Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Wenbo WANG, Quanqing LI In this paper, we investigate nonlinear Hamiltonian elliptic system {(-Δ)u=b→(x)⋅∇u+(V(x)+τ)u=K(x)g(v)inℝℕ,(-Δ)v=b→(x)⋅∇v+(V(x)+τ)v=K(x)f(u)inℝℕ,u(x)→0andv(x)→0as x →∞,where N ≥ 3, τ> 0 is a positive parameter and V, K are nonnegative continuous functions, f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Tarun Kumar CHAKRA, Gorachand CHAKRABORTY, Tarakanta NAYAK All possible arrangements of cycles of three periodic as well as four periodic Herman rings of transcendental meromorphic functions having at least one omitted value are determined. It is shown that if p = 3 or 4, then the number of p-cycles of Herman rings is at most one. We have also proved a result about the non-existence of a 3-cycle and a 4-cycle of Herman rings simultaneously. Finally some examples of functions having no Herman ring are discussed.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Dengfeng LI, Yanting LI The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Huanbin XUE, Jiye ZHANG The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and unbounded delay is addressed. On the assumption that the interconnected functions of the studied systems satisfy the Lipschitz condition, by resorting to vector Lyapunov approach and M-matrix theory, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems under arbitrary switching are obtained. The proposed method, which neither require the individual subsystems to share a Common Lyapunov Function (CLF), nor need to involve the values of individual Lyapunov functions at each switching time, provide a new way of thinking to study the stability of arbitrary switching. In addition, the proposed criteria are explicit, and it is convenient for practical applications. Finally, two numerical examples are given to illustrate the correctness and effectiveness of the proposed theories.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Yunbai DONG, Pei-Kee LIN, Bentuo ZHENG In this article, we study the preservation properties of (Šilov) boundary of multiplicative subgroups in C(X) spaces for non-surjective norm-preserving multiplicative maps. We also show a sufficient condition for surjective maps between groups of positive continuous functions to be a composition operator.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Guijuan LIN We give the sharp lower bound for Ricci curvature on the real ellipsoid in Cn+1, and prove the Lichnerowicz-Obata theorem for Kohn Laplacian.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Hanbing LIU, Haijun XIAO The aim of this work is to design oblique boundary feedback controller for stabilizing the equilibrium solutions to Boussinesq equations on a bounded and open domain in ℝ2. Two kinds of such feedback controller are provided, one is the proportional stabilizable feedback control, which is obtained by spectrum decomposition method, while another one is constructed via the Ricatti operator for an infinite time horizon optimal control problem. An example of periodic Boussinesq flow in 2-D channel is also given.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Xuejun ZHANG, Shenlian LI, Qingli SHANG, Yuting GUO In this article, the authors give a typical integral's bidirectional estimates for all cases. At the same time, several equivalent characterizations on the F(p, q, s, k) space in the unit ball of Cn are given.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Qiaozhen ZHU, Engui FAN, Jian XU The Gelfand-Levitan-Marchenko representation is used to analyze the initial-boundary value problem of two-component nonlinear Schrödinger equation on the half-line. It has shown that the global relation can be effectively analyzed by the Gelfand-Levitan-Marchenko representation. we also derive expressions for the Dirichlet-to-Neumann map to characterize the unknown boundary values.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Defu CHEN, Xia YE In this paper, we study the three-dimensional incompressible magnetohydrodynamic equations in a smooth bounded domains, in which the viscosity of the fluid and the magnetic diffusivity are concerned with density. The existence of global strong solutions is established in vacuum cases, provided the assumption that ( ∇μ(ρ0) Lp + ∇ν(ρ0) Lq + b0 L3 + ρ0 L∞) (p, q> 3) is small enough, there is not any smallness condition on the velocity.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Junhui XIE, Xiaozhong HUANG, Yiping CHEN In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions (0.1){(-Δ)psu=a(x) u q-2u+2αα+βc(x) u α-2u v β,in Ω,(-Δ)psu=b(x) v q-2v+2βα+βc(x) u α v β-2v,in Ω,u=v=0,inRn\Ω,where O is a smooth bounded domain in Rn, n> ps with s ɛ (0, 1) fixed, a(x), b(x), c(x) ≥ 0 and a(x), b(x), c(x) ɛ L∞(0), 1 < q < p and α, β> 1 satisfy p < α + β < p*, p* =npn-ps. By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem (0.1).

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Ogonnaya Michael ROMANUS, Ukamaka Victoria NNYABA, Monday Ogudu NNAKWE In this paper, relaxed iterative algorithms of Krasnoselskii-type and Halpern-type that approximate a solution of a system of a generalized mixed equilibrium problem and a common fixed point of a countable family of totally quasi-ϕ-asymptotically nonexpansive multi-valued maps are constructed. Strong convergence of the sequence generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with Kadec-Klee property. Furthermore, several applications of our theorems are also presented. Finally, our theorems are significant improvements on several important recent results for this class of nonlinear problems.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Zhizheng ZHANG, Junli HUANG The purpose of this paper is to introduce the concept of Cn WP-Bailey pairs. The Cn WP-Bailey transform is obtained by applying the Cn6ϕ5 summation formula. From this result, the Cn WP-Bailey lemma is deduced by making use of the Cn q-Dougall summation formula. Some applications are investigated. Finally, the case of elliptic Cn WP-Bailey pairs is discussed.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Lin YU, Ruhui WANG, Shoujiang ZHAO In this paper, the so-called (p, ϕ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces ℒp, ϕ(X) and the (p, ϕ)-Carleson measures is investigated. Specifically, it is proved that for q ɛ [2, ∞), the measure dμ := dfk qdℙ ⊗ dm is a (q, ϕ)-Carleson measure on Ω × ℕ for every f ɛ ℒq, ϕ(X) if and only if X has an equivalent norm which is q-uniformly convex; while for p ɛ (1, 2], the measure dμ:= dfk pdℙ ⊗ dm is a (p, ϕ)-Carleson measure on Ω × ℕ implies that f ɛ ℒp, ϕ(X) if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Sahbi BOUSSANDEL In this paper, we establish the existence of solutions for gradient systems of evolution under some type (M) and semi-coerciveness conditions. The main result is applied in order to solve nonlinear diffusion equations involving nonconvex energies.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Jae-Myoung KIM We investigate the local existence of smooth solutions of a 3D ideal magnetohydrodynamics (MHD) equations in a bounded domain and give a blow-up criteria to this equations with respect to vorticists.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Yong LIN, Hongye SONG We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Liya LIU, Daqing JIANG, Tasawar HAYAT, Bashir AHMAD In this article, we present a hepatitis B epidemic model with saturated incidence. The dynamic behaviors of the deterministic and stochastic system are studied. To this end, we first establish the local and global stability conditions of the equilibrium of the deterministic model. Second, by constructing suitable stochastic Lyapunov functions, the sufficient conditions for the existence of ergodic stationary distribution as well as extinction of hepatitis B are obtained.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Kun CHENG, Qi GAO In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution ub for the given problem. Moreover, we show that the energy of ub is strictly larger than twice the ground state energy. We also give a convergence property of ub as b ↘ 0, where b is regarded as a positive parameter.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Haixiang ZHANG, Xuehua YANG, Da XU In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Yezhou LI, Ningfang SONG The growth of entire functions under the q-difference operators is studied in this paper, and then some properties of Julia set of entire functions under the higher order q-difference operators are obtained.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Zhipeng ZHANG In this paper, we establish the existence of the global weak solutions for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions for the velocity field and the magnetic field in a bounded domain Ω ⊂ R3. Furthermore, we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible magnetohydrodynamic equations in energy space.

Abstract: Publication date: November 2018Source: Acta Mathematica Scientia, Volume 38, Issue 6Author(s): Abdul-Majeed AL-IZERI, Khalid LATRACH This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on Lp-spaces (1 ≥ p

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Huanyao WEN, Lei YAO, Changjiang ZHU The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Tao TAO, Liqun ZHANG We show the existence of dissipative Hölder continuous solutions of the Boussinesq equations. More precise, for any β ∈(0,⅕), a time interval [0,T] and any given smooth energy profile e:[0,T] → (0,∞), there exist a weak solution (v, θ) of the 3d Boussinesq equations such that (v,θ) ∈ Cβ × [0,T] with e(t) = ∫T3 v(x,t) 2 dx for all t ∈ [0,T]. This extend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Yanyan LI, Bo WANG In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ∇2 ψ + L(x, ∇ ψ), including the conformal hessian operator.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Congming LI, Zhigang WU In this paper, we consider systems of fractional Laplacian equations in ℝn with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Feimin HUANG, Yong WANG The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of (x,t) in the region ℝ3 × (0, + ∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Lin HE, Yongkai LIAO, Tao WANG, Huijiang ZHAO This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Gui-Qiang G. CHEN, Matthew RIGBY We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Binglong CHEN, Xiping ZHU The well-known Yau's uniformization conjecture states that any complete noncompact Kähler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress. In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on Kähler manifolds with minimal volume growth.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Daomin CAO, Shuangjie PENG, Shusen YAN In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations {-Δu=λ∑j=1k1Bδ(xO,j)(u-kj)+p, in Ω,u=0,on ∂Ωwhere 0 < p < 1, ⊂ ℝ2 is a bounded simply-connected smooth domain, κ (i, 1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x0 = (x0,1, … x0, k) of the Kirchhoff-Routh function defined on Ωk corresponding to (κ1, …, κk), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → + ∞, the vorticity set {y: uλ> κj} ∩ Bδ(x0, j) shrinks to {x0, j}, and the local vorticity strength near each x0, j approaches κj, j = 1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p 1, p = 1, p = 0 have already been studied in [11, 12] and [13] respectively.