This paper deals with the existence of bounded plane wave fronts of reaction-diffusion systems. The main result ensures the partial invariance of a certain region, under kinetic conditions commonly used in the literature. This allows us to construct bounded plane wave fronts taking their values in that domain. We also give an estimate of the minimum permissible value of the propagation velocity of those plane wave fronts. Some examples are given.

In this paper we prove that every precompact subset in any (LF)-space has a metrizable completion. As a consequence every (LF)-space is angelic and in this way the answer to a question posed by K. Floret [3] is given. Some contributions to the general problem of regularity in inductive limits posed by K. Floret [3] are also given. Particularly, extensions of well-known results of H. Neuss and M. Valdivia are provided in the general setting of (LF)-spaces. It should also be noted that our results hold for inductive limits of an increasing sequence of metrizable spaces.

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.

The shape of radicals of semigroups algebras of commutative and cancellative semigroups is studied. The questions asto when a radical of those algebras is homogeneous and if homogeneous radicals have more regular form are examined.

We give conditions on pairs of non-negative weight functions u and v which are sufficient that, for 1<p, q <∞,

where T is the Hankel-or the K-transformation.

The proofs are based on a weighted Marcinkiewicz interpolation theorem for linear operators. In the case that T is the Hankel transformation and 1<p≦q <∞, the result is similar to a weighted estimate of Heywood and Rooney [9], but with different weight conditions.

For many purposes, and in particular for the calculation of upper and lower bounds to bilinear forms 〈g0,f〉, where f is the solution to an operator equation and g0 is known, it isuseful to obtain an approximation to the inverse of the operator.

For a normal operator A acting in a complex Hilbert space with bounded inverse A−1, we use a direct approach through fundamental results in functional analysis and derive a recipe for the ‘best formula’ for A−1 of form B = βI with β constant and I the identity operator. Examples illustrate that this leads to improved results for certain classes of operator.

The investigation is extended to the representation of A−1 by a polynomial in A withcomplex coefficients. For polynomials of first and higher orders the application is restricted to bounded self-adjoint operators; at first order, explicit formulae are given.

Application of the method to Fredholm operators is considered in detail, and its use for the point-wise solution to Fredholm integral equations is illustrated.

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

Page 58: Lemma 7.1 is incorrect. It is claimed that Ao =cocl for c =k−l when in fact the result is proved for c = (dim H)−1. Thus while Corollaries 7.4 and 7.6 still stand (as does Corollary 7.3 by alternative arguments) the remaining results in Section 7 must be withdrawn.