It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.

This note extends the results in ‘Optimal coercivity inequalities in W1,p(Ω)’ (G. Auchmuty, Proc. R. Soc. Edinb. A 135, 915–933.) describing the dependence of the optimal constant in the p-version of Friedrichs' inequality on the boundary integral term. In particular, it is shown that this constant is continuous, increasing, concave and increases to the optimal constant for the Dirichlet problem as s → ∞.

We establish some bifurcation results for the boundary-value problem −Δu = g (u) + λ|∇u|p + μf (x, u) in Ω, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth bounded domain in RN, λ, μ ≥ 0, 0 < p ≤ 2, f is non-decreasing with respect to the second variable and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u−α around the origin, for some 0 < α < 1. Our approach relies on finding explicit sub- and supersolutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u|p.

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

We obtain spatial and temporal decay rates of weak solutions of the Navier–Stokes equations, and for strong solutions. For the spatial decay rate of the weak solutions, the power of the weight given by He and Xin in 2001 does not exceed 3/2;. However, we show the power can be extended up to 5/2;.

Let D1 ⊂ R3 be a non-empty simply connected open bounded set and Γ = ∂D1 be its boundary. We assume Γ to be a C2 surface and let (Γn)n∈N be a sequence of surfaces, such that Γn is closed, , and the surface measure of Γ\Γn is positive for n ∈ N. The set D1 is the cavity associated with (Γn)n∈N. Let HΓn, n ∈ N, HΓ be the self-adjoint operators on L2 associated with the minus Laplace operator with Dirichlet boundary condition on Γn, n ∈ N and on Γ, respectively. Note that, roughly speaking, HΓ is the operator associated with minus the Laplacian on R3\Γ with the Dirichlet boundary condition on Γ. A similar statement holds for HΓn, n ∈ N. We show that the spectral measure, dEΓn (λ), λ ∈ R, associated with the operator HΓn, n ∈ N, when n → +∞ converges to the spectral measure dEΓ (λ), λ ∈ R, associated with the operator HΓ. That is in a sense made precise later we prove that Σ (HΓn) → Σ (HΓ) when n → +∞, where Σ (HΓ) and Σ (HΓn), n ∈ N, are the spectra of HΓ and HΓn, n ∈ N, respectively. Moreover, we show that the point spectrum of HΓ is made of an infinite number of positive eigenvalues of finite multiplicity and that the point spectrum of HΓn, n ∈ N, is empty and that the essential spectrum of HΓn, n ∈ N, and of HΓ is continuous and is contained in [0, +∞). Under the extra hypothesis that Γ is a C4 surface and that the Gaussian curvature of Γ is positive at every point of Γ, we prove that the essential spectrum of HΓ is a continuous spectrum and is given by [0, +∞). So that the convergence of dEΓn (λ), λ ∈ R to dEΓ (λ), λ ∈ R, when n → +∞ can be interpreted as a spectral concentration phenomenon that consists in the fact that the limit of the spectrum of operators with purely continuous spectrum is the spectrum of an operator that is made of eigenvalues embedded in a continuous spectrum.

Let S be a 0-bisimple inverse semigroup and let denote the contracted l1-algebra of S; that is, the Banach algebra l1 (S)/Z, where Z is the ideal spanned by the zero of S. It is shown that, if l1 (G) is primitive for some non-zero maximal subgroup G of S, then is primitive. The same result also holds with primitivity replaced by a stronger condition, here called *-primitivity

This note extends the results in ‘Optimal coercivity inequalities in W1,p(Ω)’ (G. Auchmuty, Proc. R. Soc. Edinb. A 135, 915–933.) describing the dependence of the optimal constant in the p-version of Friedrichs' inequality on the boundary integral term. In particular, it is shown that this constant is continuous, increasing, concave and increases to the optimal constant for the Dirichlet problem as s → ∞.

Maz'ja and Sinnamon proved a characterization of the boundedness of the Hardy operator from Lp(v) into Lq(w) in the case 0 < q < p, 1 < p < ∞. We present here a new simple proof of the sufficiency part of that result.

We establish the existence of global-in-time weak solutions to a system of equations describing the motion of a compressible, viscous and heat conducting fluid undergoing a simple chemical reaction. Similar systems of equations arise as simple models in astrophysics.

For the family of truncations of the Gaussian Riesz transforms and Poisson integral we study their rate of convergence through the oscillation and variation operators. More precisely, we search for their Lp (dγ)-boundedness properties, where dγ denotes the Gauss measure. We achieve our results by looking at the oscillation and variation operators from a vector-valued point of view.

We discuss some ill-posedness results for solutions arising from the Picard iterations algorithm (i.e. the Banach fixed-point theorem) in the case of the nonlinear heat equation, the viscous Hamilton–Jacobi equation, the convection–diffusion equation and the incompressible Navier–Stokes system.

In this paper we consider Beurling-type distributions in the Hankel setting. The Hankel transform and Hankel convolution are studied on Beurling-type distributions. We also introduce a class of ultra-differential operators that allows us to show a Hankel version of the second structure theorem of Komatsu and Braun. Necessary and sufficient conditions are established in order that a Beurling distribution generates a surjective Hankel convolution operator.

We introduce the notion of the boundary flow for minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence on the initial data.

We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows.

Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.

We study the balanced Allen–Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions uε(x) with an increasing number of layers as ε → 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.

For n canonical systems of differential equations, the corresponding n copies of their domain (0, ∞) are thought of as a graph with vertex 0. An interface condition at 0 is given by a so-called Nevanlinna pair. Explicit formulae are deduced for the spectral representation of the corresponding underlying self-adjoint relation and the generalized Fourier transformation. Furthermore, results on compressions of the Fourier transformation to closed linear subspaces and the multiplicity of the eigenvalues if the spectrum is discrete are presented

We characterize those holomorphic symbols ϕ: D → D for which the induced composition operator Cϕ: Bω → Bμ (respectively, Bω,0 → Bμ,0) is bounded or compact, where D is the unit disc in the complex plane C, ω is a normal function on [0, 1) and μ is a non-negative function on [0, 1) with μ(tn) > 0 for some sequence satisfying limn→∞tn= 1.

We prove uniform Morrey–Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted L2-estimates for the solution. We also prove a uniform L2-estimate without weight for the trace of the solution on the interface.