Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

We introduce a class of tent‐type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well‐posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.     

Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small‐cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , and δ_2,2 ≡ 1 and δ_2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

In this paper, we consider the nonlocal non‐autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non‐negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease‐free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo‐rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.     

Riemann boundary value problems on half space in Clifford analysis

In this paper, we discuss some properties of the Cauchy type integral operator defined over the half space of . As applications, we study a type of Riemann boundary value problem for solutions to polynomially generalized Cauchy–Riemann equations including with and as special cases over the half space of . Making use of Fischer‐type decomposition and the Clifford calculus for solutions to these equations, we will obtain explicit expressions of solutions to the kind of boundary value problems over the half space of . Copyright © 2012 John Wiley & Sons, Ltd.     

Linear statistics of matrix ensembles in classical background

Given a joint probability density function of N real random variables, , obtained from the eigenvector–eigenvalue decomposition of N × N random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, . For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment‐generating function , where denotes expectation value over the orthogonal (β = 1) and symplectic (β = 4) ensembles, in the form one plus a Schwartz function, none vanishing over for the Gaussian ensembles and for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large N asymptotic of the linear statistics from suitably scaled F(·). Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems where , , and . The novelty of this paper is that, relaxing the conditions on the potential function W(t,x), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed virus infection model with Beddington–DeAngelis infection function and cytotoxic T‐lymphocyte immune response

In this paper, a class of virus infection model with Beddington–DeAngelis infection function and cytotoxic T‐lymphocyte immune response is investigated. Time delay in the immune response term is incorporated into the model. We show that the dynamics of the model are determined by the basic reproduction number and the immune response reproduction number . If , then the infection‐free equilibrium is globally asymptotically stable. If , then the immune‐free equilibrium is globally asymptotically stable. If , then the stability of the interior equilibrium is investigated. We conclude that Hopf bifurcation occurs as the time delay passes through a critical value. Numerical simulations are carried out to support our theoretical conclusion well. Copyright © 2015 John Wiley & Sons, Ltd.     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of solutions for an anisotropic elliptic problem with variable exponent and singularity

In this paper, we study the following anisotropic problem with singularity: where is a bounded domain with smooth boundary. Using the critical point theory, we obtain the existence of weak solutions for the problem under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

Using the general formalism of 12 , a study of index theory for non‐Fredholm operators was initiated in 9 . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) as , respectively. The interesting feature is that violates the relative trace class condition introduced in 9 , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of 9 enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by whenever this limit exists. In the concrete example at hand, we prove Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , .     

Determinants of classical SG‐pseudodifferential operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG‐pseudodifferential operators on and suitable manifolds, using a finite‐part integral regularization technique. This allows us to define a zeta‐regularized determinant for parameter‐elliptic operators , , . For , the asymptotics of as and of as are derived. For suitable pairs we show that coincides with the so‐called relative determinant .     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Circular symmetrization,subordination and arclength problems on convex functions

We study the class of univalent analytic functions f in the unit disk of the form satisfying where Ω will be a proper subdomain of which is starlike with respect to . Let be the unique conformal mapping of onto Ω with and and . Let denote the arclength of the image of the circle , . The first result in this paper is an inequality for , which solves the general extremal problem , and contains many other well‐known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class .     

From Bolzano‐Weierstraß to Arzelà‐Ascoli

We show how one can obtain solutions to the Arzelà‐Ascoli theorem using suitable applications of the Bolzano‐Weierstraß principle. With this, we can apply the results from 10 and obtain a classification of the strength of instances of the Arzelà‐Ascoli theorem and a variant of it. Let be the statement that each equicontinuous sequence of functions contains a subsequence that converges uniformly with the rate and let be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that is instance‐wise equivalent, over , to the Bolzano‐Weierstraß principle and that is instance‐wise equivalent, over , to , and thus to the strong cohesive principle (). Moreover, we show that over the principles , and are equivalent.     

The threshold for combs in random graphs

For let denote the tree consisting of an ‐vertex path with disjoint ‐vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 794–802, 2016     

Asymptotics for the best Sobolev constants and their extremal functions

Let , where Ω is a bounded domain of , , and . We prove that , where ρ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of converge (as ) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured domain , for some .