We consider a family of bounded dissipative asymptotically compact semigroups depending on a parameter, and study the continuity properties of the corresponding family of its global attractors. We exploit the idea of the uniform exponential attraction property to discuss the continuity properties of the family of attractors and estimate the rate of convergence of the approximating attractors to the limit one. Showing a range of applications of an abstract framework, we focus much of our attention on a perturbed damped wave equation. In this latter case our results involve nonlinearities with critical exponents, for which the continuity of the family of attractors is concluded, including the rate of convergence and the regularity of the limit attractor. This complements the results in the literature.

We study the asymptotic behaviour for ɛ→+0 of the solution Φ of the elliptic boundary value problem

is a bounded domain in ℝ2, 2 is asecond-order uniformly elliptic operator, 1 is a first-order operator, which has critical points in the interior of , i.e. points at which the coefficients of the first derivatives vanish, ɛ and μ are real parameters and h is a smooth function on . We construct firstorder approximations to Φ for all types of nondegenerate critical points of 1 and prove their validity under some restriction on the range of μ.

In a number of cases we get internal layers of nonuniformity (which extend to the boundary in the saddle-point case) near the critical points; this depends on the position of the characteristics of 1 and their direction. At special values of the parameter μ outside the range in which we could prove validity we observe ‘resonance’, a sudden displacement of boundary layers; these points are connected with the spectrum of the operator ɛ2 + 1 subject to boundary conditions of Dirichlet type.

Suppose that f(t, λ) is, for fixed t in an open interval (a, b), a regular (or analytic or holomorphic) function of λ, when λ lies in a domain D of the complex plane. We assume that f is integrable with respect to t for fixed λ ∈ D, and consider the parametric integral

By hypothesis F(λ) exists for λ ∈ D and Professor W. N. Everitt has asked whether F(λ) is necessarily regular as a function of λ.

We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ ℝ, g ∈ C(ℝ) is a superlinear and subcritical function, and Ω is a domain in ℝ2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not.

In this paper, we construct three types of symmetric peaked solutions for a Neumann problem involving critical Sobolev exponent: the interior peaked solution, the boundary peaked solution and the interior-boundary peaked solution.

Using a non-linear transformation of the state space, the dynamic behaviour of an adaptive thirdorder relay system is analysed. The control strategy yields state paths close to the time-optimal trajectories.

Bifurcations appearing in generic one-parameter families of vector fields on a compact planar region are conceptually characterised in terms of their stability and regularity properties.

Let X be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of X-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents p ≥ 2 if and only if the space X is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

Keller's phase-interchange relation, which computes the overall conductivity of a two-dimensional checkerboard with alternating conductivity, is revisited in the context of nonlinear incompressible elasticity. A general phase-interchange relation is obtained in a monotone setting through H-convergence and in a convex setting through variational methods. Several Keller-like applications are presented.

For a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, m ≠ q + 2 or for 2m ⩾ 2(p + q) + max{p, q} + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2p + q + 3 we introduce an abelian group structure on Em(Sp × Sq) and describe this group ‘up to an extension problem’. This result has corollaries which, under stronger dimension restrictions, more explicitly describe Em(Sp × Sq). The proof is based on relations between sets Em(N) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

In this paper, it is proved that the problem (∂2u/∂t2) + ∆xu = v, u(x, 0) = u0(x), ut(x, 0) = u1(x), with homogeneous Dirichlet conditions on the boundary, is well posed provided v, u0, u1 belong to a suitable space of functions.

Let Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application, we give a dilation theoretic proof of Boca's Theorem.

Consider an open bounded connected set Ω in Rn and a Lebesgue measurable set E ⊂⊂ Ω of positive measure. Let u be a solution of the strictly elliptic equation Di (aij Dj u) = 0 in Ω, where aij ∈ C0, 1 (Ω̄) and {aij} is a symmetric matrix. Assume that |u| ≤ ε in E. We quantify the propagation of smallness of u in Ω.

A variational formulation of the Hele—Shaw flow model of the point injection of fluid into a laminar cell is introduced. The analysis concerning the existence, uniqueness and regularity of a solution to the variational problem is presented.

We study the phenomenon of the infinite-time stabilization of classical global solutions of nonlinear reaction–diffusion equations to an unbounded (singular) stationary state and we present a new case of such an asymptotic singularity pattern formation. We concentrate on the most famous parabolic model, namely the semi-linear Frank-Kamenetskii equation from combustion theory,where B is a ball in RN. Our goal is to show that a new asymptotic problem arises precisely in dimension 10, not being available in other dimensions (which were studied earlier). For N = 10, we fix the ball B = {|x| < 4} and take bounded initial data u0 below the singular stationary solution Us(x) = ln(16/|x|2), which is unbounded at the origin x = 0.

We establish a sharp estimate on the rate of convergence u(x, t) → Us(x) as t → ∞ on compact subsets bounded away from x = 0 and also at the singularity point. We show that u(0, t) = α0t + O(ln t) → ∞, where the positive constant α0 is given by the first eigenvalue of the associated linear differential operator. We present formal asymptotic results showing that a detailed asymptotic analysis depends on a quite involved balance between various linear and nonlinear terms. Moreover, similar critical asymptotic behaviour is shown to exist in various related nonlinear second- and higher-order parabolic equation.

We consider the possible disentanglements of holomorphic map germs f: (ℂn, 0) → (ℂN, 0), 0 < n < N, with nonisolated locus of instability Inst (f). The aim is to achieve lower bounds for their (homological) connectivity in terms of dim Inst (f). Our methods apply in the case of corank 1.

The following research was undertaken for the purpose of determining the freezing points of solutions under conditions which would involve a different systematic error from that encountered when the ordinary method of procedure is followed; and, secondly, for the purpose of obtaining ionisation values for electrolytic solutions by the cryoscopic method which should be made by compensation as far as possible independent of any systematic error in determining the freezing points.

I should like to take this opportunity of thanking the Committee of the Carnegie Trust for the Universities of Scotland for their courtesy in furthering and facilitating the work, the results of which are submitted in the following pages.

In some experiments on sodamide carried out at 200°, in which an attempt was made to use hydrogen as an indifierent atmosphere, it was found that a slow but constant evolution of ammonia took place. The sodamide being isolated from the glass tube in which it was heated, by silver foil, the formation of ammonia could not be due to a reaction between the glass and the sodamide.