We study the homogenization and localization of high-frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch-wave packets, i.e. that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale . We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e. of the corresponding Bloch eigenvalue. Upon rescaling at size we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.

Let Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

We study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

The semi-classical regime of standing wave solutions of a Schrödinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

We study the orbital stability of standing waves for the Klein–Gordon–Schrödinger system in two spatial dimensions. It is proved that the standing wave is stable if the frequency is sufficiently small. To prove this, we obtain the uniqueness of ground state and investigate the spectrum of the appropriate linearized operator by using the perturbation method developed by Genoud and Stuart and Lin and Wei. Then we apply to our system the general theory of Grillakis, Shatah and Strauss.

We prove various bounds on moments

where fm is a solution of the discrete (respectively, continuous) Smoluchowski coagulation–fragmentation equations with diffusion. In a previous paper we proved similar results for all weak solutions to the discrete Smoluchowski equation provided that there is no fragmentation and certain moments are bounded in suitable Lq-spaces initially. We prove the corresponding results in the case of the continuous Smoluchowski equation. When there is also fragmentation, we need to assume that the solution f is regular in the sense that f can be approximated by solutions to the Smoluchowski equation for which the coagulation and fragmentation coefficients are 0 when the cluster sizes are large. We also need suitable assumptions on the coagulation rates to avoid gelation. On the fragmentation rate β we assume that supn supm≤l β(m, n)/n < ∞ for every positive l, and that there exist constants a0 ≥ 0 and c0 such that β(n,m) ≤ c0(n + m)a0a0.

We consider a non-autonomous competitive model with generalized functional responses for interaction among n species, the adult members of which are in competition. For each of the n species the model incorporates a distributed time delay which represents the time from birth to maturity of that species. Based on some comparison arguments, we discuss the permanence and extinction of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of a positive periodic solution. By means of constructing appropriate Lyapunov functionals, we obtain sufficient conditions for the uniqueness and the global stability of the periodic solution. Two examples are given to illustrate the feasibility of our main results.

This paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.

AC rings (rings satisfying Auslander's condition) have recently been studied by Christensen and Holm, among others. We will employ tilting theory to study Auslander's condition. In particular, we prove that the class of AC Artin algebras is closed under tilting equivalences.