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26 August 2020, Volume 40 Issue 4 Previous Issue    Next Issue

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The Existence of the Measure Solution for the Non-Isentropic Chaplygin Gas Yufeng Chen,Tingting Chen,Zhen Wang Acta mathematica scientia,Series A. 2020, 40 (4):  833-841.  Abstract ( 136 )   RICH HTML PDF (339KB) ( 197 )   Save In this paper, we consider the Riemann problem of one-dimensional non-isentropic Chaplygin gas dynamics. In the case that the pressure and internal energy are general, we construct the classical solution by the characteristic analysis under a sufficient and necessary condition on the Riemann data. As the density ρ concentrates, the δ shock waves exist. According to the theory of Radon measure, the general Rankine-Hugoniot is induced. Combining with entropy condition, we obtain the measure solution for this problem. This extends the result for the isentropic Chaplygin gas. References | Related Articles | Metrics

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Multiple Solutions for Nonlinear Equations Related to Kirchhoff Type Equations Wencui Liang,Zhengjie Zhang Acta mathematica scientia,Series A. 2020, 40 (4):  842-849.  Abstract ( 109 )   RICH HTML PDF (302KB) ( 166 )   Save In this paper, we will discuss the following Kirchhoff equation $\left\{\begin{array}{ll}-\left(a+b\int_{\mathbb{R}^{3}}{|\nabla u{{|}^{2}}}\right)\triangle u+u=\left(1+\varepsilon g(x)\right) u^{p}, x\in\mathbb{R}^{3}, \\u\in H^{1}\left(\mathbb{R}^{3}\right), \end{array}\right.$ where $\varepsilon$, $a$, $b$ are positive constants, $1＜ p＜5,g(x)\in L^{\infty}\left(\mathbb{R}^{3}\right)$. When $g(x)$ satisfy some conditions, we use perturbation method prove that there exists a $\varepsilon_{0}$, if $0＜\varepsilon ＜\varepsilon_{0}$ there are many solutions for above problem. References | Related Articles | Metrics

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Existence of Ground States for a Class of Modified Gross-Pitaevskii Equations Xiaomeng Huang,Yimin Zhang Acta mathematica scientia,Series A. 2020, 40 (4):  850-856.  Abstract ( 99 )   RICH HTML PDF (340KB) ( 136 )   Save In this paper, using scaling technique and some rearrangement inequalities, existence and classification of ground states for a class of Modified Gross-Pitaevskii equations with respect to the nonlinear exponent p. If $2＜ p ＜ 2+\frac{4}{N}$ , for any $c>0$ , there is at least a minimizer for this problem. If $p=2+\frac{4}{N}$ and $c\leq\|\phi\|_2$ or $c>\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$ (the definition of $\|\phi\|_2$ see section 1) or $p>2+\frac{4}{N}$ , there is no minimizer for this problem. But it is unclear if $p=2+\frac{4}{N}$ and $\|\phi\|_2＜c＜\left(\frac{3}{2}\right)^{\frac{N}{4}}\|\phi\|_2$. References | Related Articles | Metrics

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Classical Solutions for a Kind of New Kirchhoff-Type Problems Without Boundary Constraint Yue Wang,Hongmin Suo,Wei Wei Acta mathematica scientia,Series A. 2020, 40 (4):  857-868.  Abstract ( 163 )   RICH HTML PDF (391KB) ( 133 )   Save The existence of classical solutions for a class of new Kirchhoff-type problems with unadulterated exponential item at right are considered on boundary cuboid in this article, and all results on the theoretical basis are based on constructors of functions. We show that the exact expressions of classical solutions with all exponents except the minus one by using the techniques of analysis associated with it. At the same time, we give some examples to explaining and verifying our conclusion. References | Related Articles | Metrics

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Multiple Pertubations to a Quasilinear Schrödinger Equation Xiaoli Han Acta mathematica scientia,Series A. 2020, 40 (4):  869-881.  Abstract ( 105 )   RICH HTML PDF (292KB) ( 114 )   Save In this paper, we will deal with the Cauchy problem of a class of quasilinear Schrödinger equations. The main goal is to obtain some sufficient conditions on the blow up in finite time and global existence of the solution. References | Related Articles | Metrics

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Existence and Uniqueness of Fixed Points for a Class of Abstract Binary Nonlinear Operators Ping Shi Acta mathematica scientia,Series A. 2020, 40 (4):  882-890.  Abstract ( 94 )   RICH HTML PDF (324KB) ( 136 )   Save In this paper, we study the existence and uniqueness of fixed points for a class of abstract binary nonlinear operators, by means of the properties of cone, monotone iterative methods and spectral radius conditions of related positive bounded linear operators in partially ordered Banach spaces; and then generalize and improve a classical theorem, and so obtain several new results. Finally, we present an application to initial value problems of first order nonlinear ordinary differential equations. References | Related Articles | Metrics

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An Inverse Initial Value Problem for Degenerate Parabolic Equations Liu Yang,Zuicha Deng Acta mathematica scientia,Series A. 2020, 40 (4):  891-903.  Abstract ( 104 )   RICH HTML PDF (2054KB) ( 146 )   Save This paper investigates an inverse problem of reconstructing the initial value in a degenerate parabolic equation. Problems of this type have important applications in several fields of applied science. The key to numerically solve such problem is to construct highorder difference schemes for corresponding forward problem. However, the dumping point method which is widely-used for numerically solving classical heat conduction equations cannot be applied to degenerate parabolic equations, because the principal coefficients are zero on degenerate boundaries. In this paper, a new but quite simple technique is proposed to construct a difference scheme of second order accuracy, and the stability and convergence of the scheme are proved. In order to accelerate the convergence rate, the conjugate gradient method is adopted to obtain numerical solutions of the inverse problem. Numerical verification on the efficiency and accuracy of the proposed algorithm is also performed. Figures and Tables | References | Related Articles | Metrics

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The Landau Equation with Inflow Boundary Condition in a Finite Channel Liping Liu,Hang Yang,Xuan Ma Acta mathematica scientia,Series A. 2020, 40 (4):  904-917.  Abstract ( 84 )   RICH HTML PDF (380KB) ( 101 )   Save This paper is concerned with the inflow boundary value problem of the Landau equation in a finite channel. Based on an elementary energy method, a global strong solution is established for the corresponding problem in a new function space which has mild regularity in normal direction. Moreover, the large time behaviors and the regularity propagation of the solution are also obtained. References | Related Articles | Metrics

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Global Regularity for the 3D Liquid Crystal Equations with Fractional Diffusion Qiang Li Acta mathematica scientia,Series A. 2020, 40 (4):  918-924.  Abstract ( 93 )   RICH HTML PDF (280KB) ( 104 )   Save In this paper, the focus is the global regularity of three-dimensional liquid crystal equations with fractional dissipations $(-\Delta)^{\alpha}u$ and $(-\Delta)^{\beta}d$. The objective is to establish the global regularity of the fractional liquid crystal equations with the minimal amount of dissipations. And it is obtained that the equations have a global classical solution with sufficiently smooth data if $\alpha\geq\frac{5}{4}$ and $\beta\geq\frac{5}{4}$. References | Related Articles | Metrics

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Homogenization of Higher-Order Equations with Mixed Boundary Condition Juan Wang,Jie Zhao Acta mathematica scientia,Series A. 2020, 40 (4):  925-933.  Abstract ( 93 )   RICH HTML PDF (326KB) ( 97 )   Save The paper is concerned with the convergence rates of solutions for homogenization of m$-order elliptic equations with the mixed Dirichlet-Neumann boundary conditions. Our approach, which involves smoothing operator and thus avoids the estimates of the boundary discrepancies terms. As a consequence, we establish the rates of convergence in $H_{0}^{m}$ as well as $L^{2}$. This work may be regarded as an extension of the usage smoothing operator to the case of higher-order equations with mixed boundary condition settings. References | Related Articles | Metrics

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Oscillation Criteria of Second Order Nonlinear Neutral Emden-Fowler Differential Equations with Damping Zhiyu Zhang,Feifei Song,Tongxing Li,Yuanhong Yu Acta mathematica scientia,Series A. 2020, 40 (4):  934-946.  Abstract ( 76 )   RICH HTML PDF (414KB) ( 122 )   Save In this paper, by using the methods of exponential function transformation, Riccati transformation and inequality techniques, we study the oscillation behavior for a class of second order nonlinear neutral delay Emden-Fowler differential equations with the most extensive and physically meaningful damping term. Several new oscillation theorems which extend and improve some known results in the literature recently are established and some examples are provided to illustrate the relevance of new theorems. References | Related Articles | Metrics

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Small-Amplitude Solitary Interfacial Traveling Waves in a Gravity-Capillary Two-Layered Fluid with Vorticity Wang Lingjun, Wang Qiusi Acta mathematica scientia,Series A. 2020, 40 (4):  947-976.  Abstract ( 60 )   RICH HTML PDF (686KB) ( 29 )   Save Considered in this paper is a two-dimensional rotational two-layered fluid with finite thickness, rigid bottom and upper lid, and acted upon by gravity and interfacial tension. Under the assumption that in both layers of the fluid the horizontal velocity of the fluid particle does not exceed the wave speed, we prove the existence of small-amplitude solitary waves traveling at the interface of the fluid. The argument is mainly based on the center-manifold reduction technique and the dynamical systems methods. By studying the spectrum of the linearization at the equilibrium of the Hamiltonian system formulated from the hydrodynamic problem, and its bifurcation diagram in the parameter plane, we obtain the homoclinic solutions of the reduced system which gives the solitary interfacial water waves. References | Related Articles | Metrics

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Another Proof of Variational Principle for Sub-Additive Potentials Lan Xu Acta mathematica scientia,Series A. 2020, 40 (4):  977-982.  Abstract ( 79 )   RICH HTML PDF (302KB) ( 119 )   Save The variational principle of topological pressure for sub-additive potentials was proved by Cao etal. This generalized variational principle plays an important role in the theory of dimension and equilibrium state for some expansive dynamical systems such as, for example, self-affine fractals and non-conformal repellers. The main purpose of this paper is to give another proof for the variational principle in the sub-additive case when the entropy map is upper semi-continuous. References | Related Articles | Metrics

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Global Stability of the Nonmonotone Critical Traveling Waves for Reaction Diffusion Equations Yonghui Zhou Acta mathematica scientia,Series A. 2020, 40 (4):  983-992.  Abstract ( 109 )   RICH HTML PDF (319KB) ( 131 )   Save In this paper, by using the Fourier's transform method combining with the weighted energy method with some new skills, the global stability of the nonmonotone critical traveling waves for a delayed equation is established. References | Related Articles | Metrics

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Numerical Oscillation Analysis of the Mixed Type Impulsive Differential Equation Zhaolin Yan,Jianfang Gao Acta mathematica scientia,Series A. 2020, 40 (4):  993-1006.  Abstract ( 91 )   RICH HTML PDF (381KB) ( 80 )   Save The purpose of this paper is to study oscillation of θ-methods for linear mixed type impulsive differential equations with piecewise constant arguments. We obtain conditions under which θ-methods can preserve the oscillation and non-oscillation for linear mixed type impulsive differential equations with piecewise constant arguments. The numerical examples are given to confirm the theoretical results. Figures and Tables | References | Related Articles | Metrics

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Optimality of Robust Approximation Solutions for Uncertain Convex Optimization Problems Meng Li,Guolin Yu Acta mathematica scientia,Series A. 2020, 40 (4):  1007-1017.  Abstract ( 73 )   RICH HTML PDF (342KB) ( 99 )   Save This paper is devoted to study the optimality conditions of robust approximate solutions for a convex optimization problem, in which the objective and constraint functions are carried with uncertain data. First of all, under the assumption of a closed convex cone constraint qualification for the constraint functions, the optimality conditions of approximate solutions for the concerned uncertain optimization problem are presented. Then, the concept of robust approximate saddle point is introduced, and the characterization of robust approximate solutions are proposed by saddle points. References | Related Articles | Metrics

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Numerical Analysis of a Class of Fractional Langevin Equation by Predictor-Corrector Method Linjuan Pu,Xiaozhong Yang,Shuzhen Sun Acta mathematica scientia,Series A. 2020, 40 (4):  1018-1028.  Abstract ( 67 )   RICH HTML PDF (573KB) ( 110 )   Save This paper extends the Langevin equation on the fractional order to make it time-memory, and a class of fractional-order Langevin equation is solved numerically using the predictor-corrector method. Firstly, the estimated value is obtained by R0 algorithm, then the estimated value is substituted into the R2 algorithm to correct the numerical solution. Finally, the numerical solution of a class of fractional-order Langevin equation by predictor-corrector method is obtained. The error analysis proves that under the condition of $0 < \alpha < 1$ of the equation, the result of predictor-corrector method is $(1+\alpha)$ order convergence. Numerical experiments also show that the numerical solution of the predictor-corrector method is convergent under different values of $\alpha$ and step size $h$. Figures and Tables | References | Related Articles | Metrics

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Iterative Learning Control for Fourth Order Partial Differential Multi-Agent Systems Pengfei Yu,Qin Fu Acta mathematica scientia,Series A. 2020, 40 (4):  1029-1042.  Abstract ( 84 )   RICH HTML PDF (450KB) ( 95 )   Save In this paper, the problem of iterative learning control for multi-agent systems is studied based on consensus. All the considered agents are constructed by the fourth-order beam equations. Based on the network topology, a consensus iterative learning control protocol is designed by using the information of adjacent agents. When the iterative learning law is applied to the systems, the consensus errors are bounded, and furthermore, the consensus errors on ${L^2}$ space tend to zero as the iteration index tends to infinity in the absence of initial errors. The effectiveness of the algorithm is verified by simulation examples. Figures and Tables | References | Related Articles | Metrics

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Bifurcation of Periodic Orbits of an n-Dimensional Piecewise Smooth Differential System Jihua Yang,Erli Zhang Acta mathematica scientia,Series A. 2020, 40 (4):  1043-1052.  Abstract ( 65 )   RICH HTML PDF (311KB) ( 110 )   Save In this paper, we study the following $n$-dimensional piecewise smooth differential system $\begin{eqnarray*} \ \left\{\begin{array}{ll} \dot{x}_1=x_2+\varepsilon g^+_1({\bf x}),\\ \dot{x}_2=-x_1+\varepsilon g^+_2({\bf x}),\\ \dot{x}_3=\varepsilon g^+_3({\bf x}),\\ \cdots\\ \dot{x}_n=\varepsilon g^+_n({\bf x}),\\ \end{array}\right.x_1\geq0,\quad \quad \left\{\begin{array}{ll} \dot{x}_1=x_2+\varepsilon g^-_1({\bf x}),\\ \dot{x}_2=-x_1+\varepsilon g^-_2({\bf x}),\\ \dot{x}_3=\varepsilon g^-_3({\bf x}),\\ \cdots\\ \dot{x}_n=\varepsilon g^-_n({\bf x}),\\ \end{array}\right.x_1<0, \end{eqnarray*}$ where ${\bf x}=(x_1,x_2,\cdots,x_n)^T$, $0<\varepsilon\ll1$, and $g^\pm_i({\bf x})$, $i=1,2,\cdots,n$ are real polynomials of ${\bf x}$ with degree $m$. By using the first order Melnikov vector function, we obtain the upper bound of the number of periodic orbits bifurcating from the unperturbed system. References | Related Articles | Metrics

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Global Mittag-Leffler Stability of Fractional Order Nonlinear Impulsive Differential Systems with Time Delay Jian Liu,Zhixin Zhang,Wei Jiang Acta mathematica scientia,Series A. 2020, 40 (4):  1053-1060.  Abstract ( 77 )   RICH HTML PDF (322KB) ( 91 )   Save In this paper, the global Mittag-Leffler stability of fractional-order nonlinear differential systems with impulsive and time-delay factors is studied. By using the fractional Lyapunov method and Mittag-Leffler function, sufficient conditions for global Mittag-Leffler stability of fractional-order nonlinear differential systems with impulsive time-delay are given. Finally, an example is given to demonstrate the effectiveness of the results. References | Related Articles | Metrics

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A Note About Syndetic Sensitivity and Multi-Sensitivity on Frechet Space Quanquan Yao,Peiyong Zhu Acta mathematica scientia,Series A. 2020, 40 (4):  1061-1071.  Abstract ( 114 )   RICH HTML PDF (313KB) ( 95 )   Save Let $(X, T)$ be a linear dynamical system, where $X$ is a separable Frechet space and $T:X \to X$ is a operator. first we prove the following assertions are equivalent:(1) $(X, T)$ is sensitive; (2) $(X, T)$ is multi-sensitive; (3) $(X, T)$ is multi-thickly-sensitive; (4) $(X, T)$ is thickly sensitive. Then we prove the following propositions:$(X, T)$ is syndetically sensitive if and only if $(X, T)$ is thickly syndetically sensitive. If $(X, T)$ is syndetic transitive, then $(X, T)$ is syndetically sensitive. If $(X, T)$ is frequently hypercyclic, then $(X, T)$ is thickly syndetically sensitive. If $(X, T)$ is syndetically transitive, then $(X, T)$ is transitively sensitive. Moreover, the iterated function system has the similar results. References | Related Articles | Metrics

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Hurst Parameter Under Finite Second Moment and Under Heavy Tails Liang Wu Acta mathematica scientia,Series A. 2020, 40 (4):  1072-1082.  Abstract ( 95 )   RICH HTML PDF (386KB) ( 115 )   Save Hurst parameter is widely used to characterize long memory and self-similarity in series. This paper introduces the relations between the Hurst parameter originally calculated by the $R/S$ statistic and the self-similarity, long memory, heavy tails under finite second moment and under heavy tails. In the case of finite second moment, the Hurst parameter calculated by the $R/S$ statistic is consistent with the self-similar parameter, and can describe the long memory defined by covariance. In the case of heavy tails with infinite second moment, the covariance linking Hurst parameter with long memory is infinite, and their relationship cannot be discussed. The $R/S$statistic is not necessarily related to self-similar parameters and tail index either. These contents can make the practical meaning of Hurst parameter clear. References | Related Articles | Metrics

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Spectral Analysis of a Transport Operator in Structured Bacterial Population Shenghua Wang,Jiangshan Ma Acta mathematica scientia,Series A. 2020, 40 (4):  1083-1094.  Abstract ( 76 )   RICH HTML PDF (339KB) ( 84 )   Save In this paper, we study a class of structured bacterial population models under the boundary condition of total transition rule in $L^{1}$ space. The spectral analysis of the transport operators in this model is discussed, and the weak compactness of the positive irreducible $C _0$ semigroup generated by the transport operators is proved. It is concluded that the spectrum of the transport operator consists of only by at most countable isolate eigenvalues with finite algebraic multiplicities, and $-\infty$ is the only possible accumulation point, and the asymptotic behavior of the solution of the model in the topological sense of the uniform operator, so the asynchronous growth characteristics of the bacterial population are given. References | Related Articles | Metrics

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Lagrange Dualities for Robust Composite Optimization Problems Dongping Ye,Donghui Fang Acta mathematica scientia,Series A. 2020, 40 (4):  1095-1107.  Abstract ( 90 )   RICH HTML PDF (355KB) ( 96 )   Save By using the properties of the epigraph of the conjugate functions, two new constraint qualifications are given. Under these constraint qualifications, the zero duality, the strong duality, the stable zero duality and the stable strong duality between robust composite optimization problem and its Lagrange dual problems are established, which extend and improve the corresponding results in the previous papers. References | Related Articles | Metrics

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The Precise Large Deviations of a Bidimensional Risk Model Based on Customer Arrival Hongmin Xiao,Zhankui Wang Acta mathematica scientia,Series A. 2020, 40 (4):  1108-1120.  Abstract ( 77 )   RICH HTML PDF (337KB) ( 101 )   Save In this paper, we discuss the two-dimensional risk model based on the entrance process. Assuming that $\overrightarrow{X^{i}}=(X_{1}^{i}, X_{2}^{i})^{\top}$ is a two-dimensional random vector sequence with the same distribution, $X_{1}^{i}$ and $X_{2}^{i}$ are dependent, and the precise large deviation between the partial sum and the random sum of loss precess is obtained under the heavy tail distribution family C. References | Related Articles | Metrics