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.     

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.     

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.     

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.     

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.     

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.     

Persistence properties and wave‐breaking criteria for the Geng‐Xue system

Considered herein is a two‐component Novikov equations (called Geng‐Xue system for short) with cubic nonlinearities. The persistence properties and some unique continuation properties of the solutions to the system in weighted L_p spaces are established. Moreover, a wave‐breaking criterion for strong solutions is determined in the lowest Sobolev space by using the localization analysis in the transport equation theory, and we also give a lower bound for the maximal existence time.     

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

We consider the problem Δ²u = V(x)u^{p + ?} in with u,Δu→0 as |x|→ + ∞, where , N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method.     

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.     

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.     

Global well‐posedness for the compressible magnetohydrodynamic system in the critical Lp framework

The present work is dedicated to the well‐posedness issue of strong solutions (away from vacuum) to the compressible viscous magnetohydrodynamic (MHD) system in (d ≥ 2). We aim at extending those results in previous studies to more general L^p critical framework. Precisely, by recasting the whole system in Lagrangian coordinates, we prove the local existence and uniqueness of solutions by means of Banach fixed‐point theorem. Furthermore, with the aid of effective velocity, we employ the energy argument to establish global a priori estimates, which lead to the unique global solution near constant equilibrium. Our results hold in case of small data but large highly oscillating initial velocity and magnetic field.     

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.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

Structural stability and stabilization of solutions of the reversible three‐component Gray‐Scott system

This paper is concerned with the structural stability and stabilization of solutions to the three‐component reversible Gray‐Scott system under the Dirichlet or Neumann boundary conditions defined in a bounded domain of for 1 ≤ n ≤ 3. We prove that each solution depends on changes in a coefficient of the ratio of the reverse and forward reaction rates for the autocatalytic reaction as well as proving the continuous dependence on the initial data. We also prove that under Dirichlet's boundary conditions, the system is stabilized to the stationary solution by finitely many Fourier modes.     

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.     

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.     

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 spectral analysis and exponential stability of a bi‐directional coupled wave‐ODE system

In this paper, an unstable linear time invariant (LTI) ODE system is stabilized exponentially by the PDE compensato—a wave equation with Kelvin‐Voigt (K‐V) damping. Direct feedback connections between the ODE system and wave equation are established: The velocity of the wave equation enters the ODE through the variable v_t(1,t); meanwhile, the output of the ODE is fluxed into the wave equation. It is found that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point , and there are two branches of asymptotic eigenvalues: the first branch approaches to , and the other branch tends to ?∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum‐determined growth condition and exponential stability of the system are concluded.