Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

In this paper, we consider the following Schrödinger‐Poisson system: where parameters α,β∈(0,3),λ>0, , , and are the Hardy‐Littlewood‐Sobolev critical exponents. For α<β and λ>0, we prove the existence of nonnegative groundstate solution to above system. Moreover, applying Moser iteration scheme and Kelvin transformation, we show the behavior of nonnegative groundstate solution at infinity. For β<α and λ>0 small, we apply a perturbation method to study the existence of nonnegative solution. For β<α and λ is a particular value, we show the existence of infinitely many solutions to above system.     

A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions

In bounded smooth domains , N ∈ {2,3}, we consider the Keller‐Segel‐Stokes system and prove global existence of generalized solutions if These solutions are such that blow‐up into a persistent Dirac‐type singularity is excluded.     

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form where 0 ∈ Ω is a smooth bounded domain in , , and λ > 0 is a real parameter. Combining with the variational method and Nehari manifold method, two positive solutions of the system are obtained.     

Fibrations of 2‐crossed modules

In this study, we mainly show that the functor from the category X₂Mod of 2‐crossed modules of groups to the category Groups of groups assigning to each 2‐crossed module the group P, and to each 2‐crossed module morphism the group homomorphism f₀ is a fibration. In addition, we study some related properties.     

Semiclassical asymptotic behavior of ground state for the two‐component Hartree system

We consider the semiclassical asymptotic behaviors of ground state solution for the following two‐component Hartree system: which is originated from the study on cold atoms of boson and fermion system with long‐range interaction. Under the assumption by detailed compactness analysis, we prove that there is a β₀>0 such that if β<β₀, the system has a ground state solution. For this solution, the energy estimates and the decay rates are presented, and the asymptotic profiles as ε→0 are displayed in details for β<0 and β>0, respectively. Furthermore, we show that for β<0, the phase separation phenomenon may occur.     

The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation on the line. The initial value q(x,0) is given and satisfies the symmetric, nonzero boundary conditions at infinity, that is, q(x,0)→q_± as x→±∞, and |q_±|=q₀>0. The goal of this paper is to study the asymptotic behavior of the solution of this initial value problem as t→∞. The main tool is the asymptotic analysis of an associated matrix Riemann‐Hilbert problem by using the steepest descent method and the so‐called g‐function mechanism. We show that the solution q(x,t) of this initial value problem has a different asymptotic behavior in different regions of the xt‐plane. In the regions and , the solution takes the form of a plane wave. In the region , the solution takes the form of a modulated elliptic wave.     

Reciprocity formulae for generalized Dedekind‐Vasyunin‐cotangent sums

Recently, several works are done on the generalized Dedekind‐Vasyunin sum where and q are positive coprime integers, and ζ(a,x) denotes the Hurwitz zeta function. We prove explicit formula for the symmetric sum which is a new reciprocity law for the sum . Our result is a complement to recent results dealing with the sum studied by Bettin‐Conrey and then by Auli‐Bayad‐Beck. Accidentally, when a = 0, our reciprocity formula improves the known result in a previous study.     

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c^?. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium at t = ?∞, but the traveling waves need not connect the final disease‐free equilibrium at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.     

Numerical calculation of the discrete spectra of one‐dimensional Schrödinger operators with point interactions

In this paper, we consider one‐dimensional Schrödinger operators S_q on with a bounded potential q supported on the segment and a singular potential supported at the ends h₀, h₁. We consider an extension of the operator S_q in defined by the Schrödinger operator and matrix point conditions at the ends h₀, h₁. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator . Moreover, we provide closed‐form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self‐adjoint and nonself‐adjoint problems involving general point interactions described in terms of δ‐ and δ′‐distributions.     

More on a hyperbolic‐cotangent class of difference equations

We present a natural method for solving the difference equation where , parameter a, and initial values x_?j, , , are real or complex numbers. As a concrete example, the case k = 1, l = 2 is studied in detail, and several interesting formulas for solutions and objects used in the analysis of the equation are given. A useful remark on solvability of difference equations on complex domains is presented and used here.     

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.     

Global existence and time decay estimate of solutions to the Keller–Segel system

We consider the chemotaxis‐Navier–Stokes system 1.1-1.4 (Keller–Segel system) in the whole space, which describes the motion of oxygen‐driven bacteria, eukaryotes, in a fluid. We proved the global existence and time decay estimate of solutions to the Cauchy problem 1.1-1.2 in with the small initial data. Moreover, when the fluid motion is described by the Stokes equations, we established the global weak solutions to 1.3-1.4 in with the potential function ? is small and the initial density n₀(x) has finite mass.     

Blow‐up dynamics of L2−critical inhomogeneous fractional nonlinear Schrödinger equation

The purpose of this work is to investigate the blow‐up dynamics of L²?critical focusing inhomogeneous fractional nonlinear Schrödinger equation: with 0<b<1. For this, we establish a new compactness lemma related to the equation. By applying this lemma, we study the dynamical behavior for blow‐up solutions for initial data satisfying , where Q is the ground state solution of our problem.     

An Osgood type regularity criterion for the 2D Oldroyd‐B model

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, where denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2^j}.     

Global well‐posedness of classical solutions to 3D compressible magnetohydrodynamic equations with large potential force

We consider the Cauchy problem of isentropic compressible magnetohydrodynamic equations with large potential force in . When the initial data (ρ₀,u₀,H₀) is of small energy, we investigate the global well‐posedness of classical solutions where the flow density is allowed to contain vacuum states.     

Blow‐up of solutions for a viscoelastic wave equation with variable exponents

In this paper, we consider a viscoelastic wave equation with variable exponents: where the exponents of nonlinearity p(·) and m(·) are given functions and a,b > 0 are constants. For nonincreasing positive function g, we prove the blow‐up result for the solutions with positive initial energy as well as nonpositive initial energy. We extend the previous blow‐up results to a viscoelastic wave equation with variable exponents.     

Finite time blowup for Klein‐Gordon‐Schrödinger system

We investigate the blowup solutions to the Klein‐Gordon‐Schrödinger (KGS) system with power nonlinearity in spatial dimensions (N ≥ 2). Relying on a Lyapunov functional, we establish a perturbed virial‐type identity and prove the existence of blowup solutions for the system with a negative energy and small mass. Moreover, we obtain a new finite‐time blowup result of solutions to KGS system in the energy space by constructing a differential inequality.     

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain . We consider a growth term of logistic type in the equation of “u” in the form μu(1 ? u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense where f ^? is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ^?, if the constant chemotactic sensitivity χ satisfies we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.     

Two‐dimensional solvable system of difference equations with periodic coefficients

We show that the following two‐dimensional system of difference equations: where , , , and are periodic sequences, is solvable, considerably extending some results in the literature. In the case when all these four sequences are periodic with period 2 or with period 3, we present closed‐form formulas for the general solutions to the corresponding systems of difference equations. Some comments regarding theoretical and practical solvability of the system, connected to the value of the period of the sequences, are given.