We study the initial growth of the interfaces of non-negative local solutions of the equation ut = (um)xx−λuq when m ≧ 1 and 0<q <1. We show that if with C < C0, for some explicit C0 = C0(λ, m, q), then the free boundary Ϛ(t) = sup {x:u(x, t) > 0} is a ‘heating front’. More precisely Ϛ(t) ≧at(m−q)/2(1−q) for any t small enough and for some a>0. If on the contrary, with C<C0, then Ϛ(t) is a ‘cooling front’ and in fact Ϛ(t) ≧ −atm−q)/2(1−q) for any t small enough and for some a > 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.

In this paper we characterise left orders in strongly regular rings, both for classical left orders and left orders in the sense of Fountain and Gould [2] where the ring of quotients need not have an identity.

The confluent hypergeometric function Φ2(β,β′, γ, x, y) satisfies a system of partial differential equations which possesses the singular loci x = 0, y = 0, x − y = 0 of regular type and x = ∞, y = ∞ of irregular type. Near x = ∞ (|y| is bounded) and near y = ∞ (|x| is bounded), asymptotic expansions and Stokes multipliers of linearly independent solutions of the system are obtained. By a connection formula, the asymptotic behaviour of Φ2(β,β′, γ, x, y) itself is also clarified near these singular loci.

A simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.

Any solution to the problem

is shown to decay to zero in the strong topology of the energy space as t→ ∞. The function β is allowed to be nonmonotone and d is a nonnegative function strictly positive on a nonvoid subset of (0, л).

This paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in ℝn. The equation is derived from the blow-up problem in the non-linear Schrödinger equation.

The Dirichlet, Neumann and mixed boundary-value problems for the two-dimensional Helmholtz equation in the interior or exterior of a quadrant are considered in a Sobolev space setting. It is shown that the potential operators arising in the interior problems can be used to derive systems of boundary integral equations to the exterior problems, which can be solved explicitly.

We consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar, J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be an nth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.

There exists a relation (1.5) between any n + 2 distinct particular solutions of the differential equation

In this paper, we show that when and only when n = 0, 1 and 2, this relation can be represented by the following form:

provided the form of this relation function Φn depends only on n and is independent of the coefficients of the equation. This result reveals interesting properties of these non-linear differential equations.

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.

In this paper we prove an extension of inequality (1.1) due to A. Meir.

In this paper we consider the Stefan problem with a heating term. We study the continuity of the interfaces between the mush, the liquid and the solid for solutions of finite lapnumber.

The equilibrium of a Tokamak plasma not confined inside a conducting shell is governed by a free boundary value problem. The existence of solutions of the free boundary value problem is discussed.

Most of the convergence results appearing so far for delayed Lotka–Volterra-type systems require that undelayed negative feedback dominate both delayed feedback and interspecific interactions. Such a requirement is rarely met in real systems. In this paper we present convergence criteria for systems without instantaneous feedback. Roughly, our results suggest that in a Lotka–Volterra-type system if some of the delays are small, and initial functions are small and smooth, then the convergence of its positive steady state follows that of the undelayed system or the corresponding system whose instantaneous negative feedback dominates. In particular, we establish explicit expressions for allowable delay lengths for such convergence to sustain.

For the reaction diffusion equation

with homogeneous Neumann boundary conditions, we give results on the generic hyperbolicity of equilibria with respect to a for fixed f and with respect to f for fixed a.

We present a geometric approach for systems of ordinary differential equations which generate an order preserving flow. One of our main goals is to describe qualitatively the asymptotic behaviour of trajectories of dynamical systems enjoying a uniformly monotone principle.

Multiple integrals with polyconvex integrands are studied on the class of all sense-preserving diffeomorphisms from W1,p(Ω, Rn) where Ω is an open subset of Rn. They are proved to be sequentially weakly lower semicontinuous if 1 < p = n –1. An example is presented showing that a similar result is not valid if p <n –1.

We consider the nonlinear Sturm–Liouville problem with two parameters on the general level set

We establish asymptotic formulae of the n-th variational eigenvalue λ = λn(μ, α) as α→∞ and α↓(nπ)2.