This paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper concentrates on the regularity of solutions of the Poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted Sobolev spaces and countably normed spaces. These results can be generalised to elliptic problems arising from mechanics and engineering, for instance, the elasticity problem on polyhedral domains. Hence, the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges, but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems.

In this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.

A fundamental prerequisite for the numerical computation of optimal controls is to show that sequences of suboptimal (that is, close-to-optimal) controls converge. We show this in a version that applies to hyperbolic and parabolic distributed parameter systems, the latter including the Navier–Stokes equations. The optimal problems include control and state constraints; in the parabolic case, the constraints may be pointwise on the solution and the gradient.

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each point x ∊ Ω we have a pair of ODE's: the kinetic. If for some x ∊ Ω kinetic has a unique stable coexistence state, we show that there exist such that for every the RD-model is persistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there exist xu, xv ∊ Ω such that the semitrivial coexistence states (u, 0) and (0, v) of the kinetic are globally asymptotically stable at x = xu and x = xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.

Singular perturbation techniques, monotone schemes, fixed point index, global analysis of persistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.

We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model has three or more coexistence states.

In this paper we study the Poisson geometry of the second Hamiltonian structure for the periodic N Toda lattice, around a certain family of singularities. We show that their singular leaves are not isolated and that the regular codimension of the leaves at points of this kind is always equal to three. This result is based on a rather unexpected result about a certain Toeplitz matrix.

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

In this paper, the existence and uniqueness of the global smooth solution are proved for an evolutionary Ginzburg–Landau model for superconductivity under the Coulomb and Lorentz gauge.

In this paper, we give some new examples of the energy gap phenomenon for functionals defined in Sobolev spaces. Our result is independent of that of Giaquinta, Modica and Soucek. We also give some new characterisations of Sobolev maps which can be approximated by smooth maps.

We prove the existence of an arbitrarily large number of periodic solutions for a class of nonlinear differential equations generalising the dynamics of a forced pendulum with small length.

Positive definite temperature functions u(x, t) in ℝn+1 = {(x, t)| x ∈ ℝn,t > 0} are characterised by

where μ is a positive measure satisfying that for every ℰ > 0,

is finite. A transform is introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions in Then correspondence given by generalises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.

We describe all pairs of semigroups S and radicals ρ, such that ρ is invariant in S-graded rings. This generalises several known results due to Amitsur and Sands.

The special unitary group SU(n) has the stable homotopy type of a wedge of n − 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whether n is finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 when n is finite, but splits into a wedge of two pieces when n is infinite.

We study entropy travelling wave solutions for first-order hyperbolic balance laws. Results concerning existence, regularity and asymptotic stability of such solutions are proved for convex fluxes and source terms with simple isolated zeros.

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for

is locally well posed in the largest space where the three conservations

can be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

We discuss the stability and instability of steady-state solutions for a hydrodynamic model of semiconductors. We study the case where the doping profile is close to a positive constant and depends on the special variable x. We shall show that a given steady-state solution is asymptotically stable or unstable, depending on whether or not the density of the initial data satisfies P = 0, where P is defined in (3.12).

We study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension-five components in the intersection of two strata in the centre manifold, and Q4. The study of limit cycles for this degenerate case requires us to investigate not the first but the second variation M2 of the displacement function. We prove that up to three limit cycles can emerge from the period annulus surrounding the centre. This implies that the cyclicity of period annuli of nearby systems in and Q4 is at most three as well. Our approach relies upon the possibility of deriving appropriate Picard–Fuchs equations satisfied by the four independent integrals included in M2.

We consider the existence of positive solutions to a class of singular nonlinear boundary value problems for P-Laplacian-like equations. Our approach is based on the Schauder Fixed-Point Theorem.