#### Table of Content

26 April 2021, Volume 41 Issue 2
 A Partial Inverse Problem for the Sturm-Liouville Operator on Quantum Graphs with a Loop Shengyu Guan,Dongjie Wu,Sat Murat,Chuanfu Yang Acta mathematica scientia,Series A. 2021, 41 (2):  289-295.  Abstract ( 45 )   RICH HTML PDF (278KB) ( 55 )   This deals with the Sturm-Liouville operator on the quantum graphs with a loop. Given the potential on a part of edges, we try to recover the remaining potential from the subspectrum. The uniqueness theorem and a constructive algorithm for the solution of this partial inverse problem are provided.
 The Bonnesen-Style Isoperimetric Inequalities of the Tetrahedral in $\mathbb{R}^3$ Chunna Zeng,Lu Peng,Lei Ma,Xinxin Wang Acta mathematica scientia,Series A. 2021, 41 (2):  296-302.  Abstract ( 14 )   RICH HTML PDF (360KB) ( 25 )   This paper mainly studies Bonnesen-type isoperimetric inequalities and reverse Bonnesen-type isoperimetric inequalities of tetrahedron in $\mathbb{R}^3$. For a given tetrahedron in $\mathbb{R}^3$, by appling the relations of the surface area, volume, radius of inscribed sphere and radius of circumscribed radius, two important geometric inequalities are constructed, some Bonnesen-type isoperimetric inequalities of tetrahedron are obatainand, and a new simple proof of isoperimetric inequality of tetrahedron is achieved. Furthermore, by using the upper bound estimates of the isoperimetric deficit of tetrahedral, we obtain two reverse Bonnesen-type isoperimetric inequalities of tetrahedron of the expressions radius of inscribed sphere and radius of circumscribed radius.
 The Boundedness of Pseudodifferential Operators on $H^p(\omega)$ Yongming Wen,Xianming Hou Acta mathematica scientia,Series A. 2021, 41 (2):  303-312.  Abstract ( 23 )   RICH HTML PDF (325KB) ( 22 )   This paper gives the boundedness of a class of pseudodifferential operators $T_\sigma$ on weighted Hardy spaces $H^p(\omega)$, which improves the previous known results.
 ${A_{p}(\varphi)}$ Weights, Pseudo-Differential Operators and Their Commutators Yulong Deng,Shunchao Long Acta mathematica scientia,Series A. 2021, 41 (2):  313-325.  Abstract ( 19 )   RICH HTML PDF (348KB) ( 14 )   In this paper, we establish weighted $L^{p}$ inequalities for pseudo-differential operators $T$ and their commutators $[b, T]$ with smooth symbols in $S^{m}_{\rho, \delta}(\mathbb{R}^n)$, where $b\in BMO(\mathbb{R}^n)$ and the weight $\omega$ belongs to the new class $A_{p}(\varphi)$. It is well known that the Muckenhoupt class $A_{p}$ falls within the class $A_{p}(\varphi)$. This results extend the class of pseudo-differential operator to a wide range of $m$.
 Second Order Recursion Operator Structure of Constant Coefficient Infinite Dimension Hamiltonian System Wanpeng Geng,Wenxiu Ren,Yisu Cheng Acta mathematica scientia,Series A. 2021, 41 (2):  326-335.  Abstract ( 13 )   RICH HTML PDF (354KB) ( 18 )   By virtue of limited and formal constant coefficient Hamiltonian operator, it applies the method of general system recursion operator to Hamiltonian canonical system of infinite dimensional form. As to the result, the general structure of the recursion operator allowed by the next-order constant coefficient Hamiltonian operator under constraint condition and specific form of its coefficient are obtained. And then, it verifies the correctness and convenience of the conclusion by means of calculating example.
 Degenerate Regularity of Trajectory Statistical Solutions for the 3D Incompressible Navier-Stokes Equations Mingyue Xu,Caidi Zhao,Caraballo Tomás Acta mathematica scientia,Series A. 2021, 41 (2):  336-344.  Abstract ( 17 )   RICH HTML PDF (351KB) ( 12 )   In this article, the authors prove that if the (generalized) 3D Grashof number of the 3D autonomous incompressible Navier-Stokes equations is less than 2.057, then its weak trajectory statistical solutions possess (partial) degenerate regularity in the sense that they are supported by a set in which the weak solutions are in fact (partially) strong solutions. Also, they reveal that if the 3D Grashof number is less than 2.057, then the 3D incompressible Navier-Stokes equations possess only one complete and bounded strong solution which not only forward attracts but also pullback attracts its trajectories.
 Optimal Time Decay Rate of the Highest Derivative of Solutions to the Compressible Navier-Stokes Equations Qing Chen Acta mathematica scientia,Series A. 2021, 41 (2):  345-356.  Abstract ( 14 )   RICH HTML PDF (348KB) ( 14 )   In this paper, we are concerned with the time decay rates of smooth solutions to the Cauchy problem for the compressible Navier-Stokes equations. Under the assumptions that the initial data are close to the constant equilibrium state in $H^l(\mathbb{R}^3)$ with $l\geq3$ and belong to $\dot{H}^{-s}(\mathbb{R}^3)$ with $0 \le s < \frac52$, via decomposing the solutions into the low- and high-frequency parts, we establish the optimal convergence rates of all the derivatives of the solution by combining spectral analysis and the energy method.
 Pullback Attractors for Navier-Stokes-Voigt Equations with Nonlinear Damping Xiaoming Peng,Xiaoxiao Zheng,Yadong Shang Acta mathematica scientia,Series A. 2021, 41 (2):  357-369.  Abstract ( 6 )   RICH HTML PDF (381KB) ( 12 )   In this paper, we are concerned with the long-time behavior of solutions to the non-autonomous Navier-Stokes-Voigt equations with nonlinear damping. Firstly, we prove the existence and uniqueness of global weak solutions by the Galerkin method. Then, we focus on studying the existence of pullback attractors by using an energy method, which is more simple than the weak continuous method to establish the uniformly asymptotical compactness. In addition, some relationships between the attractors for the universe of fixed bounded sets and those associated to a universe given by another tempered condition are established.
 On a Nonlocal Alice-Bob-Schrödinger Equation: Bilinear Bäcklund and Darboux Transformations and Nonlinear Waves Yali Shen,Ruoxia Yao,Yarong Xia Acta mathematica scientia,Series A. 2021, 41 (2):  370-381.  Abstract ( 5 )   RICH HTML PDF (699KB) ( 6 )   Many events entangled or connected inherently however not recognized by human may happy at different spaces or different times in natural and social sciences can be described by using nonlocal Alice and Bob systems. In this paper, a newly proposed nonlinear Schrödinger equation (AB-NLS) derived from the well-known AKNS system is investigated, which is a real integrable two-place system. We obtain not only a bilinear Bäcklund transformation for the unreduced AB-NLS by the bilinear method, but also an $n$-fold Darboux transformation for the reduced AB-NLS. Armed with them we present some nonlinear waves for the nonlocal AB-NLS, which are quite different from that of the NLS equation and the solutions are analyzed about the singularity.
 Existence of Nonoscillatory Solutions Tending to Zero of Second-Order Neutral Dynamic Equations on Time Scales Yangcong Qiu,Qiru Wang Acta mathematica scientia,Series A. 2021, 41 (2):  382-387.  Abstract ( 9 )   RICH HTML PDF (291KB) ( 12 )   In this paper, a class of second-order nonlinear neutral dynamic equations on time scales are considered. We present some sufficient conditions for the existence of nonoscillatory solutions tending to zero of the equations by employing a Banach space and Krasnoselskii's fixed point theorem. In addition, two examples are provided to illustrate the applications of the conclusions.
 Liouville Type Theorem for Hartree Equations in Half Spaces Hongqiao Li Acta mathematica scientia,Series A. 2021, 41 (2):  388-401.  Abstract ( 6 )   RICH HTML PDF (405KB) ( 8 )   In this paper, we study the nonexistence of nontrivial positive solution for the following Hartree equation in half space $\left\{ \begin{array}{l} \displaystyle -\Delta u_i(y)=\sum\limits_{j=1}^n\int_{\partial\mathbb{R} _+^N} \frac{ F(u_j(\bar x, 0))}{|(\bar x, 0)-y|^{N-\alpha}}{\rm d}\bar xg(u_i(y)), &y\in\mathbb{R} _+^N, \\\displaystyle \frac{\partial u_i}{\partial \nu}(\bar x, 0)=\sum\limits_{j=1}^N\int_{\mathbb{R} _+^N} \frac{G(u_j(y))}{|(\bar x, 0)-y|^{N-\alpha}}\, {\rm d}y f(u_i(\bar x, 0)), &(\bar x, 0)\in\partial \mathbb{R} _+^N. \end{array} \right.$ Under some assumptions on the nonlinear functions F, G, f, g, we will show that the positive solutions of the above equation must be constants. We use moving plane method in an integral form to prove our result.
 Positive Solutions for a Class of Fractional Difference Equations Boundary Value Problems with p-Laplacian Operator Jiafa Xu,Honglin Luo,Lishan Liu Acta mathematica scientia,Series A. 2021, 41 (2):  402-414.  Abstract ( 14 )   RICH HTML PDF (399KB) ( 17 )   In this paper, we study a class of fractional difference equations boundary value problems with p-Laplacian operator. By virtue of the discrete Jensen inequalities, some relations between the considered problem and the corresponding problem without the p-Laplacian are established, and using the theory of fixed point index, the existence of positive solutions for the considered problem is obtained.
 Analysis of Impulsive Tempered Fractional Differential System via Variational Approach Jing Ren,Chengbo Zhai Acta mathematica scientia,Series A. 2021, 41 (2):  415-426.  Abstract ( 6 )   RICH HTML PDF (378KB) ( 9 )   The aim of this paper is to discuss a Dirichlet boundary value problem for tempered fractional differential system with instantaneous and non-instantaneous impulses in an appropriate admissible function space. By using variational method, we obtain the sufficient condition for the existence and uniqueness of weak solution, moreover, we show that every weak solution is a classical solution. In the end, an example is presented to highlight the feasibility of the theoretical results.
 Fractional Landweber Iterative Regularization Method to Identify Source Term for the Rayleigh-Stokes Equation Fan Yang,Qianchao Wang,Xiaoxiao Li Acta mathematica scientia,Series A. 2021, 41 (2):  427-450.  Abstract ( 3 )   RICH HTML PDF (772KB) ( 3 )   In this paper, the inverse problem of identifying the unknown sources for the Rayleigh-Stokes equation with a Riemann-Liouville fractional derivative in time is considered. We prove that such a problem is ill-posed and apply the fractional Landweber method to solve this inverse problem. Based on the results of conditional stability, under the priori and posteriori regularization parameters choice rules, the error estimates between the exact solution and the regularization solution are given respectively. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of these methods.
 A Modular grad-div Stabilized Finite Element Method for Nematic Liquid Crystal Flow Ting Li,Pengzhan Huang Acta mathematica scientia,Series A. 2021, 41 (2):  451-467.  Abstract ( 4 )   RICH HTML PDF (439KB) ( 5 )   In this paper, we presents a modular grad-div stabilized finite element method for nematic liquid crystal flow, which adds to the backward Euler scheme a post precessing step. This method can penalize for lack of mass conservation but it does not increase computational time for increasing stabilized parameters. Moreover, error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown. Finally, the theoretical findings and numerical efficiency are verified by numerical experiments.
 Error Estimates for Expanded Mixed Finite Element Methods for Nonlinear Hyperbolic Equation Keyan Wang,Qisheng Wang Acta mathematica scientia,Series A. 2021, 41 (2):  468-478.  Abstract ( 6 )   RICH HTML PDF (688KB) ( 21 )   In this paper, expanded mixed finite element method is developed for a class of nonlinear hyperbolic equation. A priori error estimate for the space discrete scheme is discussed in L∞(L2) norm. Centered finite differences are used to advance in time, a fully discrete scheme is proposed. Further, a priori error estimate for the fully discrete scheme is established. Finally, a numerical example is presented to confirm the theoretical results.
 An Effective Algorithm for Generalized Sylvester Equation Minimization Problem Under Columnwise Orthogonal Constraints Yueyuan Liu,Kai Wang,Shujuan Qin,Jiaofen Li Acta mathematica scientia,Series A. 2021, 41 (2):  479-495.  Abstract ( 7 )   RICH HTML PDF (2266KB) ( 12 )   This paper presents an efficient algorithm for solving the generalized Sylvester equation minimization problem under columnwise orthogonal constraints. Based on some geometric properties of the Stiefel manifold and the MPRP conjugate gradient method in Euclidean space, a Riemannian MPRP conjugate gradient algorithm with Armijo-type line search is proposed to solve the presented problem, and its global convergence is also established. An attractive property of the proposed method is that the direction generated by the method is always a descent direction for the objective function. Some numerical tests are given to show the efficiency of the proposed method. Comparisons with some existing methods are also given.
 Dynamics of Traveling Wave Solutions to Fully Nonlinear Heavy Ion-Acoustic Degenerate Relativistic Quantum Plasmas Leta Temesgen Desta,Wenjun Liu,Jian Ding Acta mathematica scientia,Series A. 2021, 41 (2):  496-506.  Abstract ( 13 )   RICH HTML PDF (2055KB) ( 11 )   Based on recent advancements on ion-acoustic wave models, a fully nonlinear heavy ion-acoustic waves (HIAWs), in astrophysical degenerate relativistic quantum plasmas (ADRQP), which is studied by Sultana and Schlickeiser, is investigated. We study the dynamical behavior of the model to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the model has bounded solutions (including kink and anti-kink wave solutions, periodic peakon, pseudo-peakon wave solutions, and compact solutions). These results completely improve the study of traveling wave solutions for the mentioned model.
 A Regularized Nonmonotone Inexact Smoothing Newton Algorithm for Weighted Symmetric Cone Complementarity Problems Xiaoni Chi,Rong Zeng,Sanyang Liu,Zhibin Zhu Acta mathematica scientia,Series A. 2021, 41 (2):  507-522.  Abstract ( 9 )   RICH HTML PDF (445KB) ( 12 )   In this paper, we propose a regularized nonmonotone inexact smoothing Newton algorithm for solving the weighted symmetric cone complementarity problem(wSCCP). In the algorithm, we consider the regularized parameter as an independent variable. Therefore, it is simpler and easier to implement than many available algorithms. At each iteration, we only need to obtain an inexact solution of a system of equations. Moreover, the nonmonotone line search technique adopted in our algorithm includes two popular nonmonotone search schemes. We prove that the algorithm is globally and locally quadratically convergent under suitable assumptions. Finally, some preliminary numerical results indicate the effectiveness of our algorithm.
 Invariant Borel Probability Measures for the Discrete Three Component Reversible Gray-Scott Model Qiaoyi Xiao,Chunqiu Li Acta mathematica scientia,Series A. 2021, 41 (2):  523-537.  Abstract ( 4 )   RICH HTML PDF (368KB) ( 4 )   In this paper, we study the Borel probability measures that can be associated to the time averaged observation of the process generated by the nonautonomous three component reversible Gray-Scott model on infinite lattices. We first show that there exists a pullback-${\cal D}$ attractor for the process. Furthermore, we establish the existence of a unique family of invariant Borel probability measures carried by this pullback attractor.
 On Optimal Dividend and Reinsurance Problems in the Diffusion Risk Model with Random Time Horizon Xiao Liu,Peng Yao,Zhenlong Chen Acta mathematica scientia,Series A. 2021, 41 (2):  538-547.  Abstract ( 7 )   RICH HTML PDF (372KB) ( 8 )   This paper studies the optimal dividend and reinsurance problems in the diffusion risk model with random time horizon. Assume that the proportional reinsurance strategy is applied, the random time is exponentially distributed, and if the random time comes before ruin, a fixed nonnegative value exists, otherwise another fixed nonnegative value exists, the optimal dividend and reinsurance strategies and the explicit expressions of the value function are obtained, and a numerical example is presented.
 Parisian Ruin for Spectrally Negative Lévy Processes Under a Hybrid Observation Scheme Xiaofeng Yang,Hua Dong,Hongshuai Dai Acta mathematica scientia,Series A. 2021, 41 (2):  548-561.  Abstract ( 5 )   RICH HTML PDF (443KB) ( 6 )   In this paper, we consider Parisian ruin problem based on a hybrid observation scheme for a spectrally negative Lévy process. We obtain the joint Laplace transform of the time to ruin and the deficit at ruin time by using the method of Laplace transform and the change of measure technique.
 Stability Analysis of Fractional-Order Hepatitis B Virus Infection Model With Immune Delay Xiling Li,Fei Gao,Wenqin Li Acta mathematica scientia,Series A. 2021, 41 (2):  562-576.  Abstract ( 12 )   RICH HTML PDF (547KB) ( 16 )   In this paper, we study the stability of fractional-order HBV (Hepatitis B Virus) infection model with immune delay and nonlinear incidence. Initially, the existence, uniqueness, positivity and boundedness of the model solutions are discussed. In addition, with the stability theory of functional differential equation, combining some new lemmas about Caputo fractional derivatives and some theories about fractional dynamic system, we discuss the in?uence of time delay on the stability of equilibrium point by analyzing the distribution of the characteristic equation roots on the equilibrium point. The results show that time delay does not a?ect the stability of disease-free equilibrium, while induces the stability of endemic equilibrium and produces periodic solutions with small amplitude nearby. Meanwhile, global asymptotic stability of the disease-free equilibrium is investigated by constructing a suitable Lyapunov function. Finally, using the fractional order delay stability principle, the corresponding linear controller is designed to effectively control the fractional order HBV infection model.