic behaviour guarantees the existence of a purely absolutely continuous spectrum outside a gap containing the origin.

In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.

We study decay estimates for the solutions to the initial value problem for a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system. The method is integral estimation.

The effective conductivity tensor σ* of a two-dimensional polycrystalline material depends on the conductivity tensor σ0 of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal, represented by a rotation field R(x) giving the orientation of the crystal at each point x. Here it is established that the dependence of σ* on σ0 in any polycrystal, with R (x) held fixed, can be mimicked exactly by a polycrystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we truncate the Hilbert space of fields in the polycrystal to a finitedimensional space. Then the structure of this finite-dimensional space of fields is shown to be isomorphic to the structure of the finite-dimensional space of fields associated with the sequential laminate. In particular, there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields.

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton–Jacobi–Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.

The switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a Dirichlet space approach and is applied to prove the existence theorem of the Cauchy initial problem for the system. A representation theorem for the solution of the Dirichlet boundary value problem and a generalised Skorohod decomposition for the reflecting switched diffusion process are obtained.

A class of semilinear elliptic systems of two equations is considered. Sufficient conditions are given for the existence of different types of sign-definite solutions. These conditions relate the larger eigenvalues of certain 2 × 2 real matrices associated with the system to the first eigenvalue of − ∆ under the homogeneous Dirichlet boundary condition. A special case provides a complementary result to some of the recent works.

For scalar variational problems

subject to linear boundary values, we determine completely those integrands W: ℝn → ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.

As a corollary, we show that in case of nonattainment (and provided W grows superlinearly at infinity), every minimising sequence converges weakly but not strongly in W1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.

Connections with solid–solid phase transformations are indicated.

We consider the system (∂/∂t)u = ∆u + σ(u)|∇φ|2, div (σ(u)∇φ) = 0 in a bounded region of ℝN coupled with initial and boundary conditions, where σ(s) ∈ C(ℝ) is nonnegative and σ(u) = 0 if and only if u ≧ a for some a > 0. Owing to the degeneracy involved, solutions of the problem display new phenomena that cannot be incorporated into the classical weak formulation. The notion of a capacity solution introduced in [14,15] is employed to study the problem. It turns out that this notion of a solution is just general enough to encompass the new phenomena involved.

We introduce algorithms for calculating minimum length factorisations of order-preserving mappings on a finite chain into products of idempotents, and into products of idempotents of defect one. The least upper bounds for these lengths are given.

The Laplacian operator Δ on a bounded domain Ω in ℝn containing 0, with Dirichlet boundary condition, is perturbed by a pseudopotential δ, the Dirac measure at 0. Such a perturbation will be defined in Lp(ℝ) for n = 2, 1 <lt; p < ∞, and for n = 3, < p < 3, and is shown to be the generator of an analytic semigroup. Thus solutions of the corresponding evolutionary system are well defined. The necessary estimates involve the Gagliardo– Nirenberg inequality and the Kato inequality.

This paper is devoted to the study of integral functional denned on the space SBV(Ω ℝk) of vector-valued special functions with bounded variation on the open set Ω⊂ℝn, of the form

We suppose only that f is finite at one point, and that g is positively 1-homogeneous and locally bounded on the sets ℝk⊗vm, where {v1,…, vR} ⊂ Sn−1 is a basis of ℝn. We prove that the lower semicontinuous envelope of F in the L1(Ω;ℝk)-topology is finite and with linear growth on the whole BV(Ω;ℝk), and that it admits the integral representation

A formula for ϕ is given, which takes into account the interaction between the bulk energy density f and the surface energy density g.