This paper is concerned with quasilinear hyperbolic systems of conservation type and establishes two main results. The first is that when a general system is considered in one space dimension and time, then the exceptional nature of a characteristic field implies the coincidence of a shock with one of the characteristics of that field. The second result involves the demonstration by example that quasilinear hyperbolic systems of conservation type may possess solutions that become unbounded after only a finite time, even though they are exceptional.

We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.

Hilbert boundary value problems for a half-space are considered for analytic representations of Schwartz distributions: given data g ∈D'(ℛ) and a coefficient x we seek functions F(z) analytic for Jmz≠0 whose limits exist in D'(ℛ) and satisfy F+—XF– = g on an open subset U of the real line R. U is the complement of a finite set which contains the singular support and the zeros of X·X and its reciprocal satisfy certain growth conditions near the boundary points of U. Solutions F(z) are shown to exist, and their general form is determined by obtaining a suitable factorisation of x.

The aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.

The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

The principal concern here is with conditions on f or on special solutions of the equation

which ensure that the higher differences of the zeros and related quantities of solutions of (1) are regular in sign. In particular, by choosing f(x)= 2v−2x1/v−2, it is shown that if ⅓ ≦|v|<½, then

where cvk denotes the kth positive zero of a Bessel function of order v and Δµk = Δk+1 − µk. Lorch and Szego [15] conjectured that (2) should hold for the larger range | v | < ½ but the methods used here do not apply to the range | v <| ⅓.

Some continuous dependence theorems are presented for classical solutions of initial-boundary value problems in viscoelastic materials which occupy the exterior of a bounded domain in Euclidean three space. Two broad classes of anisotropic viscoelastic materials are considered. The theorems are proved by a combination of the Protter and the Graffi methods.

The existence of a variational solution is shown for the strongly non-linear elliptic boundary value problem in unbounded domains. The proof is a generalisation to Orlicz-Sobolev space setting of the idea introduced in [15] for the equations involving polynomial non-linearities only.

This note is concerned with the spaces F'p,µ of generalised functions introduced in a previous paper. A necessary and sufficient condition for an inclusion of the form

to hold is established. The case p = ∞ leads to consideration of a class G''∞µ whose simple properties are noted. Some consequences of relevance to fractional integrals and Hankel transforms are indicated.

A general formalism for the description of high-frequency gravitational radiation is presented, and a justification is provided for the often made assumption that the energy of an isotropic field of gravitational radiation in a cosmological model may be represented by a perfect fluid.

The Stieltjes transformation is extended to generalised functions both by the direct approach and the method of adjoints, and the resulting extensions are correlated. Inversion formulae are developed, as is the application of fractional integration to these transforms. An integral transformation with a hypergeometric kernel is also briefly considered.

Using the theory of the weighted Sobolev space H1,2(μ), on a bounded domain Ω, in Rn, the existence and regularity of solutions u in K to the variational inequality

is established for various convex subsets K of H1, 2(μ). The growth conditions imposed on the functions A and B give the differential inequality degenerate elliptic structure, extending the results on regularity for inequalities of elliptic type.

Existence and uniqueness theorems are obtained for a class of mixed boundary value problems associated with the three-dimensional Helmholtz equation. In this context the boundary of the region of interest is assumed to consist of the union of a finite number of disjoint, closed, bounded Lyapunov surfaces on some of which are imposed Dirichlet conditions whilst Neumann conditions are imposed on the remainder. An integral equation method is adopted throughout. The required boundary integral equations are generated by a modified layer theoretic approach which extends the work of Brakhage and Werner [1] and Leis [2, 3].

Since the appearance of the article [15] to which this is a sequel, considerable progress has been made in the study of the groups F(r, n) of the title. It is therefore our intention to give a brief account of these developments before proceeding to our main theme, which is to apply the elegant and powerful methods of small cancellation theory to these groups. This has a variety of consequences, perhaps the most important of which is the generalisation to arbitrary r of Lyndon's proof that F(2, n) is infinite for n ≧ 11.

It is established that under certain restrictions the solution u of the characteristic initial value problem uxy+g(x, y)u = 0, u(x, 0) = p(x) and u(0, y) = q(y), where p(x) > 0 and q(y) > 0, in [0, ∞) x [0, ∞) changes sign along a monotonic decreasing curve which is asymptotic to the axes.

Let Ω be an open set of ℝm; we consider some symmetric singular systems:

We study the existence and uniqueness of a solution of the problem Lu = f plus boundary conditions when the characteristic matrix is of constant rank only on the boundary.

Dual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

This report deals with the asymptotic behaviour of solutions of the wave equation in a domain Ω ⊆Rn. The boundary, Γof Ωft consists of two parts. One part reflects all energy while the other part absorbs energy to a degree. If the energy-absorbing part is non-empty we show that the energy tends to zero as t→∞. With stronger assumptions we are able to obtain decay rates for the energy. Certain relationships with controlability are discussed and used to advantage.

In this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

This paper is concerned with the effect of the delays on the bounded solutions of the nth order n ≧ 1) differential equation

Sufficient conditions involving the retardations τJ(j = 1,2, …, m) which insure that every bounded solution of the considered equation is oscillatory are given. The results obtained generalise recent ones in [3 and 4].

Sufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.