Acta Mathematica Scientia (Series A)
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Edited by  Editorial Committee of Acta Mathematica
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ISSN 1003-3998
CN  42-1226/O
26 June 2024, Volume 44 Issue 3 Previous Issue   
Hankel Operators on Vector-Valued Bergman Space with Exponential Type Weights
Dong Jianxiang
Acta mathematica scientia,Series A. 2024, 44 (3):  513-524. 
Abstract ( 61 )   RICH HTML PDF(747KB) ( 60 )   Save

In this paper, we study some characterizations of Hankel operators on vector-valued exponential type weights Bergman spaces $A^{2}_{\varphi}(\mathcal{H})$ induced by operator-valued function symbols and co-analytic operator-valued function symbols. Main results including the boundedness and compactness of Hankel operators.

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EC-tractability of Multivariate $\mathbb{L}_{\infty}$-approximation in Weighted Korobov Spaces
Zhang Jie, Sun Yiming, Liu Yongping
Acta mathematica scientia,Series A. 2024, 44 (3):  525-538. 
Abstract ( 35 )   RICH HTML PDF(797KB) ( 31 )   Save

In this paper we study exponential tractability of multivariate $\mathbb{L}_{\infty}$-approximation for weighted Korobov spaces in the worst case setting. We consider all algorithms that use the class $\Lambda^{\text{all}}$ of all linear functionals and the class $\Lambda^{\text{std}}$ of only function evaluations as information. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability which have not been discussed before in terms of two weight parameters of the problem.

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Two-Weight Inequalities for $ {n}$-dimentional Hardy Operator and Commutators
Wang Yaoyao, Lv Meichuan, Li Wenming
Acta mathematica scientia,Series A. 2024, 44 (3):  539-546. 
Abstract ( 27 )   RICH HTML PDF(617KB) ( 28 )   Save

Let $ P $ be the Hardy operator on $ \mathbb{R}^n $ and $ Q $ be the adjoint operator. In this paper, we get the two-weight inequalities for $ P $, $ Q $ and the commutators of $ P $ and $ Q $ with $ CMO $ functions.

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Properties and Computations of the $\mathfrak{m}$-WG Inverse
Wei Huaquan, Wu Hui, Liu Xiaoji, Jin Hongwei
Acta mathematica scientia,Series A. 2024, 44 (3):  547-562. 
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In this paper, the properties and computations of the $\mathfrak{m}$-WG inverse in Minskowski space are presented. Firstly, the characterization of the $\mathfrak{m}$-WG inverse is given by using the range and null space. Secondly, the relationship between the $\mathfrak{m}$-WG inverse and an invertible bordered matrix is given. Moreover, the perturbation bounds of the $\mathfrak{m}$-WG inverse is discussed. Finally, the successive matrix squaring algorithm is used to compute the $\mathfrak{m}$-WG inverse.

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On Convergence Sets of Power Series with Holomorphic Coefficients
Liu Hua, Basma Al-Shutnawi
Acta mathematica scientia,Series A. 2024, 44 (3):  563-574. 
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We consider convergence sets of formal power series $ f(z,t)=\sum\limits_{n=0}^{\infty} f_n(z)t^n $, where $ f_n(z) $ are holomorphic functions on a domain $ \Omega $ in $ \mathbb{C} $. A subset $ E $ of $ \Omega $ is said to be a convergence set in $ \Omega $ if there is a series $ f(z,t) $ such that $ E $ is exactly the set of points $ z $ for which $ f(z,t) $ converges as a power series in a single variable $ t $ in some neighborhood of the origin. A $ \sigma $-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. We prove that a subset of $ \mathbb{C} $ is a convergence set if and only if it is $ \sigma $-convex.

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Inequality and Asymptotic Approximation Generalized Nielsen Beta Function with Two Parameters
Yin Li, Huang Liguo, Zhang Jumei
Acta mathematica scientia,Series A. 2024, 44 (3):  575-585. 
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In this paper, we introduces a two parameters generalization of the classical Nielsen beta function, discusses its monotonicity, concavity and convexity, complete monotonicity, and asymptotic properties of integrals, and establishes some new inequalities.

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Global Small Solutions of 3D Hall-Magnetohydrodynamic Equations
Yu Yanghai, Wang Hui, Wu Xing
Acta mathematica scientia,Series A. 2024, 44 (3):  586-594. 
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In this paper, we establish the global solutions to the three-dimensional incompressible Hall-MHD equations provided that the initial horizontal velocity is sufficiently small, which improves the result given by Chae and Lee (J. Differ. Equ. 2014).

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Convergence Problem and Dispersive Blow-up for the Modified Kawahara Equation
Wang Weimin, Yan Wei
Acta mathematica scientia,Series A. 2024, 44 (3):  595-608. 
Abstract ( 13 )   RICH HTML PDF(718KB) ( 16 )   Save

In this paper, we consider the convergence problem and dispersive blow-up for the modified Kawahara equation. Firstly, we prove that $ u(x,t)\rightarrow u_0(x),$ a.e. $ x\in\mathbb{R} $ as $ t\rightarrow 0 $ by the Fourier restriction norm method, high-low frequency technique and Strichartz estimate, respectively. Here $ u(x,t) $ is the solution of the modified Kawahara equation, and the initial value $ u_0(x)\in H^{s}(\mathbb{R}) $ $ (s\geq\frac{1}{4}) $. Secondly, using the Fourier restriction norm method, we show that $ u(x,t)\rightarrow U(t)u_0(x) $ as $ t\rightarrow 0 $ with $ u_0(x)\in H^{s}(\mathbb{R}) $ $ (s>0) $. Finally, we establish the dispersive blow-up of the modified Kawahara equation.

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Persistence Property and Propagation Speed for the mCH-CH Equation
Li Yaohong, Tian Shoufu
Acta mathematica scientia,Series A. 2024, 44 (3):  609-620. 
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Initial value problem of the mCH-CH equation with cubic and quadratic nonlinearities is studied, which is an important integrable equations obtained by applying the bi-Hamiltonian duality approach to reduce the Gardner equation. Firstly, by constructing the weight function and using the energy method and Gronwall's inequality, the persistence property of the strong solution at infinity was obtained when the initial data decays exponentially or algebraically. Secondly, we prove that the strong solution $ m(x, t) $ has compact support when the initial data $ m_{0}, u_{0} $ has compact support, and the nontrivial solution $ u(x, t) $ no longer has compact support, but has exponential decay property at infinity.

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Positive Solutions of Nonlocal Beam Equations with Fully Nonlinear Terms
Guo Zhiyuan, Gao Yuan, Sun Jiacheng, Zhang Guowei
Acta mathematica scientia,Series A. 2024, 44 (3):  621-636. 
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Two classes of beam equations with fully nonlinear terms are studied that the boundary conditions involve Stieltjes integrals. An a priori bound on the third-order derivative term is evaluated by using a Gronwall -type inequality, and the existence of positive solutions is obtained based on the theory of fixed-point index on suitable open sets. The third-order derivative terms in nonlinearities have quadratic growth.

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Infinitely Many Large Energy Solutions for the Klein-Gordon-Born-Infeld Equation on $\mathbf{R}^3$ with Concave and Convex Nonlinearities
Chen Shangjie
Acta mathematica scientia,Series A. 2024, 44 (3):  637-649. 
Abstract ( 10 )   RICH HTML PDF(728KB) ( 17 )   Save

In this paper, we obtained the existence of infinitely many large energy solutions for the Klein-Gordon equation with concave and convex nonlinearities coupled with Born-Infeld theory on $\mathbb{R}^3$ by using $\mathbf{Z}_2$-Mountain Pass Theorem in critical point theory.

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On the Schrödinger-Poisson System with Asymptotically Linear Term and Coulomb Potential
Chang Jinhua, Wei Na
Acta mathematica scientia,Series A. 2024, 44 (3):  650-660. 
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The purpose of this paper is to study the following Schrödinger-Poisson system

$\begin{equation} \left\{\begin{array}{ll} -\Delta u + \Big(\omega-\sum\limits_{i=1}^m\frac{1}{|x-x_i|}\Big)u+\lambda\phi (x)u =f(u)\,\,\, &x\in \mathbb{R}^3, \\ -\Delta\phi = |u|^{2},\, \ &u\in H^1(\mathbb{R}^3), \end{array}\right. \end{equation}$

where $\omega>0$, $\lambda>0$, $x_i\in\mathbb{R}^3$, $ m\in\mathbb{N}$, $f(u)\sim lu$ (as $u\rightarrow+\infty$) is the asymptotically linear term. We study the effect of values of parameters $\omega$, $\lambda$ and asymptotic coefficient $l$ on the existence of ground state, multiple solutions to system (P), by using the variational method.

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Anisotropic Liouville Type Theorem for the Stationary Nematic Liquid Crystal Equations in $\mathbb{R}^{3}$
Chen Hao, Deng Xuemei, Bie Qunyi
Acta mathematica scientia,Series A. 2024, 44 (3):  661-669. 
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This paper investigates a Liouville type theorem for three-dimensional stationary liquid crystal equations. We show that if $u\in{L^6}({\mathbb{R}^3}) \cap {L^ q}({\mathbb{R}^3})$, $\nabla d\in{L^2}({\mathbb{R}^3}) \cap {L^q}({\mathbb{R}^3})$ and the anisotropic integrability conditions of ${u_i} \in L_{{x_i}}^{\frac{q}{{q - 2}}}L_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}), \forall i = 1,2,3$, $\frac{2}{q} + \frac{1}{s} \ge \frac{1}{2}$, $2 < q < \infty,1 \le s \le \infty $ are satisfied, then $u=0, \nabla d = 0$.

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Solitary and Periodic Solutions of the Generalized b-equation
Yang Jiaopeng, Liang Yong
Acta mathematica scientia,Series A. 2024, 44 (3):  670-686. 
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The research of generalized $b$-equation mainly focuses on the case of $b\ge0$. This paper uses the bifurcation method to investigate the bifurcation, nonlinear wave solutions and dynamical characteristics of generalized $b$-equation with $b=-3$. Under certain parameter conditions, one obtains the bifurcation phase diagram of the equation. Meanwhile, a new phenomenon is found different from the case of $b>0$, in which an infinite number of periodic trajectories in the traveling wave system cross the singular line $\varphi=c$. The existence of solitary and periodic solutions is given, and the 15 exact expressions for nonlinear wave solutions are obtained.

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Time Optimal Control for Semilinear Riemann-Liouville Fractional Evolution Feedback Control Systems
Bin Maojun, Shi Cuiyun
Acta mathematica scientia,Series A. 2024, 44 (3):  687-698. 
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In this article, we investigate a class of time feedback optimal control systems governed by Riemann-Liouville fractional semilinear differential equations in Banach space. At first, we discuss the existence and uniqueness of the mild solution for the equations by using fixed point theorem. Secondly, we show that the admissible trajectories set is nonempty involving the compactness of semigroup $ T(t)(t > 0) $ with the help of the Cesari roperty and the Fillippove theorem. Moreover, we also give an existence result for time eedback optimal control. In the end, an example is given to illustrate our main results.

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Multiple Positive Solutions of Kirchhoff Type Linearly Coupled System with Critical Exponent
Duan Xueliang, Wu Xiaofan, Wei Gongming, Yang Haitao
Acta mathematica scientia,Series A. 2024, 44 (3):  699-716. 
Abstract ( 8 )   RICH HTML PDF(791KB) ( 11 )   Save

This paper deals with the following Kirchhoff type linearly coupled system with Sobolev critical exponent

$\left\{ \begin{array}{l} -(1+b_{1}\|u\|^{2})\Delta u+\lambda_{1}u=u^{5}+\beta v, x\in\Omega,\\ -(1+b_{2}\|v\|^{2})\Delta v+\lambda_{2}v=v^{5}+\beta u, x\in\Omega,\\ u=v=0 {\rm on} \partial \Omega, \end{array} \right.$

where $ \Omega\subset\mathbb{R}^{3} $ is an open ball, $ \|\cdot\| $ is the standard norm of $ H_{0}^{1}(\Omega) $ and $ \beta\in\mathbb{R} $ is a coupling parameter. Constants $ b_{i}\geq0 $ and $ \lambda_{i}\in(-\lambda_{1}(\Omega),-\frac{1}{4}\lambda_{1}(\Omega)), i=1,2 $, where $ \lambda_{1}(\Omega) $ is the first eigenvalue of $ (-\Delta,H^{1}_{0}(\Omega)) $. Under the effects of Kirchhoff terms, we prove that the system has a positive ground state solution and a positive higher energy solution for some $ \beta>0 $ by using variational method. Moreover, we study the asymptotic behaviours of these solutions as $ \beta\rightarrow0 $.

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Blow-up Solutions in a p-Kirchhoff Equation of Pseudo-Parabolic Type
Li Fengjie, Li Ping
Acta mathematica scientia,Series A. 2024, 44 (3):  717-736. 
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This paper deals with a homogeneous Dirichlet initial-boundary value problem of the p-Kirchhoff pseudo-parabolic equation involving a variable exponent. Firstly, we give the optimal classification of initial energy by using modified potential well method and the auxiliary function method and obtain the criteria of the existence of blow-up and global existence of solutions. Secondly, we show asymptotic estimates about blow-up time for blow-up solutions and large time estimate for global solutions, respectively.

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Global Well-Posedness for the Fractional Navier-Stokes Equations with the Coriolis Force
Sun Xiaochun, Wu Yulian, Xu Gaoting
Acta mathematica scientia,Series A. 2024, 44 (3):  737-745. 
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We consider the Cauchy problem of fractional Navier-Stokes equations with the Coriolis force. Combining the $L^p-L^q$ and $\dot{H}^{\frac{5}{2}-2\alpha}-L^q$ smooth estimates of semiggroup $S$, it is proved that the well-posedness of the fractional Navier-Stokes equations with the Coriolis force and these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data $u_0$ is small enough in $\dot{H}_{\sigma}^{\frac{5}{2}-2\alpha}(\mathbb{R}^3)$.

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The Asymptotic Behaviors for Parameter Estimation of Stochastic Functional Differential Equations
Wang Jiaxi, Mao Mingzhi
Acta mathematica scientia,Series A. 2024, 44 (3):  746-760. 
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This paper studies the minimum distance estimate for stochastic functional differential equations (McKean-Vlasov SDE). Under some assumptions for the drift coefficient, it obtains the consistency and the limit distribution on the estimators as the diffusion coefficient goes to zero. Further, it also discusses the asymptotic behavior of the coefficient estimators under the condition of the $ L^\gamma $ penalty function. A typical case is provided.

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An Inverse Eigenvalue Problem for a Kind of Periodic Pseudo-Jacobi Matrix
Hu Wenyu, Xu Weiru, Zeng Yu
Acta mathematica scientia,Series A. 2024, 44 (3):  761-770. 
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In this paper, we consider the inverse eigenvalue problem of a class of periodic pseudo- Jacobi matrices, relying on a signature operator, whose component changes will cause large perturbations to the entire spectra of these matrices. The distribution of their eigenvalues is firstly discussed according to the roots distribution of the secular equations of these matrices. When the last component of the signature operator changes, the necessary and sufficient conditions for the solvability of the inverse eigenvalue problem are given, and the concrete construction process is also presented. Numerical examples are finally given to verify the effectiveness and feasibility of the proposed algorithm.

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A with-in Host HIV-1 Infection Dynamics Model Based on Virus-to-cell Infection and Cell-to-cell Transmission
Xu Rui, Zhou Kaijuan, Bai Ning
Acta mathematica scientia,Series A. 2024, 44 (3):  771-782. 
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In this paper, we consider an HIV-1 infection model with cell-to-cell transmission, intracellular delay, saturated CTL immune response and immune impairment. By calculation, we get immunity-inactivated and immunity-activated reproduction ratios. By analyzing the characteristic equations, the local stability of each of feasible equilibria is established. By means of suitable Lyapunov functional and LaSalle's invariance principle, it is proved that the global asymptotic stability of each of feasible equilibria is determined by immunity-inactivated and immunity-activated reproduction ratios: If the immunity-inactivated reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; if the immunity-inactivated reproduction ratio is greater than unity and the immunity-activated reproduction ratio is less than unity, the immunity-inactivated infection equilibrium is globally asymptotically stable; if the immunity-activated reproduction ratio is greater than unity, the chronic infection equilibrium is globally asymptotically stable. In addition, numerical simulation is carried out to illustrate the effects of immune impairment and cell-to-cell transmission on dynamics of the model.

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Autoregressive Model for Large-scale Three-mode Networks
Wei Yibing, Zhu Fukang
Acta mathematica scientia,Series A. 2024, 44 (3):  783-803. 
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Based on the two-mode network autoregressive (NAR) model, the specific form of the three-mode NAR model is given. This model considers three types of nodes in large-scale social networks, and edges are only allowed to occur between different types of nodes. First, the definition of the model, the reversibility and parameter identification of the model are introduced, and the estimation methods of quasi-maximum likelihood and conditional least squares and the large sample properties of the corresponding estimators are considered. Second, numerical simulations are carried out in multiple cases, the accuracy of the estimation methods and computational efficiency are compared, and finally a practical example is analyzed.

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Study on Comprehensive Control Strategies of an Age Structured Influenza Model
Yang Junyuan, Zhang Chenlin, Yang Li
Acta mathematica scientia,Series A. 2024, 44 (3):  804-814. 
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In this paper, we combine the age heterogeneity and the mechanisms of influenza to build an age structured SIR influenza model. Then we introduce two control variables-vaccination and treatment and propose the benefitial functional. Using the Pontryagin maximum principle and Ekland variant principle, we obtain the existence and uniqueness of the optimal control problem. Moreover, we employ forward and backward algorithms to do numerical experiments. Compared the results of many control measures, it is concluded that the combination of vaccination and treatment has the best control effect and furthermore, low-cost strategy is the best. Finally, through comprehensive analysis, vaccination has a best benefit from cost points of view and a combined measure has a best effect from public health perspectives.

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