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    26 October 2021, Volume 41 Issue 5 Previous Issue    Next Issue
    Bonnesen-Style Inequalities for Star Bodies
    Zengle Zhang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1249-1262. 
    Abstract ( 82 )   RICH HTML PDF (351KB) ( 153 )   Save

    Motivated by works of Lutwak and Petty[25-26, 37], a new star body ${\cal G}K$ associated with a given convex body $K$ is constructed. The isoperimetric inequality for ${\cal G}K$ and the reverse Bonnesen-style inequalities for K are established.

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    Generalized Transversality Theorem for Fredholm Operator in Global Analysis
    Qiang Li
    Acta mathematica scientia,Series A. 2021, 41 (5):  1263-1269. 
    Abstract ( 38 )   RICH HTML PDF (820KB) ( 18 )   Save

    Generalized transversality theorem for $ C^r $ mapping $ F(u, s):M\times S\rightarrow N $ is established in infinite dimensional Banach manifolds $ M, S, N $. If the mapping $ F(u, s) $ is generalized transversal to a single point set $ \{\hat{\theta}\} $, and $ f_s(u)=F(u, s) $ is a Fredholm operator in the sense of parameter s, then there exists a residual set $ \Sigma\subset S, $ such that $ f_s(u) $ are generalized transversal to $ \{\hat{\theta}\} $, for all $ s\in \Sigma. $

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    The Two-Dimensional Steady Chaplygin Gas Flows Passing a Straight Wedge
    Jia Jia
    Acta mathematica scientia,Series A. 2021, 41 (5):  1270-1282. 
    Abstract ( 40 )   RICH HTML PDF (349KB) ( 20 )   Save

    The purpose of this paper is to investigate the two-dimensional steady supersonic chaplygin gas flows passing a straight wedge. By the definition of Radon measure solution, the accurate expressions are obtained for all cases where the Mach number is greater than 1. It is quite different from the polytropic gas, for the chaplygin gas flows passing problems, there exists a Mach number $ M^{\ast}_{0} $, when the Mach number of incoming flows is greater than or equal to $ M^{\ast}_{0} $, the quality will be concentrated on the surface of the straight wedge. At this time, there are not piecewise smooth solutions in the Lebesgue sense. The limit analysis is used to prove that the limit obtained by Lebesgue integral is consistent with the solution obtained in the sence of Radon measure solution.

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    General Propagation Lattice Boltzmann Model for a Variable-Coefficient Compound KdV-Burgers Equation
    Zongning Zhang,Chunguang Li,jianqiang Dong
    Acta mathematica scientia,Series A. 2021, 41 (5):  1283-1295. 
    Abstract ( 28 )   RICH HTML PDF (816KB) ( 25 )   Save

    This paper studies the numerical calculation method of a kind of general Kdv-Burgers equation with variable coefficients. Firstly, a lattice Boltzmann model of the generalized KdV-Burgers equation with variable coefficients is obtained by selecting the equilibrium distribution function and adding the correction function. The model could accurately recover the KdV-Burgers equation without any assumptions. Secondly, this paper studies the temporal and spatial change trend of the nonlinear high-order derivative term in the equation, and compares it with the analytical solution, and then gives an error analysis. Finally, This paper analyzes the precision of the space and time of the equation. According to the simulation experiment results, the model could reach 2nd order accuracy. Numerical results show that the current lattice Boltzmann model is a satisfactory and efficient algorithm.

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    High Order Sign Preserving Entropy Stable Schemes
    Supei Zheng,Xia Xu,Jianhu Feng,Dou Jia
    Acta mathematica scientia,Series A. 2021, 41 (5):  1296-1310. 
    Abstract ( 17 )   RICH HTML PDF (1553KB) ( 18 )   Save

    Ensuring the sign preserving on entropy variable after the high-order reconstruction is fundamental in constructing the high-order entropy stable schemes. In this paper, we construct the 3rd order compact CWENO-type entropy stable schemes (Fjordholm's the 3rd order entropy conservative schemes with sign preserving compact CWENO reconstruction for the entropy variable) and prove the sign preserving on entropy variable with the 3rd order compact CWENO reconstruction. Numerical results show that the schemes can achieve third-order accuracy, and have high resolution, robustness and non-oscillation.

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    Finite Difference Scheme for the Nonhomogeneous Initial Boundary Value Problem of Critical Schrödinger Map
    Haiyun Deng,Hui Liu,Wenjing Song
    Acta mathematica scientia,Series A. 2021, 41 (5):  1311-1322. 
    Abstract ( 31 )   RICH HTML PDF (752KB) ( 27 )   Save

    In this paper, we study finite difference scheme for the nonhomgeneous initial boundary value problem of critical Schrödinger map in two-dimensional space. We get the convergence and stability of the difference scheme. At the same time, we prove that this difference scheme has good effectiveness and stability by numerical experiments.

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    A p-Laplace Eigenvalue Problem with Coercive Potentials
    Jingran He,Helin Guo,Wenqing Wang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1323-1332. 
    Abstract ( 37 )   RICH HTML PDF (345KB) ( 33 )   Save

    In this paper, we are concerned with the asymptotic behavior of solutions for a p-Laplace eigenvalue problem with coercive potentials. The bottom of the potential (The set of global minimum points of the potential) is an ellipsoid, we prove that the solutions of the problem will blow up at one of the endpoints of the major axis of the ellipsoid as the related parameter tends to a threshold value, and we also give the exact blow-up rate.

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    Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations
    Hui Yang,Yuzhu Han
    Acta mathematica scientia,Series A. 2021, 41 (5):  1333-1346. 
    Abstract ( 28 )   RICH HTML PDF (405KB) ( 29 )   Save

    In this paper, blow-up properties of solutions to an initial-boundary value problem for a parabolic type Kirchhoff equation are studied. The main results contain two parts. In the first part, we consider this problem with a general diffusion coefficient $M(\|\nabla u\|_2^2)$ and general nonlinearity $f(u)$. A new finite time blow-up criterion is established, and the upper and lower bounds for the blow-up time are also derived. In the second part, we deal with the case that $M(\|\nabla u\|_2^2)=a+b\|\nabla u\|_2^2$ and $f(u)=|u|^{q-1}u$, which was considered in[Computers and Mathematics with Applications, 2018, 75:3283-3297] with $q>3$, where global existence and finite time blow-up of solutions were obtained for subcritical, critical and supercritical initial energy. Their results are complemented in this paper in the sense that $q=3$ will be shown to be critical for the existence of finite time blow-up solutions to this problem.

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    Ground State Solutions of Nehari-Pohozaev Type for a Class of Reaction-Diffusion System
    Peng Chen
    Acta mathematica scientia,Series A. 2021, 41 (5):  1347-1356. 
    Abstract ( 26 )   RICH HTML PDF (339KB) ( 24 )   Save

    In this paper, we consider a class of nonlinear reaction diffusion systems, by using the non Nehari manifold method in strongly indefinite functional theory, a more direct and simple method to prove the ground state solution is given when the nonlinear term is superlinear. The existence of Nehari pankov type ground state solution is proved without strict monotone condition, and some new results are obtained.

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    Existence and Asymptotic Behavior of Solutions of a Class of k-Hessian Equation
    Lihong Zhang,Zedong Yang,Guotao Wang,Dumitru Baleanu
    Acta mathematica scientia,Series A. 2021, 41 (5):  1357-1371. 
    Abstract ( 30 )   RICH HTML PDF (389KB) ( 31 )   Save

    In this paper, we consider the following boundary blow-up $k$-Hessian problem where $\Omega \subset \mathbb{R} ^{N}$ is a smooth, bounded, strictly convex domain. We are concerned with the existence of the radially symmetric positive solutions of the $k$-Hessian equation and obtain new boundary asymptotic behavior of strictly convex blow-up positive solutions of the $k$-Hessian equation. Our approach mainly relies on the monotone iterative method, the upper and lower solution method and Karamata regular variation theory.

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    Nonexistence of Global Solutions for a Semilinear Double-Wave Equation with a Nonlinear Memory Term
    Baiping Ouyang,Shengzhong Xiao
    Acta mathematica scientia,Series A. 2021, 41 (5):  1372-1381. 
    Abstract ( 12 )   RICH HTML PDF (325KB) ( 14 )   Save

    In this paper, we investigate the blow-up of solutions to a semilinear double-wave equation with a nonlinear memory term. By establishing some auxiliary functions and using iteration methods associated with a nonlinear integral inequality, the estimate of upper bound for the lifespan is obtained.

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    Global Boundedness in a Chemotaxis-Haptotaxis Model with Nonlinear Diffusion and Signal Production
    Zhe Jia,Zuodong Yang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1382-1395. 
    Abstract ( 12 )   RICH HTML PDF (375KB) ( 12 )   Save

    This paper is concerned with an initial-boundary value problem for the following chemotaxis-haptotaxis model under homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R} ^{3} $, with $ \chi, \xi, \mu,\lambda,$ $\gamma >0$, $k>1$, $a \in \mathbb{R} $, and $D(u)\geq C_{D} (u+1)^{m-1}$ for $C_{D}>0, m\in \mathbb{R} $. It is shown that(i) For $0<\gamma\leq\frac{2}{3}$, if $\alpha>\gamma-k+1$ and $\beta>1-k$, there is a classical solution $(u, v, w)$ which is globally bounded to the above problem.(ii) For $\frac{2}{3}<\gamma\leq1$, if $\alpha>\gamma-k+\frac{1}{e}+1$ and or $\alpha>\gamma-k+1$ and there is a classical solution $(u, v, w)$ which is globally bounded to the above problem.

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    Regularity Criteria in Lorentz Spaces for the Three Dimensional Navier-Stokes Equations
    Daoguo Zhou
    Acta mathematica scientia,Series A. 2021, 41 (5):  1396-1404. 
    Abstract ( 25 )   RICH HTML PDF (287KB) ( 16 )   Save

    We prove regularity criteria for weak solutions to the three dimensional Navier-Stokes equations, via horizontal part of the velocity, or the vorticity, or the gradient of velocity in scaling invariant Lorentz spaces. Our results improve almost all known regularity criteria involving Lorentz spaces or two components.

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    Regularizing Effect of L1 Interplay Between Coefficients in Nonlinear Degenerate Elliptic Equations
    Weilin Zou,Yuanchun Ren,Meipin Xiao
    Acta mathematica scientia,Series A. 2021, 41 (5):  1405-1414. 
    Abstract ( 14 )   RICH HTML PDF (340KB) ( 37 )   Save

    In this paper, we consider a class of nonlinear degenerate elliptic equations of the form $-\mbox{div}(a(x,u,\nabla u))+b(x)g(u)+B(x,u,\nabla u)=f(x)$, where the principal part degenerates on $\{u=0\}$. Even if $f$ only belongs to $L^{1}(\Omega)$, the existence of bounded weak solutions are proven. This generalizes, to some extent, previous results.

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    Asymptotic Behavior for the Damped Boussinesq
    Xiaojing Cai,Yanjie Zhou
    Acta mathematica scientia,Series A. 2021, 41 (5):  1415-1427. 
    Abstract ( 18 )   RICH HTML PDF (348KB) ( 33 )   Save

    This paper mainly focus on the asymptotic behavior of the solutions for the 3D Boussinesq equations with damping term. We obtain the explicit time decay rate of the solutions by means of the Fourier splitting method. In addition, we get the upper-bound of the time decay results compared with the heat equation.

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    General Decay for the Transmission Problem of Viscoelastic Waves with not Necessarily Decreasing Kernel
    Zhiqing Liu,Zhongbo Fang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1428-1444. 
    Abstract ( 12 )   RICH HTML PDF (410KB) ( 12 )   Save

    In this paper, we are concerned with the asymptotic behavior for a transmission problem of viscoelastic waves with not necessarily decreasing kernel. We construct a new Lyapunov functional to derive the general decay estimate. Meanwhile, we present an example to illustrate that the decay rate we obtained includes exponential, algebraic and logarithmic decay etc.

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    Low Mach Number Limit to One-Dimensional Non-Isentropic Compressible Viscous Micropolar Fluid Model
    Xin Liu,Xiaolei Dong
    Acta mathematica scientia,Series A. 2021, 41 (5):  1445-1464. 
    Abstract ( 15 )   RICH HTML PDF (418KB) ( 10 )   Save

    In this paper, we consider the one dimensional non-isentropic compressible micropolar fluid model with general initial data, and justify rigorously the low Mach number limit of this system. The limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. Moreover, the difference between the states at ±∞ can be arbitrary large in this case.

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    Stability and Optimality of 2-D Mindlin-Timoshenko Plate System
    Chunguo Zhang,Yuzhi Fu,Yubiao Liu
    Acta mathematica scientia,Series A. 2021, 41 (5):  1465-1491. 
    Abstract ( 14 )   RICH HTML PDF (462KB) ( 11 )   Save

    In this paper, 2-D Mindlin Timoshenko plate system with local boundary control is studied. By using the receding horizon control method, the infinite time domain optimality problem is transformed into the finite time domain optimality problem. With the help of the multiplier technique, a priori estimation is made for any finite time domain system, and the observability inequality is obtained, which proves that the energy of the system is uniformly exponentially decay. Furthermore, with the aid of dual system, by means of the variational principle and Bellman optimality principle, the suboptimal conditions of the system in infinite time domain are obtained, and it is proved that the optimal trajectory is also exponential decay.

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    Oscillation Theorems of Second-Order Variable Delay Dynamic Equations with Quasilinear Neutral Term
    Guijiang Qin,Jiashan Yang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1492-1503. 
    Abstract ( 14 )   RICH HTML PDF (384KB) ( 9 )   Save

    The objective of this paper is to discuss the oscillation of a class of second-order dynamic equations with a quasi-linear neutral term on the time measurement chain. Under the regularity condition, by using the generalized Riccati transformation and the classical inequality, and combining with the time scales theory, some new oscillation theorems for the equations are established. The results obtained extend, improve and enrich the part of the study results established in previous literatures. Finally, examples are given to illustrate the applications of the obtained theorems.

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    The Averaging Method of Set Impulsive Differential Equations with Initial and Boundary Value Conditions
    Peiguang Wang,Kaiyu Yang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1504-1515. 
    Abstract ( 10 )   RICH HTML PDF (324KB) ( 6 )   Save

    This paper uses the scheme of full, and partially additive averaging method to study the set impulsive differential equations with the initial and multipoint boundary value problems in Euclidean space $ \mathbb{R}^{n} $, and proves the approximate relationship of the solutions between the original equations and the average equations.

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    The Influence of Splitting Index on Heteroclinic Orbit Bifurcation Under Periodic Perturbation
    Bin Long,Shanshan Xu,Hui Cao,Jianquan Li
    Acta mathematica scientia,Series A. 2021, 41 (5):  1516-1528. 
    Abstract ( 21 )   RICH HTML PDF (373KB) ( 13 )   Save

    By using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we consider the degenerate heteroclinic orbit bifurcation with $m$ dimensional periodic perturbations. The variational equation along the heteroclinic orbit has $d (d\ge1)$ bounded solutions. The splitting index of the unperturbed heteroclinic orbit is $s$. The bifurcation equation has $d+m$ variables and $d-s$ equations. The zeros of bifurcation function correspond to the existence of heteroclinic orbits for perturbed equation. If the splitting index $s<0$, it needs at least $1-s$ dimensional periodic perturbation can break the unperturbed heteroclinic orbit. If the splitting index $s\geq0$, there is a small perturbation can break the unperturbed heteroclinic orbit.

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    On the Observer Design for Fractional Singular Linear Systems
    Zaiyong Feng,Ning Chen,Yongpeng Tai,Zhengrong Xiang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1529-1544. 
    Abstract ( 45 )   RICH HTML PDF (607KB) ( 18 )   Save

    The paper discussed the Caputo fractional derivatives of impulse function $\delta(t) $, i.e., ${^C_0}{D}_t^\alpha\delta(t) $ and its Laplace transform, the distributional solution of Fractional Singular Linear Systems (FSLS) was consequently obtained. Based on the distributional solution of the system, the asymptotic stability theorem for FSLS was given, and the existence theorem of state observer for FSLS was proved. A simplified design method of full state observer for FSLS was investigated and summarized by pole assignment only for the slow subsystem. Finally, a state observer as an example was designed to verify the effectiveness of the proposed method.

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    The Collectively Sensitivity and Accessible in Non-Autonomous Composite Systems
    Xiaofang Yang,Xiao Tang,Tianxiu Lu
    Acta mathematica scientia,Series A. 2021, 41 (5):  1545-1554. 
    Abstract ( 9 )   RICH HTML PDF (371KB) ( 8 )   Save

    In this paper, collectively sensitivity, collectively infinity sensitivity, collectively Li-Yorke sensitivity and collectively accessible are defined in the non-autonomous discrete system. First of all, it is showed that, on compact metric spaces, mapping sequence $(f_k)^\infty_{k=1}$ is ${\cal P}$-chaos if and only if $ \forall n\in {\Bbb N}$ ($N$ is the set of natural numbers and does not contain 0). Then, under the condition that $f_{1, \infty}$ is uniformly convergence, it is proved that $f_{1, \infty}$ is ${\cal CP}$-chaos if and only if for any $m\in {\Bbb N}$, $f_{1, \infty}^{[m]}$ is ${\cal CP}$-chaos. Where ${\cal P}$-chaos denote one of the five properties: transitivity, sensitivity, infinitely sensitivity, accessibility and exact, ${\cal CP}$-chaos denote one of the four properties: collectively sensitivity, collectively infinity sensitivity, collectively Li-Yorke sensitivity and collectively accessible.

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    Exact Support Recovery of Sparse Signals from Noisy Measurements
    Min Fu,Jiajun Hao,Liejun Xie,Jinping Wang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1555-1565. 
    Abstract ( 16 )   RICH HTML PDF (627KB) ( 118 )   Save

    In this paper, we presents an improved OMP algorithm. Then the sparse reconstruction problem of OMP algorithm under the influence of the noise is studied. The conditions of SNR parameters for sparse reconstruction are obtained. Finnally, numerical simulation is used to verify the above conclusions.

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    Approximation Theorem and General Convergence of Population Games
    Huaxin Chen,Wensheng Jia
    Acta mathematica scientia,Series A. 2021, 41 (5):  1566-1573. 
    Abstract ( 22 )   RICH HTML PDF (366KB) ( 40 )   Save

    In this paper, we study whether the approximate solution of bounded rationality converges to the exact solution of complete rationality, which provides a theoretical support for the algorithm of population games. Firstly, under certain assumptions, the approximation theorem of population games under bounded rationality is proved. Then, by using the method of set-valued analysis and in the sense of Baire classification, we obtain the result that the solution of population games with perturbations on the objective function has generic convergence.

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    On Stochastic Accelerated Gradient with Convergence Rate of Regression Learning
    Yiyuan Cheng,Xingxing Zha,Yongquan Zhang
    Acta mathematica scientia,Series A. 2021, 41 (5):  1574-1584. 
    Abstract ( 29 )   RICH HTML PDF (384KB) ( 68 )   Save

    This paper studies the regression learning problem from given sample data by using stochastic approximation (SA) type algorithm, namely, the accelerated SA.We focus on problems without strong convexity, for which all well known algorithms achieve a convergence rate for function values of $O(1/n)$. We consider and analyze accelerated SA algorithm that achieves a rate of $O(1/n)$ for classical least square regression and logistic regression problems respectively. Comparing with the well known results, we only need fewer conditions to obtain the tight convergence rate for least square regression and logistic regression problems.

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