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    NONLINEAR SEMIGROUP APPROACH TO TRANSPORT EQUATIONS WITH DELAYED NEUTRONS
    Abdul-Majeed AL-IZERI, Khalid LATRACH
    Acta mathematica scientia,Series B    2018, 38 (6): 1637-1654.  
    Abstract84)      PDF       Save
    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<+∞). Local and non local evolution problems are discussed.
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    A LIOUVILLE THEOREM FOR STATIONARY INCOMPRESSIBLE FLUIDS OF VON MISES TYPE
    Martin FUCHS, Jan MÜLLER
    Acta mathematica scientia,Series B    2019, 39 (1): 1-10.   DOI: 10.1007/s10473-019-0101-1
    Abstract67)      PDF       Save
    We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of u implies its constancy. A positive answer is true for p-fluids in the case p>1 (including the classical Navier-Stokes system for the choice p=2), and recently we established this Liouville property for the Prandtl-Eyring fluid model, for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is governed by a von Mises-type stress-strain relation formally corresponding to the case p=1. It turns out that, for dissipative potentials of linear growth, the condition of μ-ellipticity with exponent μ<2 is sufficient for proving the Liouville theorem.
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    THE SCHWARZ LEMMA AT THE BOUNDARY OF THE NON-CONVEX COMPLEX ELLIPSOIDS
    Le HE, Zhenhan TU
    Acta mathematica scientia,Series B    2019, 39 (4): 915-926.   DOI: 10.1007/s10473-019-0401-5
    Abstract49)      PDF       Save
    Let B2,p:={z∈C2:|z1|2+|z2|p<1} (0 < p < 1). Then, B2,p (0 < p < 1) is a non-convex complex ellipsoid in C2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ∂B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2,p, where z0 is any smooth boundary point of B2,p.
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    THE COMBINED INVISCID AND NON-RESISTIVE LIMIT FOR THE NONHOMOGENEOUS INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS
    Zhipeng ZHANG
    Acta mathematica scientia,Series B    2018, 38 (6): 1655-1677.  
    Abstract45)      PDF       Save
    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.
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    RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS
    Tian CHONG, Yuxin DONG, Hezi LIN, Yibin REN
    Acta mathematica scientia,Series B    2019, 39 (2): 339-356.   DOI: 10.1007/s10473-019-0201-y
    Abstract42)      PDF       Save
    We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kähler manifolds. We derive some elliptic differential inequalities from Weitzenbock formulas for the traceless Ricci tensor of Kähler manifolds with constant scalar curvature and the Bochner tensor of Kähler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature and complex space forms among Kähler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kähler manifolds and noncompact Kähler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.
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    LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS
    Xianming HOU, Huoxiong WU
    Acta mathematica scientia,Series B    2019, 39 (1): 11-25.   DOI: 10.1007/s10473-019-0102-0
    Abstract38)      PDF       Save
    In this paper, we establish the following limiting weak-type behaviors of Littlewood Paley g-function gφ:for nonnegative function fL1(Rn),

    where ft(x)=t-nf(t-1x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying Lαq-Dini condition are also given.
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    SIGN-CHANGING SOLUTIONS FOR THE STATIONARY KIRCHHOFF PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN IN RN
    Kun CHENG, Qi GAO
    Acta mathematica scientia,Series B    2018, 38 (6): 1712-1730.  
    Abstract36)      PDF       Save
    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.
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    CONVERGENCE RATES TO NONLINEAR DIFFUSIVE WAVES FOR SOLUTIONS TO NONLINEAR HYPERBOLIC SYSTEM
    Shifeng GENG, Yanjuan TANG
    Acta mathematica scientia,Series B    2019, 39 (1): 46-56.   DOI: 10.1007/s10473-019-0105-x
    Abstract35)      PDF       Save
    This article is involved with the asymptotic behavior of solutions for nonlinear hyperbolic system with external friction. The global existence of classical solutions is proven, and Lp convergence rates are obtained. Compared with the results obtained by Hsiao and Liu, better convergence rates are obtained in this article.
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    A NOTE ON q-DIFFERENCE OPERATOR AND RELATED LIMIT DIRECTION
    Yezhou LI, Ningfang SONG
    Acta mathematica scientia,Series B    2018, 38 (6): 1678-1688.  
    Abstract33)      PDF       Save
    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.
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    PRECISE MOMENT ASYMPTOTICS FOR THE STOCHASTIC HEAT EQUATION OF A TIME-DERIVATIVE GAUSSIAN NOISE
    Heyu LI, Xia CHEN
    Acta mathematica scientia,Series B    2019, 39 (3): 629-644.   DOI: 10.1007/s10473-019-0302-7
    Abstract33)      PDF       Save
    This article establishes the precise asymptotics
    Eum(t, x) (t→∞ or m→∞)
    for the stochastic heat equation

    with the time-derivative Gaussian noise W/t (t, x) that is fractional in time and homogeneous in space.
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    LIOUVILLE RESULTS FOR STABLE SOLUTIONS OF QUASILINEAR EQUATIONS WITH WEIGHTS
    Phuong LE, Vu HO
    Acta mathematica scientia,Series B    2019, 39 (2): 357-368.   DOI: 10.1007/s10473-019-0202-x
    Abstract33)      PDF       Save
    This paper is devoted to the quasilinear equation

    where p ≥ 2, Ω is a (bounded or unbounded) domain of RN, w1, w2 are nonnegative continuous functions and f is an increasing function. We establish a Liouville type theorem for nontrivial stable solutions of the equation under some mild assumptions on Ω, w1, w2 and f, which extends and unifies several results on this topic.
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    GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS
    Ting ZHANG, Wancheng SHENG
    Acta mathematica scientia,Series B    2019, 39 (1): 57-82.   DOI: 10.1007/s10473-019-0106-9
    Abstract30)      PDF       Save
    The Riemann problem for the chromatography equations in a conservative form is considered. The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states. The wave interaction problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann problem. In addition, it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.
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    FINITE EXTENSIONS OF GENERALIZED BESSEL SEQUENCES TO GENERALIZED FRAMES
    Dengfeng LI, Yanting LI
    Acta mathematica scientia,Series B    2018, 38 (6): 1939-1950.  
    Abstract29)      PDF       Save
    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.
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    NUMERICAL ANALYSIS FOR VOLTERRA INTEGRAL EQUATION WITH TWO KINDS OF DELAY
    Weishan ZHENG, Yanping CHEN
    Acta mathematica scientia,Series B    2019, 39 (2): 607-617.   DOI: 10.1007/s10473-019-0222-6
    Abstract29)      PDF       Save
    In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1, 1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L and Lωc2-norm. In the end, we give numerical test to confirm the conclusion.
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    HERMAN RINGS WITH SMALL PERIODS AND OMITTED VALUES
    Tarun Kumar CHAKRA, Gorachand CHAKRABORTY, Tarakanta NAYAK
    Acta mathematica scientia,Series B    2018, 38 (6): 1951-1965.  
    Abstract28)      PDF       Save
    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.
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    APPROXIMATE SOLUTION OF A p-th ROOT FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN (2, β)-BANACH SPACES
    Iz-iddine EL-FASSI, Hamid KHODAEI, Themistocles M. RASSIAS
    Acta mathematica scientia,Series B    2019, 39 (2): 369-381.   DOI: 10.1007/s10473-019-0203-9
    Abstract27)      PDF       Save
    In this paper, using the Brzd?k’s fixed point theorem [9, Theorem 1] in non-Archimedean (2, β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation

    where p ∈ {1, …, 5}, a1, …, ak are fixed nonzero reals when p ∈ {1, 3, 5} and are fixed positive reals when p ∈ {2, 4}.
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    ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL
    Haixiang ZHANG, Xuehua YANG, Da XU
    Acta mathematica scientia,Series B    2018, 38 (6): 1689-1711.  
    Abstract27)      PDF       Save
    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.
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    HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS
    Yong LIN, Hongye SONG
    Acta mathematica scientia,Series B    2018, 38 (6): 1751-1758.  
    Abstract26)      PDF       Save
    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.
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    EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS
    Wenbo WANG, Quanqing LI
    Acta mathematica scientia,Series B    2018, 38 (6): 1966-1980.  
    Abstract24)      PDF       Save
    In this paper, we investigate nonlinear Hamiltonian elliptic system

    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.
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    NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY
    Le CHEN, Kunwoo KIM
    Acta mathematica scientia,Series B    2019, 39 (3): 645-668.   DOI: 10.1007/s10473-019-0303-6
    Abstract24)      PDF       Save
    In this article, we study the nonlinear stochastic heat equation in the spatial domain Rd subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Zd to that on Rd. Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan[9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al[17] using different techniques.
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