We consider the free linear Schrödinger equation on a torus 𝕋d, perturbed by a Hamiltonian nonlinearity, driven by a random force and subject to a linear damping:

Here u = u(t, x), x ∈ 𝕋d, 0 < ν ≪ 1, q∗ ∈ ℕ, f is a positive continuous function, ρ is a positive parameter and are standard independent complex Wiener processes. We are interested in limiting, as ν → 0, behaviour of distributions of solutions for this equation and of its stationary measure. Writing the equation in the slow time τ = νt, we prove that the limiting behaviour of them both is described by the effective equation

where the nonlinearity F(u) is made out of the resonant terms of the monomial |u|2q∗u.

This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrizing such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.

We study the dynamics of a domain wall under the influence of applied magnetic fields in a one-dimensional ferromagnetic nanowire, governed by the Landau–Lifshitz–Gilbert equation. Existence of travelling-wave solutions close to two known static solutions is proven using implicit-function-theorem-type arguments.

We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation polynomials. Based on the non-vanishing property of these determinants, we apply the functional equations to get an infinite sequence of approximations that is used to produce the linear independence measures.

We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.

We construct countably infinitely many non-radial singular solutions of the problem

of the form

where v(σ) depends only on σ ∈𝕊N−1. To this end we construct countably infinitely many solutions of

using ordinary differential equation techniques.

We work in the smooth category. The following problem was suggested by E. Rees in 2002: describe the precomposition action of self-diffeomorphisms of Sp × Sq on the set of isotopy classes of embeddings Sp × Sq → ℝm.

Let G: Sp × Sq → ℝm be an embedding such that

is null-homotopic for some pair of different points a, b ∈ Sp. We prove the following statement: if ψ is an autodiffeomorphism of Sp × Sq identical on a neighbourhood of a × Sq for some a ∈ Sp and p ⩽ q and 2m ⩾ 3p +3q + 4, then G◦ ψ is isotopic to G.

Let N be an oriented (p + q)-manifold and let f, g be isotopy classes of embeddings N → ℝm, Sp × Sq → ℝm, respectively. As a corollary we obtain that under certain conditions for orientation-preserving embeddings s: Sp × Dq → N the Sp-parametric embedded connected sum f#sg depends only on f, g and the homology class of s|Sp × 0.

This paper is concerned with the global dynamics and spreading speeds of a partially degenerate non-local dispersal system with monostable nonlinearity in periodic habitats. We first obtain the existence of the principal eigenvalue for a periodic eigenvalue problem with partially degenerate non-local dispersal. Then we study the coexistence and extinction dynamics. Finally, the existence and characterization of spreading speeds are considered. In particular, we show that the spreading speed is linearly determinate. Overall, we extend the existing results on global dynamics and spreading speeds for the degenerate reaction–diffusion system to the degenerate non-local dispersal case. The extension is non-trivial and meaningful.

We consider the finite-time blow-up of solutions for the following two kinds of nonlinear wave equation in de Sitter spacetime:

This proof is based on a new blow-up criterion, which generalizes that by Sideris. Furthermore, we give the lifespan estimate of solutions for the problems.

We consider weak solutions to

with p > 1, q ≥ max{p − 1, 1}. We exploit the Moser iteration technique to prove a Harnack comparison inequality for C1 weak solutions. As a consequence we deduce a strong comparison principle.

The KP-II equation was derived by Kadmotsev and Petviashvili to explain stability of line solitary waves of shallow water. Recently, Mizumachi proved nonlinear stability of 1-line solitons for exponentially localized perturbations. In this paper, we prove stability of 1-line solitons for perturbations in (1 + x2)−1/2−0H1(ℝ2) and perturbations in H1(ℝ2) ∩ ∂xL2(ℝ2).

We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

Set differential equations are usually formulated in terms of the Hukuhara differential. As a consequence, the theory of set differential equations is perceived as an independent subject, in which all results are proved within the framework of the Hukuhara calculus. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of ℝd with their support functions. Using this representation, standard existence and uniqueness theorems for ordinary differential equations can be applied to set differential equations. We provide a geometric interpretation of the main result, and demonstrate that our approach overcomes the heavy restrictions that the use of the Hukuhara differential implies for the nature of a solution.

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can be obtained from the coefficients of the systems without using explicit solutions of their linearizations at the given branch. Our constructions are based on a comparison principle for the spectral flow and a generalization of a bifurcation theorem due to Szulkin.

Recently, Sun posed a series of conjectures on the log-concavity of the sequence , where is a familiar combinatorial sequence of positive integers. Luca and Stănică, Hou et al. and Chen et al. proved some of Sun's conjectures. In this paper, we present a criterion on the log-concavity of the sequence . The criterion is based on the existence of a function f(n) that satisfies some inequalities involving terms related to the sequence . Furthermore, we present a heuristic approach to compute f(n). As applications, we prove that, for the Zagier numbers , the sequences are strictly log-concave, which confirms a conjecture of Sun. We also prove the log-concavity of the sequence of Cohen–Rhin numbers.

In this paper, we study the singular fourth-order differential equation with a deviating argument:

By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.