Acta Mathematica Scientia (Series B)
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Mathematics, CAS, China
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Scientia
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ISSN 0252-9602
CN 　42-1227/O
25 August 2020, Volume 40 Issue 4
Articles
 ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS Chunlei CAO, Yuefei WANG Acta mathematica scientia,Series B. 2020, 40 (4):  903-909.  DOI: 10.1007/s10473-020-0401-5 In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of ${\bf C^*}$ can neither contain a free Jordan arc nor be contained in any finite set of straight lines.
 BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS Zongguang LI, Rui LIU Acta mathematica scientia,Series B. 2020, 40 (4):  910-920.  DOI: 10.1007/s10473-020-0402-4 In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.
 UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION Wenqiang ZHAO, Yijin ZHANG Acta mathematica scientia,Series B. 2020, 40 (4):  921-933.  DOI: 10.1007/s10473-020-0403-3 In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.
 GROUND STATE SOLUTIONS FOR A SCHRÖDINGER-POISSON SYSTEM WITH UNCONVENTIONAL POTENTIAL Yao DU, Chunlei TANG Acta mathematica scientia,Series B. 2020, 40 (4):  934-944.  DOI: 10.1007/s10473-020-0404-2 We consider the Schrödinger-Poisson system with nonlinear term $Q(x)|u|^{p-1}u$, where the value of $\displaystyle\lim_{|x|\rightarrow\infty} Q(x)$ may not exist and $Q$ may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks.
 MIXED HARDY SPACES AND THEIR APPLICATIONS Wei DING, Yueping ZHU Acta mathematica scientia,Series B. 2020, 40 (4):  945-969.  DOI: 10.1007/s10473-020-0405-1 Multi-parameter mixed Hardy space $H^{p}_{\rm mix}$ is introduced by a new discrete Calderón's identity. As an application, we obtain the $H^{p}_{\rm mix} \rightarrow L^{p}(\mathbb{R}^{n_{1}+n_{2}})$ boundedness of operators in the mixed Journé's class.
 UNIQUENESS THEOREMS OF L-FUNCTIONS IN THE EXTENDED SELBERG CLASS Junfan CHEN, Chunhui QIU Acta mathematica scientia,Series B. 2020, 40 (4):  970-980.  DOI: 10.1007/s10473-020-0406-0 We establish uniqueness theorems of L-functions in the extended Selberg class, which show how an L-function and a meromorphic function are uniquely determined by their shared values in two finite sets. This can be seen as a new solution of a problem proposed by Gross.
 THE UNIQUENESS FOR ALGEBROID FUNCTIONS OF FINITE ORDER Yang TAN, Yinying KONG Acta mathematica scientia,Series B. 2020, 40 (4):  981-990.  DOI: 10.1007/s10473-020-0407-z In this article, we discuss, by Nevanlinna theory, the influence of multiple values and deficiencies on the uniqueness problem of algebroid functions. We get several uniqueness theorems of algebroid functions which include an at most 3v-valued theorem. These results extend the existing achievements of some scholars.
 LANDESMAN-LAZER TYPE (p, q)-EQUATIONS WITH NEUMANN CONDITION Nikolaos S. PAPAGEORGIOU, Calogero VETRO, Francesca VETRO Acta mathematica scientia,Series B. 2020, 40 (4):  991-1000.  DOI: 10.1007/s10473-020-0408-y We consider a Neumann problem driven by the (p,q)-Laplacian under the Landesman-Lazer type condition. Using the classical saddle point theorem and other classical results of the calculus of variations, we show that the problem has at least one nontrivial weak solution.
 POINTWISE ESTIMATES OF SOLUTIONS FOR THE NONLINEAR VISCOUS WAVE EQUATION IN EVEN DIMENSIONS Nianying LI Acta mathematica scientia,Series B. 2020, 40 (4):  1001-1019.  DOI: 10.1007/s10473-020-0409-x In this article, we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions (n≥4). We use the Green's function method. Our approach is on the basis of the detailed analysis of the Green's function of the linearized system. We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions. It is shown that the solution exhibits a generalized Huygens principle.
 REMARKS ON THE CAUCHY PROBLEM OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS Yongkai LIAO Acta mathematica scientia,Series B. 2020, 40 (4):  1020-1034.  DOI: 10.1007/s10473-020-0410-4 This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas. Based on the elaborate energy estimates, we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t) and W(t) introduced by Liao and Zhao [J. Differential Equations, 2018, 265(5): 2076-2120]. Our results also improve upon the results obtained in Liao and Zhao [J. Differential Equations, 2018, 265(5): 2076-2120].
 ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES Wei CHEN, Qiong WANG Acta mathematica scientia,Series B. 2020, 40 (4):  1035-1044.  DOI: 10.1007/s10473-020-0411-3 This article investigates the algebraic differential independence concerning the Euler $\Gamma$-function and the function $F$ in a certain class $\mathbb{F}$ which contains Dirichlet $\mathcal{L}$-functions, $\mathcal{L}$-functions in the extended Selberg class, or some periodic functions. We prove that the Euler $\Gamma$-function and the function $F$ cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions $\phi$ with $\rho(\phi)<1$.
 SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM Yekini SHEHU Acta mathematica scientia,Series B. 2020, 40 (4):  1045-1063.  DOI: 10.1007/s10473-020-0412-2 We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.
 GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR A FRACTIONAL SCHRÖ DINGER-POISSON SYSTEM WITH CRITICAL GROWTH Wentao HUANG, Li WANG Acta mathematica scientia,Series B. 2020, 40 (4):  1064-1080.  DOI: 10.1007/s10473-020-0413-1 We study the following nonlinear fractional Schrödinger-Poisson system with critical growth: $$\label{eqS0.1} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \left \{ \begin{array}{ll} (-\Delta )^s u+u+\phi u=f(u)+|u|^{2^*_s-2}u,\quad &x\in \mathbb{R}^3, \\ (-\Delta )^t \phi=u^2,& x\in \mathbb{R}^3, \\ \end{array} \right . \end{array}$$ where $0 < s,t < 1$, $2s+2t > 3$ and $2^*_s=\frac{6}{3-2s}$ is the critical Sobolev exponent in $\mathbb{R}^3$. Under some more general assumptions on $f$, we prove that (0.1) admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.
 ON THE INSTABILITY OF GROUND STATES FOR A GENERALIZED DAVEY-STEWARTSON SYSTEM Yuanping DENG, Xiaoguan LI, Qian SHENG Acta mathematica scientia,Series B. 2020, 40 (4):  1081-1090.  DOI: 10.1007/s10473-020-0414-0 In this paper, we give a simpler proof for Ohta's theorems [1995, Ann. Inst. Henri Poincare, 63, 111; 1995, Diff. Integral Eq., 8, 1775] on the strong instability of the ground states for a generalized Davey-Stewartson system. In addition, a sufficient condition is given to ensure the nonexistence of a minimizer for a variational problem, which is related to the stability of the standing waves of the Davey-Stewartson system. This result shows that the stability result of Ohta [Diff. Integral Eq., 8, 1775] is sharp.
 A STOCHASTIC GALERKIN METHOD FOR MAXWELL EQUATIONS WITH UNCERTAINTY Lizheng CHENG, Bo WANG, Ziqing XIE Acta mathematica scientia,Series B. 2020, 40 (4):  1091-1104.  DOI: 10.1007/s10473-020-0415-z In this article, we investigate a stochastic Galerkin method for the Maxwell equations with random inputs. The generalized Polynomial Chaos (gPC) expansion technique is used to obtain a deterministic system of the gPC expansion coefficients. The regularity of the solution with respect to the random is analyzed. On the basis of the regularity results, the optimal convergence rate of the stochastic Galerkin approach for Maxwell equations with random inputs is proved. Numerical examples are presented to support the theoretical analysis.
 ON SELECTIONS OF SET-VALUED EULER-LAGRANGE INCLUSIONS WITH APPLICATIONS Hamid KHODAEI, Iz-iddine EL-FASSI, Bahman HAYATI Acta mathematica scientia,Series B. 2020, 40 (4):  1105-1115.  DOI: 10.1007/s10473-020-0416-y We discuss the set-valued dynamics related to the theory of functional equations. We look for selections of convex set-valued functions satisfying set-valued Euler-Lagrange inclusions. We improve and extend upon some of the results in [13, 20], but under weaker assumptions. Some applications of our results are also provided.
 $L^{p}$ SOLUTION OF GENERAL MEAN-FIELD BSDES WITH CONTINUOUS COEFFICIENTS Yajie CHEN, Chuanzhi XING, Xiao ZHANG Acta mathematica scientia,Series B. 2020, 40 (4):  1116-1140.  DOI: 10.1007/s10473-020-0417-x In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution $(Y,Z)$ but also on the law $P_{Y}$ of $Y$. The first part of the paper is devoted to the existence and uniqueness of solutions in $L^p$, $1< p\leq2$, where the monotonicity conditions are satisfied. Next, we show that if the generator $f$ is uniformly continuous in $(\mu,y,z)$, uniformly with respect to $(t,\omega)$, and if the terminal value $\xi$ belongs to $L^{p}(\Omega,\mathcal{F},P)$ with \$1
 THE GENERALIZED LOWER ORDER OF DIRICHLET SERIES Qingyuan CHEN, Yingying HUO Acta mathematica scientia,Series B. 2020, 40 (4):  1141-1151.  DOI: 10.1007/s10473-020-0418-9 In this paper, we study the generalized lower order of entire functions defined by Dirichlet series. By constructing the Newton polygon based on Knopp-Kojima's formula, we obtain a relation between the coefficients of the Dirichlet series and its generalized lower order.
 QUASI-NEUTRAL LIMIT AND THE INITIAL LAYER PROBLEM OF THE DRIFT-DIFFUSION MODEL Shu WANG, Limin JIANG Acta mathematica scientia,Series B. 2020, 40 (4):  1152-1170.  DOI: 10.1007/s10473-020-0419-8 In this article we study quasi-neutral limit and the initial layer problem of the drift-diffusion model. Different from others studies, we consider the physical case that the mobilities of the charges are different. The quasi-neutral limit with an initial layer structure is rigorously proved by using the weighted energy method coupled with multi-scaling asymptotic expansions.