A version of the Darboux transformation is explored for Sturm-Liouville problems with eigenvalue-dependent boundary conditions, from differential-equation and operator-theoretic viewpoints. Some of the literature on Darboux's transformation is related in a historical introduction.

For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

We consider the existence of positive solutions for the boundary-value problem

(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,

au(r) − bϕ−1 (q(r))u′(r) = 0, cu(R) + dϕ−1(q(R))u′(R) = 0,

where ϕ(u′) = |u′|p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.

Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2-cluster categories, the ‘lowest’ type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E.

Following a method by Meldrum and van der Walt, near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved.

We study positive solutions to the porous media equation of degenerate logistic type −Δu = a(x)u1/m – b(x)f(u), m > 1, which blow up at the inner boundary and satisfy a homogeneous boundary condition at the outer boundary of a multiply connected domain. In particular, we investigate uniqueness and blow-up rate near the inner boundary for such solutions. We also look at the limiting behaviour as m ↘ 1 for the special case where b(x) > 0 and f(u) = up/m with p > m.

We discuss the combined semi-classical and relaxation limit of a one-dimensional isentropic quantum hydrodynamical model for semiconductors. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density, including the quantum potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the relaxation time and Planck constant tend to zero, periodic initial-value problems of a scaled one-dimensional isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the classical drift-diffusion model from the quantum hydrodynamic model.

This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.

In feedback stabilization studies, a control scheme is designed so that the ‘state’ of the system decays with a designated rate as t → ∞. Thus, any linear functional of the state also decays with at least the same decay rate. We study a class of linear parabolic systems, and show the existence of a feedback control scheme with a specific property such that some non-trivial linear functionals decay faster than the state. In constructing the control scheme, we also solve the new problem of an arbitrary allocation of the eigenvalues of some coefficient matrix which is subject to constraint, and give the necessary and sufficient condition in terms of controllability and observabililty assumptions. This is an essential extension of the celebrated 1967 theory of pole allocation by W. M. Wonham.

We characterize the boundedness of the Hardy operator between weighted amalgams, a problem studied, but not completely solved, by C. Carton-Lebrun, H. P. Heinig and S. C. Hofmann. We also characterize the weighted weak-type inequalities for modified Hardy operators on amalgams.

We investigate a non-cooperative reaction-diffusion model arising in the theory of nuclear reactors and are concerned with the associated steady-state problem. We determine the asymptotic behaviour of the coexistence states near the point of bifurcation from infinity, which exhibits the following very interesting spatial blow-up pattern: when the fuel temperature reaches a certain value, the free fast neutrons undergoing nuclear reaction will blow up in each spatial point of the interior of the reactor. Without any restriction on spatial dimensions, we also discuss the uniqueness and stability of the coexistence states. Our results complement and sharpen those derived in two recent works by Arioli and Lóopez-Gómez.

We consider the problem −Δu = c0K(x)upε, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth, bounded domain in ℝN, N ≥ 3, c0 = N(N − 2), pε = (N + 2)/(N − 2) − ε and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small ε > 0 under some assumptions on the coefficient function K. This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

Some of the proofs in the above paper are incomplete.

The pressure term in the incompressible elasticity equation needs to be considered and estimated in order to pass to the limit in the homogenization process. This will require Sections 2.2 and 3.2 to be rewritten.

A corrected version will be submitted in due course.