Authors:Dexing KONG; Qi LIU Pages: 745 - 755 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Dexing KONG, Qi LIU In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation ∂ 2 g i j ∂ t 2 + μ ( 1 + t ) λ ∂ g i j ∂ t = - 2 R i j , on Riemann surface. On the basis of the energy method, for 0 < λ ≤ 1, μ > λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t,x) of the solution metric gij remains uniformly bounded.

Authors:Xiaoli DING; Yaolin JIANG Pages: 756 - 768 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Xiaoli DING, Yaolin JIANG Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.

Authors:Xiaofei ZHANG; Jin LU; Xiaofei LI Pages: 769 - 777 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Xiaofei ZHANG, Jin LU, Xiaofei LI In this article, the sharp growth theorem for almost starlike mappings of complex order λ is given firstly. Secondly, distortion theorem along a unit direction is also established as the application of the growth theorem. In particular, using our results can reduce to some well-known results.

Authors:Jianlin ZHANG; Yuming QIN Pages: 778 - 790 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Jianlin ZHANG, Yuming QIN In this article, we establish exact solutions to the Cauchy problem for the 3D spherically symmetric incompressible Navier-Stokes equations and further study the existence and asymptotic behavior of solution.

Authors:Gholamreza ZAMANI ESKANDANI; Soheila AZARMI; Masoumeh RAEISI Pages: 791 - 804 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Gholamreza ZAMANI ESKANDANI, Soheila AZARMI, Masoumeh RAEISI In this article, we introduce and investigate the concept of multivalued hybrid mappings in CAT(0) spaces by using the concept of quasilinearization. Also, we present a new iterative algorithm involving products of Moreau-Yosida resolvents for finding a common element of the set of minimizers of a finite family of convex functions and a common fixed point of two multivalued hybrid mappings in CAT(0) spaces.

Authors:Huifang LIU; Zhiqiang MAO Pages: 819 - 828 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Huifang LIU, Zhiqiang MAO In this article, the existence of finite order entire solutions of nonlinear difference equations f n + P d ( z , f ) = p 1 e α 1 z + p 2 e α 2 z are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n − 2), p1, p2 are small meromorphic functions of ez , and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.

Authors:Qingzhai Fan; Xiaochun Fang Pages: 829 - 842 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Qingzhai Fan, Xiaochun Fang We introduce a special tracial Rokhlin property for unital C*-algebras. Let A be a unital tracial rank zero C*-algebra (or tracial rank no more than one C*-algebra). Suppose that α:G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C*-algebra. Then, the crossed product C*-algebra C*(G, A,α) has tracia rank zero (or has tracial rank no more than one). In fact, we get a more general results.

Authors:Yong LIN; Yiting WU Pages: 843 - 856 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Yong LIN, Yiting WU Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut=Δu + f(u) on G. The blow-up phenomenons for ut=Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

Authors:Yongting HUANG; Hongxia LIU Pages: 857 - 888 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Yongting HUANG, Hongxia LIU In this article, we are concerned with the nonlinear stability of the rarefaction wave for a one-dimensional macroscopic model derived from the Vlasov-Maxwell-Boltzmann system. The result shows that the large-time behavior of the solutions coincides with the one for both the Navier-Stokes-Poisson system and the Navier-Stokes system. Both the time-decay property of the rarefaction wave profile and the influence of the electromagnetic field play a key role in the analysis.

Authors:Shujun LIU; Fangqi CHEN; Zejun WANG Pages: 889 - 897 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Shujun LIU, Fangqi CHEN, Zejun WANG In this article, we give the existence of global L ∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v 1=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.

Authors:Shibin SU; Xiaokui ZHAO Pages: 898 - 914 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Shibin SU, Xiaokui ZHAO The initial boundary value problem of the one-dimensional magneto-hydrodynamics system, when the viscosity, thermal conductivity, and magnetic diffusion coefficients are general smooth functions of temperature, is considered in this article. A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1.

Authors:Rui LI; Chong LAI; Yonghong WU Pages: 915 - 925 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Rui LI, Chong LAI, Yonghong WU The existence of global weak solutions for a generalized Benjamin-Bona-Mahony-Burgers equation is established in the space C ( [ 0 , ∞ ) × R ) ∩ L ∞ ( [ 0 , ∞ ) ; H 1 ( R ) ) under the condition that its initial value belongs to the space H1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first order derivatives of the solution with respect to the space variable are derived to prove the existence.

Authors:Ling LI; Hongyi LI; Di ZHAO Pages: 926 - 934 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Ling LI, Hongyi LI, Di ZHAO In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P ( D n , B N ) is C 1 + α at z 0 ∈ E r ⊂ ∂ D n with f(0)= 0 and f ( z 0 ) = w 0 ∈ ∂ B N for any n, N ≥ 1, then there exist a nonnegative vector λ f = ( λ 1 , 0 , … , λ r , 0 , … , 0 ) T ∈ R 2 n satisfying λ i ≥ 1 2 2 n - 1 for 1 ≤ i ≤ r such that ( D f ( z ′ 0 ) ) T w ′ 0 = diag ( λ f ) z ′ 0 , where z ′ 0 and w ′ 0 are real versions of z 0 and w 0, respectively.

Authors:Pengyan WANG; Yongzhong WANG Pages: 935 - 949 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Pengyan WANG, Yongzhong WANG In this article, we study positive solutions to the system { A α u ( x ) = C n , α P V ∫ R n a 1 ( x - y ) ( u ( x ) - u ( y ) ) x - y n + α d y = f ( u ( x ) , υ ( x ) ) , B β υ ( x ) = C n , β P V ∫ R n a 2 ( x - y ) ( υ ( x ) - υ ( y ) ) x - y n + β d y = g ( u ( x ) , υ ( x ) ) . To reach our aim, by using the method of moving planes, we prove a narrow region principle and a {decay at infinity} by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.

Authors:Fei SONG; Yi QI; Guangming HU Pages: 950 - 964 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Fei SONG, Yi QI, Guangming HU The eventually distance minimizing ray (EDM ray) in moduli spaces of the Riemann surfaces of analytic finite type with 3g + n − 3 > 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmüller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.

Authors:Shaojun TANG; Lan ZHANG Pages: 973 - 1000 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Shaojun TANG, Lan ZHANG We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small. The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

Authors:Yonghui ZHOU; Yunrui YANG; Kepan LIU Pages: 1001 - 1024 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Yonghui ZHOU, Yunrui YANG, Kepan LIU This article is concerned with a population dynamic model with delay and quiescent stage. By using the weighted-energy method combining continuation method, the exponential stability of traveling waves of the model under non-quasi-monotonicity conditions is established. Particularly, the requirement for initial perturbation is weaker and it is uniformly bounded only at x = +∞ but may not be vanishing.

Authors:Azer KHANMAMEDOV; Sema YAYLA Pages: 1025 - 1042 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Azer KHANMAMEDOV, Sema YAYLA We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in R n , then the semigroup generated by the considered problem possesses a global attractor in H 2 ( R n ) × L 2 ( R n ) . We also establish the boundedness of this attractor in H 3 ( R n ) × H 2 ( R n ) .

Authors:K. Divya JOSEPH; P.A. DINESH Pages: 1043 - 1056 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): K. Divya JOSEPH, P.A. DINESH This article is concerned with the initial boundary value problem for a nonconservative system of hyperbolic equation appearing in elastodynamics in the space time domain x > 0, t > 0. The number of boundary conditions, to be prescribed at the boundary x = 0, depends on the number of characteristics entering the domain. Because our system is nonlinear, the characteristic speeds depends on the unknown and the direction of the characteristics curves are known apriori. As it is well known, the boundary condition has to be understood in a generalised way. One of the standard way is using vanishing viscosity method. We use this method to construct solution for a particular class of initial and boundary data, namely the initial and boundary datas that lie on the level sets of one of the Riemann invariants.

Authors:Weifeng JIANG; Kaitai LI Pages: 1057 - 1104 Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Weifeng JIANG, Kaitai LI In this article, we investigate three-dimensional solution with helical symmetry in a gap between two concentric rotating cylinders, inside is a helicoidal surface (screw propeller) while outside is a cylindrical surface. Establish the partial differential equations and its variational formulation satisfied by a helical solution in a helical coordinate system using tensor analysis method, we provide a computational method for the power and propulsion of the screw. The existence and uniqueness of weak helical solutions are proved.

Authors:Janusz BRZDEK; Krzysztof CIEPLIŃSKI Pages: 377 - 390 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Janusz BRZDEK, Krzysztof CIEPLIŃSKI The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

Authors:Huifang JIA; Gongbao LI Pages: 391 - 418 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Huifang JIA, Gongbao LI In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type - ∈ p M ( ∈ p - N ∫ R N ∇ u p ) Δ p u + V ( x ) u p - 2 u = f ( u ) in R N , where Δp is the p-Laplacian operator, 1 < p < N, M: R + → R + and V: R N → R + are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik-Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

Authors:Huijuan SONG; Jingxue YIN; Zejia WANG Pages: 419 - 428 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Huijuan SONG, Jingxue YIN, Zejia WANG In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system - div ( h 1 ( x ) ∇ u p - 2 ∇ u ) = d ( x ) u r - 2 u + G u ( x , u , υ ) - div ( h 2 ( x ) ∇ υ q - 2 ∇ υ ) = f ( x ) υ s - 2 υ + G υ ( x , u , υ ) u = υ = 0 in Ω in Ω on ∂ Ω where Ω is a bounded domain in R N with smooth boundary ∂Ω, N≥2, 1 < r < p < ∞, 1 < s < q < ∞; h1(x) and h2(x) are allowed to have “essential” zeroes at some points in Ω ; d ( x ) u r - 2 u and f ( x ) υ s - 2 υ are small sources with Gu(x,u,v), Gv(x,u,v) being their high-order perturbations with respect to (u,v) near the origin, respectively.

Authors:Jing FU; Daqing JIANG; Ningzhong SHI; Tasawar HAYAT; Ahmed ALSAEDI Pages: 429 - 440 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Jing FU, Daqing JIANG, Ningzhong SHI, Tasawar HAYAT, Ahmed ALSAEDI This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stochastic differential equations. Secondly, in the case of persistence, we prove that there exists a ergodic stationary distribution. Finally, numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.

Authors:Mingquan WEI; Dunyan YAN Pages: 441 - 449 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Mingquan WEI, Dunyan YAN In this article, we obtain the sharp bounds from L P ( G n ) to the space wL P ( G n ) for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L P ( G n ) to the space L PI ( G n ) are obtained.

Authors:Huoyuan DUAN; Junhua MA Pages: 450 - 470 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Huoyuan DUAN, Junhua MA On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.

Authors:Jianjun ZHANG; Liangwen LIAO Pages: 471 - 478 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Jianjun ZHANG, Liangwen LIAO In this article, we give a simple proof of Malmquist-Yosida type theorem of higher order algebraic differential equations, which is different from the methods as that of Gackstatter and Laine [2], and Steinmetz [12].

Authors:Zhou SHENG; Gonglin YUAN; Zengru CUI Pages: 479 - 496 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Zhou SHENG, Gonglin YUAN, Zengru CUI It is well known that trust region methods are very effective for optimization problems. In this article, a new adaptive trust region method is presented for solving unconstrained optimization problems. The proposed method combines a modified secant equation with the BFGS updated formula and an adaptive trust region radius, where the new trust region radius makes use of not only the function information but also the gradient information. Under suitable conditions, global convergence is proved, and we demonstrate the local superlinear convergence of the proposed method. The numerical results indicate that the proposed method is very efficient.

Authors:Jiafeng LIAO; Yang PU; Chunlei TANG Pages: 497 - 518 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Jiafeng LIAO, Yang PU, Chunlei TANG In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered, { - Δ u = g ( x ) u 2 * - 2 u + λ f ( x ) u q - 2 u , x ∈ Ω u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N (N ≥ 3) is an open bounded domain with smooth boundary, 1 < q < 2,λ > 0. 2 * = 2 N N - 2 is the critical Sobolev exponent, f ∈ L 2 * 2 * - q ( Ω ) is nonzero and nonnegative, and g ∈ C ( Ω ¯ ) is a positive function with k local maximum points. By the Nehari method and variational method, k + 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75(2012) 2660-2671].

Authors:Yirang YUAN; Aijie CHENG; Dangping YANG; Changfeng LI; Qing YANG Pages: 519 - 545 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yirang YUAN, Aijie CHENG, Dangping YANG, Changfeng LI, Qing YANG The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in l 2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.

Authors:Kamel BRAHIM; Latifa RIAHI Pages: 546 - 560 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Kamel BRAHIM, Latifa RIAHI In this article, we introduce the two dimensional Mellin transform M q ˜ ( f ) ( s , t ) , give some properties, establish the Paley-Wiener theorem and Plancherel formula, present the Hausdorff-Young inequality, and find several applications for the two dimensional Mellin transform.

Authors:P. BASKAR; S. PADMANABHAN; M. Syed ALI Pages: 561 - 579 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): P. BASKAR, S. PADMANABHAN, M. Syed ALI In this article, we investigates finite-time H ∞ control problem of Markovian jumping neural networks of neutral type with distributed time varying delays. The mathematical model of the Markovian jumping neural networks with distributed delays is established in which a set of neural networks are used as individual subsystems. Finite time stability analysis for such neural networks is addressed based on the linear matrix inequality approach. Numerical examples are given to illustrate the usefulness of our proposed method. The results obtained are compared with the results in the literature to show the conservativeness.

Authors:A.M. NAGY; N.H. SWEILAM Pages: 580 - 590 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): A.M. NAGY, N.H. SWEILAM In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann-Liouville and the Grünwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.

Authors:Yangrong LI; Lianbing SHE; Jinyan YIN Pages: 591 - 609 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yangrong LI, Lianbing SHE, Jinyan YIN A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.

Authors:Yekini SHEHU; Olaniyi. S. IYIOLA Pages: 610 - 626 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yekini SHEHU, Olaniyi. S. IYIOLA In this article, we first introduce an iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping (assuming existence) and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a nonlinear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.

Authors:Zhonglin WU; Shu WANG Pages: 627 - 642 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Zhonglin WU, Shu WANG We establish magnetic diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed viscosity. The convergence is verified under various Sobolev norms, including the L ∞(L 2) and L ∞(H1) norm.

Authors:Chunyan LIU; Zihou ZHANG; Yu ZHOU Pages: 643 - 650 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Chunyan LIU, Zihou ZHANG, Yu ZHOU In this article, we prove the following results: (1) A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; (2) A Banach space X is midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively compact k-Chebyshev set.

Authors:Lei ZHANG; Bin LIU Pages: 651 - 672 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Lei ZHANG, Bin LIU This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton (gμHS2) system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative (gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative (gμHS2) system, we show that the semi-group of the solution operator { S ( t ) } t ≥ 0 has a bounded absorbing set. Moreover, we prove that the dynamical system { S ( t ) } t ≥ 0 possesses a global attractor in the Sobolev space H 2 ( S ) × H 2 ( S ) .

Authors:Xiaojun ZHAO Pages: 673 - 680 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Xiaojun ZHAO In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation - Δ u = ∫ R N u ( y ) p x - y α d y u ( x ) p - 2 u ( x ) in R N , where N ≥ 3, 0 < α < {4,N}. Suppose that 2 < p < 2 N - α N - 2 , we will show that this problem does not possess nontrivial solution with finite Morse index. While for p = 2 N - α N - 2 , if i(u) < ∞ then we have ∫ R N ∫ R N u ( x ) p u ( y ) p x - y α d x d y < ∞ and ∫ R N ∇ u 2 d x = ∫ R N ∫ R N u ( x ) p u ( y ) p x - y α d x d y .

Authors:Yuecai HAN; Yifang SUN Pages: 681 - 694 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yuecai HAN, Yifang SUN The local existence and uniqueness of the solutions to backward stochastic differential equations(BSDEs, in short) driven by both fractional Brownian motions with Hurst parameter H∈(1/2,1) and the underlying standard Brownian motions are studied. The generalization of the Itô formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.

Authors:Hongmei ZHU Pages: 695 - 708 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Hongmei ZHU In this article, we study a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric α and a 1-form β. We determine all of Douglas general (α, β)-metrics on a manifold of dimension n > 2.

Authors:Rakesh KUMAR; Anuj Kumar SHARMA; Kulbhushan AGNIHOTRI Pages: 709 - 732 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Rakesh KUMAR, Anuj Kumar SHARMA, Kulbhushan AGNIHOTRI In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage (time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period (time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Authors:Xincai ZHU Pages: 733 - 744 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Xincai ZHU In this article, we study constrained minimizers of the following variational problem e ( ρ ) : = inf { u ∈ H 1 ( R 3 ) , ‖ u ‖ 2 2 = ρ } E ( u ) , ρ > 0 , where E(u) is the Schrödinger-Poisson-Slater (SPS) energy functional E ( u ) : = 1 2 ∫ R 3 ∇ u ( x ) 2 d x - 1 4 ∫ R 3 ∫ R 3 u 2 ( y ) u 2 ( x ) x - y d y d x - 1 p ∫ R 3 u ( x ) p d x in R 3 , and p∈ (2, 6). We prove the existence of minimizers for the cases 2 < p < 10 3 , ρ > 0, and p = 10 3 , 0 < ρ < ρ *, and show that e(ρ)= − ∞ for the other cases, where ρ* = ‖ ϕ ‖ 2 2 and ϕ(x) is the unique (up to translations) positive radially symmetric solution of - Δ u + u = u 7 3 in R 3 . Moreover, when e ( ρ * ) = - ∞ , the blow-up behavior of minimizers as ρ ↗ ρ * is also analyzed rigorously.

Authors:Xin Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Xin XU We are concerned with the zero dielectric constant limit for the full electro-magneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the well-prepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t=0. The strong convergence results only hold outside the initial layer.

Authors:Peng SUN Abstract: Publication date: May 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 3 Author(s): Peng SUN We prove a generalized Gauss-Kuzmin-Lévy theorem for the generalized Gauss transformation T p ( x ) = { p x } . In addition, we give an estimate for the constant that appears in the theorem.