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    DECAY ESTIMATE AND GLOBAL EXISTENCE OF SEMILINEAR THERMOELASTIC TIMOSHENKO SYSTEM WITH TWO DAMPING EFFECTS
    Weike WANG, Rui XUE
    Acta mathematica scientia,Series B    2019, 39 (6): 1461-1486.   DOI: 10.1007/s10473-019-0601-z
    Abstract126)      PDF       Save
    In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data is presented.
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    HILBERT PROBLEM 15 AND NONSTANDARD ANALYSIS (I)
    Banghe LI
    Acta mathematica scientia,Series B    2020, 40 (1): 1-15.   DOI: 10.1007/s10473-020-0101-4
    Abstract73)      PDF       Save
    Hilbert problem 15 required understanding Schubert's book. In this book, reducing to degenerate cases was one of the main methods for enumeration. We found that nonstandard analysis is a suitable tool for making rigorous of Schubert's proofs of some results, which used degeneration method, but are obviously not rigorous. In this paper, we give a rigorous proof for Example 4 in Schubert's book, Chapter 1. §4 according to his idea. This shows that Schubert's intuitive idea is correct, but to make it rigorous a lot of work should be done.
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    SYMMETRY OF POSITIVE SOLUTIONS FOR THE FRACTIONAL HARTREE EQUATION
    Xiangqing Liu
    Acta mathematica scientia,Series B    2019, 39 (6): 1508-1516.   DOI: 10.1007/s10473-019-0603-x
    Abstract62)      PDF       Save
    In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
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    SHUBIN REGULARITY FOR THE RADIALLY SYMMETRIC SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH DEBYE-YUKAWA POTENTIAL
    Léo GLANGETAS, Haoguang LI
    Acta mathematica scientia,Series B    2019, 39 (6): 1487-1507.   DOI: 10.1007/s10473-019-0602-y
    Abstract52)      PDF       Save
    In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.
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    GLOBAL L SOLUTIONS TO SYSTEM OF ISENTROPIC GAS DYNAMICS IN A DIVERGENT NOZZLE WITH FRICTION
    Qingyou SUN, Yunguang LU, Christian KLINGENBERG
    Acta mathematica scientia,Series B    2019, 39 (5): 1213-1218.   DOI: 10.1007/s10473-019-0501-2
    Abstract47)      PDF       Save
    In this article, we study the global L entropy solutions for the Cauchy problem of system of isentropic gas dynamics in a divergent nozzle with a friction. Especially when the adiabatic exponent γ=3, we apply for the maximum principle to obtain the L estimates w(ρδ,ε, uδ,ε) ≤ B(t) and z(ρδ,ε, uδ,ε) ≤ B(t) for the viscosity solutions (ρδ,ε, uδ,ε), where B(t) is a nonnegative bounded function for any finite time t. This work, in the special case γ ≥ 3, extends the previous works, which provided the global entropy solutions for the same Cauchy problem with the restriction w(ρδ,ε, uδ,ε) ≤ 0 or z(ρδ,ε, uδ,ε) ≤ 0.
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    INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS
    Wenchang CHU
    Acta mathematica scientia,Series B    2020, 40 (2): 293-315.   DOI: 10.1007/s10473-020-0201-y
    Abstract47)      PDF       Save
    The unified Ω-series of the Gauss and Bailey $_2F_1(\tfrac12)$-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts. Several remarkable transformation theorems for the Ω-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type, including a couple of beautiful expressions for π and the Catalan constant discovered by Guillera (2008).
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    MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY
    Lun GUO, Tingxi HU
    Acta mathematica scientia,Series B    2020, 40 (2): 316-340.   DOI: 10.1007/s10473-020-0202-x
    Abstract42)      PDF       Save
    In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity \begin{equation*} -\Delta u+(\lambda a(x)+1)u=\Big(\frac{1}{|x|^{\alpha}}\ast F(u)\Big)f(u) \ \ \text{in}\ \ \mathbb{R}^{N}, \end{equation*} where $N\geq 3$, $0<\alpha< \min\{N,4\}$, $\lambda$ is a positive parameter and the nonnegative potential function $a(x)$ is continuous. Using variational methods, we prove that if the potential well int$(a^{-1}(0))$ consists of $k$ disjoint components, then there exist at least $2^k-1$ multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as $\lambda\to +\infty$.
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    INFINITELY MANY SOLITARY WAVES DUE TO THE SECOND-HARMONIC GENERATION IN QUADRATIC MEDIA
    Chunhua WANG, Jing ZHOU
    Acta mathematica scientia,Series B    2020, 40 (1): 16-34.   DOI: 10.1007/s10473-020-0102-3
    Abstract40)      PDF       Save

    In this paper, we consider the following coupled Schr?dinger system with χ(2) nonlinearities

    which arises from second-harmonic generation in quadratic media. Here V1(x) and V2(x) are radially positive functions, 2 ≤ N < 6, α > 0 and α > β. Assume that the potential functions V1(x) and V2(x) satisfy some algebraic decay at infinity. Applying the finite dimensional reduction method, we construct an unbounded sequence of non-radial vector solutions of synchronized type.

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    COMPLEX SYMMETRIC TOEPLITZ OPERATORS ON THE UNIT POLYDISK AND THE UNIT BALL
    Cao JIANG, Xingtang DONG, Zehua ZHOU
    Acta mathematica scientia,Series B    2020, 40 (1): 35-44.   DOI: 10.1007/s10473-020-103-2
    Abstract37)      PDF       Save
    In this article, we study complex symmetric Toeplitz operators on the Bergman space and the pluriharmonic Bergman space in several variables. Surprisingly, the necessary and sufficient conditions for Toeplitz operators to be complex symmetric on these two spaces with certain conjugations are just the same. Also, some interesting symmetry properties of complex symmetric Toeplitz operators are obtained.
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    GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR SOME COUPLED SYSTEMS VIA A LYAPUNOV FUNCTIONAL
    Lamia DJEBARA, Salem ABDELMALEK, Samir BENDOUKHA
    Acta mathematica scientia,Series B    2019, 39 (6): 1538-1550.   DOI: 10.1007/s10473-019-0606-7
    Abstract37)      PDF       Save
    The purpose of this work is to study the global existence and asymptotic behavior of solutions to a coupled reaction-diffusion system describing epidemiological or chemical situations. Our analytical proofs are based on the Lyapunov functional methods.
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    STABILITY OF ε-ISOMETRIES ON L-SPACES
    Duanxu DAI
    Acta mathematica scientia,Series B    2019, 39 (6): 1733-1742.   DOI: 10.1007/s10473-019-0619-2
    Abstract31)      PDF       Save
    In this article, we discuss the stability of ε-isometries for L∞,λ-spaces. Indeed, we first study the relationship among separably injectivity, injectivity, cardinality injectivity and universally left stability of L∞,λ-spaces as well as we show that the second duals of universally left-stable spaces are injective, which gives a partial answer to a question of Bao-Cheng-Cheng-Dai, and then we prove a sharpen quantitative and generalized Sobczyk theorem which gives examples of nonseparable L-spaces X (but not injective) such that the couple (X, Y) is stable for every separable space Y. This gives a new positive answer to Qian's problem.
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    THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERÓN-ZYGMUND OPERATORS
    Jinxia LI, Baode LI, Jianxun HE
    Acta mathematica scientia,Series B    2020, 40 (1): 45-58.   DOI: 10.1007/s10473-020-104-1
    Abstract30)      PDF       Save
    Let T be an anisotropic Calderón-Zygmund operator and φ : Rn×[0, ∞) → [0, ∞) be an anisotropic Musielak-Orlicz function with φ(x, ·) being an Orlicz function and φ(·, t) being a Muckenhoupt A(A) weight. In this paper, our goal is to study two boundedness theorems for commutators of anisotropic Calderón-Zygmund operators. Precisely, when b ∈ BMOw(Rn, A) (a proper subspace of anisotropic bounded mean oscillation space BMO(Rn, A)), the commutator [b, T] is bounded from anisotropic weighted Hardy space Hw1(Rn, A) to weighted Lebesgue space Lw1(Rn) and when b ∈ BMO(Rn) (bounded mean oscillation space), the commutator [b, T] is bounded on Musielak-Orlicz space Lφ(Rn), which are extensions of the isotropic setting.
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    EPIDEMIC SPREAD ON ONE-WAY CIRCULAR-COUPLED NETWORKS
    Zhongpu XU, Xinchu FU
    Acta mathematica scientia,Series B    2019, 39 (6): 1713-1732.   DOI: 10.1007/s10473-019-0618-3
    Abstract29)      PDF       Save
    Real epidemic spreading networks are often composed of several kinds of complex networks interconnected with each other, such as Lyme disease, and the interrelated networks may have different topologies and epidemic dynamics. Moreover, most human infectious diseases are derived from animals, and zoonotic infections always spread on directed interconnected networks. So, in this article, we consider the epidemic dynamics of zoonotic infections on a unidirectional circular-coupled network. Here, we construct two unidirectional three-layer circular interactive networks, one model has direct contact between interactive networks, the other model describes diseases transmitted through vectors between interactive networks, which are established by introducing the heterogeneous mean-field approach method. Then we obtain the basic reproduction numbers and stability of equilibria of the two models. Through mathematical analysis and numerical simulations, it is found that basic reproduction numbers of the models depend on the infection rates, infection periods, average degrees, and degree ratios. Numerical simulations illustrate and expand these theoretical results very well.
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    NEW RESULTS FOR A CLASS OF UNIVALENT FUNCTIONS
    Zhigang PENG, Milutin OBRADOVI?
    Acta mathematica scientia,Series B    2019, 39 (6): 1579-1588.   DOI: 10.1007/s10473-019-0609-4
    Abstract26)      PDF       Save
    Let A denote the family of all analytic functions f(z) in the unit disk D={z ∈ C:|z|<1}, normalized by the conditions f(0)=0 and f'(0)=1. Let U denote the set of all functions fA satisfying the condition
    |(z/f(z))2 f'(z) -1|<1 for zD.
    Let Ω be the class of all fA for which
    |zf'(z) -f(z)|<1/2, z ∈ D.
    In this paper, the relations between the two classes are discussed. Furthermore, some new results on the class Ω are obtained.
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    ASYMPTOTIC BEHAVIOR OF SOLUTION BRANCHES OF NONLOCAL BOUNDARY VALUE PROBLEMS
    Xian XU, Baoxia QIN, Zhen WANG
    Acta mathematica scientia,Series B    2020, 40 (2): 341-354.   DOI: 10.1007/s10473-020-0203-9
    Abstract26)      PDF       Save
    In this article, by employing an oscillatory condition on the nonlinear term, a result is proved for the existence of connected component of solutions set of a nonlocal boundary value problem, which bifurcates from infinity and asymptotically oscillates over an interval of parameter values. An interesting and immediate consequence of such oscillation property of the connected component is the existence of infinitely many solutions of the nonlinear problem for all parameter values in that interval.
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    ON THE DIMENSIONS OF SPACES OF HARMONIC FUNCTIONS WITH POLYNOMIAL GROWTH
    Xiantao HUANG
    Acta mathematica scientia,Series B    2019, 39 (5): 1219-1234.   DOI: 10.1007/s10473-019-0502-1
    Abstract25)      PDF       Save
    In this paper, we obtain an estimate for the lower bound for the dimensions of harmonic functions with polynomial growth and a Liouville type theorem on manifolds with nonnegative Ricci curvature whose tangent cone at infinity is a unique metric cone with a conic measure.
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    GLOBAL EXISTENCE AND DECAY ESTIMATES FOR THE CLASSICAL SOLUTIONS TO A COMPRESSIBLE FLUID-PARTICLE INTERACTION MODEL
    Shijin DING, Bingyuan HUANG, Quanrong LI
    Acta mathematica scientia,Series B    2019, 39 (6): 1525-1537.   DOI: 10.1007/s10473-019-0605-8
    Abstract25)      PDF       Save
    We prove the global existence of classical solutions to a fluid-particle interaction model in R3, namely, compressible Navier-Stokes-Smoluchowski equations, when the initial data are close to the stationary state (ρ, 0, η) and the external potential satisfies the smallness assumption. Furthermore, optimal decay rates of classical solutions in H3-framework are obtained.
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    STABILITY OF MONOSTABLE WAVES FOR A NONLOCAL EQUATION WITH DELAY AND WITHOUT QUASI-MONOTONICITY
    Kepan LIU, Yunrui Yang, Yang YANG
    Acta mathematica scientia,Series B    2019, 39 (6): 1589-1604.   DOI: 10.1007/s10473-019-0610-y
    Abstract24)      PDF       Save
    By using the weighted-energy method combining continuation method, the exponential stability of monostable traveling waves for a delayed equation without quasi-monotonicity is established, including even the slower waves whose speed are close to the critical speed. Particularly, the nonlinearity is nonlocal in the equation and the initial perturbation is uniformly bounded only at x=+∞ but may not be vanishing.
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    BOUNDEDNESS OF MULTILINEAR LITTLEWOOD-PALEY OPERATORS ON AMALGAM-CAMPANATO SPACES
    Xiang LI, Qianjun HE, Dunyan YAN
    Acta mathematica scientia,Series B    2020, 40 (1): 272-292.   DOI: 10.1007/s10473-020-0119-7
    Abstract23)      PDF       Save
    In this paper, we consider the boundedness of multilinear Littlewood-Paley operators which include multilinear g-function, multilinear Lusin’s area integral and multilinear Littlewood-Paley gλ*-function. Furthermore, norm inequalities of the above operators hold on the corresponding Amalgam-Campanato spaces.
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    GLOBAL EXISTENCE, EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS
    Hua CHEN, Huiyang XU
    Acta mathematica scientia,Series B    2019, 39 (5): 1290-1308.   DOI: 10.1007/s10473-019-0508-8
    Abstract21)      PDF       Save
    In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut -△Xu=|u|p-1u, where X=(X1, X2, …, Xm) is a system of real smooth vector fields which satisfy the H?rmander's condition, and is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.
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