This paper deals with the blow-up of solutions to a class of parabolic problems with time-dependent coefficients under homogeneous Neumann boundary conditions. For one set of problems in this class we show that no global solution can exist. For another we derive lower bounds for the time of blow-up when blow-up occurs.

The author considers the solution of the Cauchy problem for an equation

giving necessary and sufficient conditions for the existence of

This paper is devoted to a spectral description of wave propagation phenomena in conservative unbounded media, or, more precisely, the fact that a time-dependent wave can often be represented by a continuous superposition of time-harmonic waves. We are concerned here with the question of the perturbation of such a generalized eigenfunction expansion in the context of scattering problems: if such a property holds for a free situation, i.e. an unperturbed propagative medium, what does it become under perturbation, i.e. in the presence of scatterers? The question has been widely studied in many particular situations. The aim of this paper is to collect some of them in an abstract framework and exhibit sufficient conditions for a perturbation result. We investigate the physical meaning of these conditions which essentially consist in, on the one hand, a stable limiting absorption principle for the free problem, and on the other hand, a compactness (or short-range) property of the perturbed problem.

This approach is illustrated by the scattering of linear water waves by a floating body. The above properties are obtained with the help of integral representations, which allow us to deduce the asymptotic behaviour of time-harmonic waves from that of the Green function of the free problem. The results are not new: the main improvement lies in the structure of the proof, which clearly distinguishes the properties related to the free problem from those which involve the perturbation.

In this paper we consider a new class of Riemannian spaces which arise in the theory of the solution of the tensor generalisation of Laplace's equation ∇2V = o. To obtain this generalisation Beltrami's second differential parameter is defined in terms of the metric

of the associated n-dimensional Riemannian space by the usual formulæ

where denotes the Christoffel symbol . The generalised Laplace's equation is then Δ2V = o. For simplicity the quadratic differential form (1.1) is taken to be positive definite, which involves no essential loss of generality.

Human populations have afforded favourite material for analysis by statisticians, and others, interested in mathematical theories of population-growth. Experimentally, however, human beings are not ideal biological material, so that other animals, such as protozoa, insects, and mammals, have been used. On account of their great reproductive capacity, rapid response to changes in their environment, and ease of handling in the laboratory, certain insects are very suitable for this type of work. The enormous reproductive capacity of insects has been the subject of comment by many biologists, but we are still comparatively ignorant of the exact rôle of the various factors involved in the limitation of their numbers.

If E is the set of idempotents and G the group of units within a full transformation semigroup ℐx, then EG = GE = ℐx if X is finite. The question of identifying the subsemigroup EG = GE = 〈G∪E〉 in the case where X is infinite leads to an investigation of interrelations among various naturally occurring subsemigroups of ℐX. In the final section it is shown that precisely two additional elements µ, v are needed in order that G∪E∪{µ, v} should generate ℐx.

This paper studies Sobolev spaces under perturbation of domains of definition. It establishes the basic concepts, methods and results for the convergence of sequences of sets in ℝn, the strong and weak convergence of sequences of Sobolev spaces, the discrete compactness of natural embeddings and the continuous convergence of continuous linear functionals, boundary integrals and trace operators for sequences of Sobolev spaces.

This note pursues two aims: the first is historical and the second is factual.

1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.

2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.

We establish that, for $n\geq3$, the elliptic equation

$$ -\Delta u=\lambda|x|^\mu|u|^{q-2}u+|x|^\nu|u|^{p-2}u $$

on a ball with zero Dirichlet data possesses a pair of nodal radial solutions for all $\lambda>0$ provided that

$$ \mu,\nu>-2,\quad\max\bigg\{2,\frac{n+2\mu+2}{n-2}\bigg\}<q<\frac{2(n+\mu)}{n-2}\quad\text{and}\quad p=\frac{2(n+\nu)}{n-2}. $$

When $q=2$ and $n>2\mu+6$, the same result holds for $\lambda>0$ small. Canonical transformations convert the equation into a quasi-linear elliptic equation and an equation with Hardy term. Then the results correspond to the results for the transformed equations. For example, the equation

$$ -\Delta w-\frac{\chi}{|y|^2}w=\tilde{\lambda}|y|^a|w|^{q-2}w +|y|^\nu|w|^{p-2}w, $$

on a ball with zero Dirichlet data, possesses a pair of nodal radial solutions for all $\tilde\lambda>0$ provided that $a,\nu>-2$ and

$$ \max\bigg\{2,\frac{n+a-\sqrt{\bar{\chi}-\chi}}{\sqrt{\bar{\chi}}}\bigg\} <q<\frac{n+a}{\sqrt{\bar{\chi}}}\quad\text{with~}\bar{\chi}=\bigg(\frac{n-2}{2}\bigg)^2. $$

When $q=2$, $n>2a+6$ and $0<\chi<\bar\chi-(a+2)^2$, the same result holds for $\tilde\lambda>0$ small.

These results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The “no flux” boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise “fast association” there is a criticalmass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.

First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-forms which satisfy the equality case of the inequality. Finally, we obtain the complete classification of Lagrangian submanifolds in nonflat complex-space-forms which satisfy this basic equality.

We use the method of the moving plane (MMP) to obtain necessary and sufficient conditions for the radial symmetry of positive solutions of the following semi-linear elliptic equation with singular nonlinearity:

$$ \Delta u-\frac{1}{u^\nu}=0\quad\text{in~}\mathbb{R}^n,\quad n\geq2, $$

where $\nu>0$. In order to apply the MMP, it is crucial to obtain the asymptotic expansion of $u$ at $\infty$.

This paper considers the Cauchy problem for the isentropic equations of gas dynamics in Euler coordinates ρt + (ρu)x = 0, (ρu)t + (ρu)2 + P(ρ))x=0 and gives the Hölder-continuous solution by applying the method of vanishing viscosity.

Given a parametrised measure and a family of continuous functions (φn), we construct a sequence of functions (uk) such that, as k→∞, the functions φn(uk) converge to the corresponding moments of the measure,in the weak * topology. Using the sequence (uk) corresponding to a dense family of continuous functions, a proof of the fundamental theorem for Young measures is given.

We apply these techniques to an optimal design problem for plates with variable thickness. The relaxation of the compliance functional involves three continuous functions of the thickness. We characterise a set of admissible generalised thicknesses, on which the relaxed functional attains its minimum.

We study the limit as ε → 0 of global minimisers of functionals of the type

where Ω is an annul us or a ball in ℝn.

This paper gives a sufficient condition for almost-everywhere injectivity for nonlinear three dimensional elasticity similar to that of Claret-Necas [8], namely.

We prove that this relation is maintained under the weak convergence of minimising sequences for nonlinear elasticity problems. The existence and partial regularity of an “inverse” function are proved.

Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2-cluster categories, the ‘lowest’ type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E.

The characteristics of social behaviour in man are conditioned by previous experience. What is observed is the product on the one hand of a certain genetic constitution and on the other of an intricate, prolonged, and at present largely obscure, process of training and physical environment, including both the environment of the fœtus and family influences, social and physical. The experimental methods for detecting differences due to single gene substitutions cannot be applied directly. Indeed, we can see no immediate prospect of applying to social behaviour methods of genetic analysis such as have led to the mapping of the chromosomes in animals and in plants. With methods available at present, genetic inquiry can undertake to detect whether any gene differences are associated with observed differences, and whether such gene differences are recognisable throughout a comparatively wide or narrow range of social and physical environment.

A general framework is presented for the proof of the existence of classical solutions of second order elliptic equations which satisfy non-linear boundary conditions. The results obtained contain many of the known theorems for such problems and the approach used unifies the various methods of study based upon upper and lower solutions.

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.