We study the system in RN, where V is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimising argument, we can prove the existence of a homoclinic orbit when the component Ω of {x ∈ RN: V(x) < V(0)} containing 0 is an arbitrary open set; in the case Ω unbounded we allow V(x) to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in Ω implies that a homoclinic solution can also be found via a minimax procedure and, comparing the critical levels of the functional associated to the system, we see that the two solutions are distinct whenever the singularity is ‘not too far’ from 0.

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.

In this paper some special entropy–entropy flux pairs of Lax type are constructed for nonlinear hyperbolic systems of types (1.1) and (1.2), in which the progression terms are functions of a single variable. The necessary estimates for the major terms are obtained by the use of singular perturbation theory. The special entropies provide a convergence theorem in the strong topology for the artificial viscosity method when applied to the Cauchy problems (1.1), (1.3) and (1.2), (1.3) and used together with the theory of compensated compactness.

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.

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

We consider the following problem:

where is continuous on RN and h(x)≢0. By using Ekeland's variational principle and the Mountain Pass Theorem without (PS) conditions, through a careful inspection of the energy balance for the approximated solutions, we show that the probelm (*) has at least two solutions for some λ* > 0 and λ ∈ (0, λ*). In particular, if p = 2, in a different way we prove that problem (*) with λ ≡ 1 and h(x) ≧ 0 has at least two positive solutions as

Nonautonomous parabolic equations of the form ut − Δu = f(u, t) on a symmetric domain are considered. Using the moving-hyperplane method, it is proved that any bounded nonnegative solution symmetrises as t → ∞. This is then used to show that for nonlinearities periodic in t, any non-negative bounded solution approaches a periodic solution.

Classes of exact solutions of the Navier–Stokes equations for incompressible fluid flow are explored. These have spatially-uniform velocity gradients at each instant, but often display complex temporal behaviour. Particular illustrative cases are described and related to previously-known solutions.

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

We consider the problem

where u ∊ Rn, n ≧ 2, and V ∊ C2(Rn∖e, R) is a potential having an absolute maximum at 0 and such that V(x) → − ∞ as x → e. We prove that, under some conditions on V, this problem has at least n − 1 geometrically distinct solutions.

For a finite semilattice S, is is proved that if every noninvertible endomorphism is a product of idempotents, then S is a chain; the converse was proved, independently, by A. Ya. Aĭzenštat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible endomorphisms are products of idempotents if and only if S is Boolean or a chain.

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 .

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

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.

We study the resonance set ∑ of pairs (α,β) ∊ ℝ2 for which the problem ∆u + αu+ − βu− = 0 has a nontrivial solution . We show that if λ0, is an eigenvalue of multiplicity two of −Δ, then has measure zero, where are the neighbouring eigenvalues of λ0. Moreover, we have that, if the operator Δ + αIu<0 + βIu < 0 has a kernel of dimension one for(α, β) ∊ ∑ and u ≠ 0 such that Δu + αu+ − βu− = 0, then (α, β) is an isolated point on ∑ ∩ L, where L is the straight line parallel to the diagonal of ℝ+ × ℝ+ through (α, β).

The problem of describing one-relator pro-p-groups of cohomological dimension two (along the lines of Lyndon's description of discrete one-relator groups of cohomological dimension two) is still open. The known method of passing by means of a suitable p-filtration to a graded Lie algebra is not applicable to the family of one-relator pro-p-groups presented in this article, since the relators cannot be separated from the p-th powers in the free pro-p-group. In terms of the p-filtrations, the relators come arbitrarily close to a p-th power, yet the groups they define have cohomological dimension two.

The paper is concerned with the asymptotic behaviour of the solutions to a nonlocal evolution equation which arises in models of phase separation. As in the Allen–Cahn equations, stationary spatially nonhomogeneous solutions exist, which represent the interface profile between stable phases. Local stability of these interface profiles is proved.

For a 2 × 2 periodic system with a perturbation P whose first moment is finite, Jy′ = [λI + R(x) + P(x)]y, we study the behaviour of the Titchmarsh–Weyl m(λ)-coefficient at the spectral gap endpoints. Assuming gap nondegeneracy, our main result is that as λ → λ0, (λ − λ0)½(m(λ) → c ≠ 0 if and only if λ0 is a φ-half-bound state, which follows from an analysis of Jost-type functions.

Necessary and sufficient conditions are obtained for the 2-skeleton of the total space of a graph of 2-complexes to be Cockcroft, or L-Cockcroft for some subgroup L of the fundamental group. These conditions are used to construct new examples of Cockcroft and absolutely Cockcroft 2-complexes.