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26 June 2023, Volume 43 Issue 3 Previous Issue    Next Issue

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Structure Expression Form of Isotropic Growth Surface Qian Jinhua,Bian Jinxin,Fu Xueshan Acta mathematica scientia,Series A. 2023, 43 (3):  657-668.  Abstract ( 104 )   RICH HTML PDF (992KB) ( 192 )   Save The isotropic growth surface in complex 3-space is investigated by evolving an isotropic curve as dictated growth velocity. The structure expression form of the isotropic growth surface is explored by the aid of the structure function of its generating isotropic curve. As an important application, the isotropic growth surface initiated by the isotropic helix is discussed deeply and explicitly. At the same time, several typical examples are constructed to characterize the generating process of such growth surfaces. Figures and Tables | References | Related Articles | Metrics

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The Multiplicities of Eigenvalues and Inverse Nodal Problem of a Vectorial Sturm-Liouville Problem Liu Xiaoyun,Shi Guoliang,Yan Jun Acta mathematica scientia,Series A. 2023, 43 (3):  669-679.  Abstract ( 82 )   RICH HTML PDF (360KB) ( 295 )   Save The $m$-dimensional vectorial Sturm-Liouville problem with Dirichlet boundary conditions on $(0,1)$ is studied. We firstly discuss the relationship between the matrix-valued potential and the multiplicities of eigenvalues. We prove that if the multiplicities of eigenvalues of $\int_{0}^{1}Q(x){\rm d}x$ are at most $k$ $(1\leq k\leq m-1)$, with finitely many exceptions, the multiplicities of eigenvalues of the vectorial problem are also at most $k$. Then, the inverse nodal problem is investigated with a different method. We show that if there exists an infinite eigenfunctions sequence $\{y_{n_{j},r}(x,\lambda_{n_{j},r})\}_{j=1}^{\infty }$ which are all vectorial functions of type $(CZ)$, then $Q$ is simultaneously diagonalizable. References | Related Articles | Metrics

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The Existence of Ground State Solutions for a Class of Equations Related to Klein-Gordon-Maxwell Systems Li Yixian,Zhang Zhengjie Acta mathematica scientia,Series A. 2023, 43 (3):  680-690.  Abstract ( 60 )   RICH HTML PDF (331KB) ( 115 )   Save In this paper, we will study the existence of ground state solutions for a class of nonlinear equations by using the theory of compactness of concentration, variational method and critical point theory. $\begin{eqnarray*} \left \{ \begin{array}{l} -\Delta u+(m+2\omega\phi)u=A(x)|u|^{p-2}u,\\ -\Delta\phi+\lambda\phi=\omega u^{2}, \lim\limits_{|x|\rightarrow\infty}u(x)=0, \lim\limits_{|x|\rightarrow\infty}\phi(x)=0. \end{array} \right. \end{eqnarray*}$ where $u\in H^{1}({\Bbb R}^{3})$, $\phi\in H^{1}({\Bbb R}^{3})$, $\lambda>0$, $m$ and $\omega$ are positive constants. Then we study the problem assuming the follwwing two cases on $A(x)$. If $A(x)$ is a positive constant function, we prove that the ground state solution $(u, \phi)$ exists for any $p\in(4,6)$; if $A(x)$ is not a constant function, we prove that the ground state solution $(u, \phi)$ exists for any $p\in(4,6)$ under the right conditions. References | Related Articles | Metrics

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Existence of Back-Flow Point for the Two-Dimensional Compressible Prandtl Equation Zou Yonghui,Xu Xin Acta mathematica scientia,Series A. 2023, 43 (3):  691-701.  Abstract ( 83 )   RICH HTML PDF (359KB) ( 279 )   Save In this paper, we study the back-flow problem of the two-dimensional unsteady compressible Prandtl boundary layer equations. By using the maximum principle, we obtain that a first back-flow point should appear on the boundary $\left\{y=0\right\}$ if back-flow occurs under Oleinik's monotonicity assumption. Moreover, when the pressure gradient of the outer flow is adverse and the initial velocity satisfies certain growth condition, we obtain the existence of a back-flow point of the compressible Prandtl boundary layer by Lyapunov functional method. Finally, an example of the existence of the back-flow point is given. References | Related Articles | Metrics

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Existence Results of Periodic Solutions for Semilinear Evolution Equation in Banach Spaces and Applications Li Yongxiang,Wei Qilin Acta mathematica scientia,Series A. 2023, 43 (3):  702-712.  Abstract ( 71 )   RICH HTML PDF (361KB) ( 341 )   Save In this paper, we deal with the existence of periodic solutions for the semilinear evolution equation in a Banach space $X$, $u'(t)+Au(t)=f(t,\,u(t)),\quad t\in{\Bbb R},$ where $A: D(A)\subset X\to X$ is a closed linear operator and $-A$ generates a $C_{0}$-semigroup $X$, $f:{\Bbb R}\times X\to X$ is a continuous mapping and $f(t,\,x)$ is $\omega$-periodic in $t$. Existence results of $\omega$-periodic mild solutions are obtained by using operator semigroup theory, estimation technique of noncompact measure and fixed point theorem. Examples of applications in parabolic partial differential equations and weakly damped wave equations are present. References | Related Articles | Metrics

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Gradient Regularity of Very Weak Solution to Elliptic Equations with Singular Convection Chen Shuhong,Tan Zhong Acta mathematica scientia,Series A. 2023, 43 (3):  713-732.  Abstract ( 103 )   RICH HTML PDF (452KB) ( 213 )   Save This paper deals with the partial regularity of very weak solutions to elliptic equations with singular convective. By the properties of Lorentz space and its relation to Lebesgue space, we conclude that the elliptic systems with singular convection have very weak solutions in $L^p$ space. Then, it can be found from Hodge decomposition that the very weak solutions of Dirichlet problem are actually the classical weak solutions. Finally, combining with A-harmonic approximation technique, we further find that the obtained weak solution has partial regularity; especially, the regularity is optimal. References | Related Articles | Metrics

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The Completely Regular Growth of Solutions of Higher Order Linear Differential Equations Chen Li,Liu Huifang Acta mathematica scientia,Series A. 2023, 43 (3):  733-742.  Abstract ( 105 )   RICH HTML PDF (349KB) ( 122 )   Save In this paper, the existence of completely regular growth solutions of higher order linear differential equations is studied, where its dominant coefficient is an exponential polynomial. By using the Nevanlinna characteristic of exponential polynomials, some conditions which guarantee the non-existence of such solutions are obtained. At the same time, for the higher order linear differential equation with exponential polynomial solutions, the relationship between the expression of its solutions and dominant coefficient is given. References | Related Articles | Metrics

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Regularity of the Solutions to the Nordström-Vlasov System Chen Ruijuan, He YingXi, Xiao Meixia Acta mathematica scientia,Series A. 2023, 43 (3):  743-753.  Abstract ( 111 )   RICH HTML PDF (340KB) ( 131 )   Save In this paper, we investigate the Nordström-Vlasov system in the whole space. The kinetic model is a relativistic generalization of the classical Vlasov-Poisson system in the gravitational case and describes the ensemble motion of collisionless particles interacting by means of a self-consistent scalar gravitational field. With the Fourier analysis and the smoothing effect of low velocity particles, we get a regularity of weak solutions for the field. References | Related Articles | Metrics

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Stability and Exponential Decay of the 3D Boussinesq Equations with Partial Dissipation Li Xiaoli,Chen Xiaoli Acta mathematica scientia,Series A. 2023, 43 (3):  754-770.  Abstract ( 76 )   RICH HTML PDF (343KB) ( 267 )   Save This paper is devoted to solving the stability and large time behavior problem on three dimensional Boussinesq equations with anisotropic dissipation and vertical thermal diffusion near the hydrostatic equilibrium. The stability of the solution with certain symmetries to the Boussinesq euations is established on the spatial domain $R^2\times \mathrm{T}$ with the periodic box $\mathrm{T}=[-\frac12,\frac12]$. In addition, the oscillators of the velocity $u$ and the temperature $\theta$ admit the exponential decay in time variable $t$. References | Related Articles | Metrics

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Cauchy Problem for the Evolution of Cells and Tissue During Curvature-Controlled Growth Wang Zenggui Acta mathematica scientia,Series A. 2023, 43 (3):  771-784.  Abstract ( 55 )   RICH HTML PDF (393KB) ( 276 )   Save In this paper, We consider Cauchy problem for the evolution of cells and tissue during curvature-controlled growth. By the definition of Riemann invariants, the evolution equation can be rewritten as a non-homogeneous quasilinear hyperbolic system. the lifespan of classical solution to the initial value problem is given by a priori estimation of the solution of the quasilinear hyperbolic system. References | Related Articles | Metrics

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Exotic Dynamics of Freak Waves in the Focusing Kundu-Eckhaus Equation Wang Xiubin, Tian Shoufu Acta mathematica scientia,Series A. 2023, 43 (3):  785-794.  Abstract ( 73 )   RICH HTML PDF (591KB) ( 243 )   Save In this work, based on Darboux transformation general higher-order freak wave solutions of the focusing Kundu-Eckhaus equation are derived by using the variable separation method. Then the dynamics of these freak wave solutions are discussed with some graphics. In particular, we observe that one four-petaled freak wave and three eye-shaped ones can coexist, in contrast to the four eye-shaped ones reported before. They demonstrate that the structure of freak waves in this paper is richer than that in the well-known nonlinear Schrödinger equation. Figures and Tables | References | Related Articles | Metrics

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Critical Fujita Exponent and Blow-up Results for the Rockland Heat Equation Yang Zhipeng Acta mathematica scientia,Series A. 2023, 43 (3):  795-807.  Abstract ( 55 )   RICH HTML PDF (380KB) ( 97 )   Save We obtain the subcritical Fujita exponent and nonexistence result for the Cauchy problem of the nonlinear Rockland heat equation $\begin{eqnarray*} \left\{\begin{array}{ll} u_{t}(t,x)+{\cal R}_{x}u(t,x)=|u(t,x)|^{p}, &(t,x) \in (0,+\infty)\times{\Bbb G}:=\Omega, \\ u(0,x)=u_{0}(x), & x \in {\Bbb G}. \end{array}\right. \end{eqnarray*}$ In this paper, we consider the critical Fujita exponent and obtain the blow-up result by an ODE method. Central to our proof is the heat kernel for Rockland operator. References | Related Articles | Metrics

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A Leap-Frog Crank-Nicolson Multidomain Legendre-Tau Collocation Spectral Method for 2D Nonlinear Maxwell's Equations in Inhomogeneous Media Niu Cuixia,Ma Heping Acta mathematica scientia,Series A. 2023, 43 (3):  808-828.  Abstract ( 87 )   RICH HTML PDF (506KB) ( 89 )   Save In this paper, numerical methods for solving 2D nonlinear Maxwell's equations in inhomogeneous media are discussed. A multidomain Legendre-tau collocation spectral method is proposed. The proposed method is of spectral accuracy in space and second order in time. In time direction, the leap-frog Crank-Nicolson method is applied, which is a three-level scheme with the nonlinear term being treated by some collocation methods explicitly in intermediate level and the linear terms being treated by the Legendre-tau spectral method implicitly. By the implicit-explicit scheme, the numerical method is of better stability and easy implementation. We construct a reasonable weak form which can deal with the interface conditions in a way just like the natural boundary condition without any additional interface conditions. The uniform scheme without any additional interface conditions is constructed by using polynomial spaces of different degrees. For the semi-discrete and fully discrete schemes, the stability and convergence are proved, and the $L^2$-norm optimal error estimates are obtained. In numerical examples, the nonlinear terms are computed at the Chebyshev points by the fast Legendre transform to improve the efficiency of the algorithm. Numerical results show the efficiency of the proposed method for solving this nonlinear problem. Moreover, the results indicate that the spectral accuracy is achieved and not affected by the discontinuity of solutions. Figures and Tables | References | Related Articles | Metrics

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Analysis of a New Time Filter Algorithm for the Unsteady Stokes/Darcy Model Wang Yang,Li Jian,Li Yi,Qin Yi Acta mathematica scientia,Series A. 2023, 43 (3):  829-854.  Abstract ( 123 )   RICH HTML PDF (4511KB) ( 101 )   Save Firstly, based on the first-order $\theta$-scheme of the Linear Multistep method for the unsteady Stokes/Darcy model, this paper combines the time filter algorithm to effectively improve the convergence order of the Linear Multistep method from first order to second order almost without increasing the amount of calculation, and a new efficient numerical algorithm is proposed. Secondly, the stability and error estimation of the coupled and decoupled Linear Multistep method plus time filter algorithm are analyzed theoretically. Finally, numerical experiments further demonstrate the effectiveness, convergence and efficiency of the coupled and decoupled algorithms. Figures and Tables | References | Related Articles | Metrics

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An Improved Berry-Esséen Bound of Least Squares Estimation for Fractional Ornstein-Uhlenbeck Processes Chen Yong,Gu Xiangmeng Acta mathematica scientia,Series A. 2023, 43 (3):  855-882.  Abstract ( 63 )   RICH HTML PDF (1194KB) ( 263 )   Save The aim of this paper is twofold. First, it offers a novel formula to calculate the inner product of the bounded variation function in the Hilbert space ${\cal H}$ associated with the fractional Brownian motion with Hurst parameter $H\in (0,\frac12)$. This formula is based on a kind of decomposition of the Lebesgue-Stieljes measure of the bounded variation function and the integration by parts formula of the Lebesgue-Stieljes measure. Second, as an application of the formula, we explore that as $T\to \infty$, the asymptotic line for the square of the norm of the bivariate function $f_T(t,s)=e^{-\theta|t-s|}1_{\{0\leq s,t\leq T\}}$ in the symmetric tensor space ${\cal H}^{\odot 2}$ (as a function of $T$), and improve the Berry-Esséen type upper bound for the least squares estimation of the drift coefficient of the fractional Ornstein-Uhlenbeck processes with Hurst parameter $H\in (\frac14,\frac12)$. The asymptotic analysis of the present paper is much more subtle than that of Lemma 17 in Hu, Nualart, Zhou(2019) and the improved Berry-Esséen type upper bound is the best improvement of the result of Theorem 1.1 in Chen, Li (2021). As a by-product, a second application of the above asymptotic analysis is given, i.e., we also show the Berry-Esséen type upper bound for the moment estimation of the drift coefficient of the fractional Ornstein-Uhlenbeck processes where the method is obvious different to that of Proposition 4.1 in Sottinen, Viitasaari(2018). Figures and Tables | References | Related Articles | Metrics

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A Kind of Numerical Algorithm for Elliptic Interface Problem Yu Yikang,Niu Jing Acta mathematica scientia,Series A. 2023, 43 (3):  883-895.  Abstract ( 93 )   RICH HTML PDF (464KB) ( 299 )   Save In this paper, a numerical algorithm based on the theory of reproducing kernel space is proposed for the one-dimensional ellipse interface problem. This method integrates the reproducing kernel function of ${W}^{1}_{2}$ space to obtain a set of new basis in the cubic spline space, on which a broken cubic spline space is established. Then we use the least squares method to get the approximate solution of this kind of interface problem. We discussed the order of convergence under the $H_2$ norm, $H_1$ norm and $L_2$ norm. Finally, our theory is verified by several numerical example. Figures and Tables | References | Related Articles | Metrics

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A Fast High Order Method for Fractional Differential Equations with the Caputo-Fabrizio Derivative Fu Bo,Wang Shiyu,Gao Tingting,Lv Xueqin Acta mathematica scientia,Series A. 2023, 43 (3):  896-912.  Abstract ( 86 )   RICH HTML PDF (629KB) ( 96 )   Save The computational work and storage of numerically solving the ODEs are generally huge for the traditional direct methods. To overcome this difficulty, we presents a fast high order numerical method for fractional ordinary differential equations with the Caputo-Fabrizio derivative based on a L2 scheme and a simply recurrence relation. The algorithm greatly reduces the storage capacity and the total calculation cost. Furthermore, we also analyzes the feasibility of algorithm, error estimation and stability analysis of the fast scheme. Illustrative examples are included to demonstrate the performance of our technique. Figures and Tables | References | Related Articles | Metrics

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Approximation Theorem of Population Games and Multi-objective Population Games Wang Chun,Yang Hui,GuangYang Hui,Wang Guoling Acta mathematica scientia,Series A. 2023, 43 (3):  913-920.  Abstract ( 76 )   RICH HTML PDF (405KB) ( 316 )   Save In population games and multi-objective population games, by perturbation of strategies, we relax rationality of agents further, which is represented by an approximate solution called approximate Nash equilibria and approximate weakly Pareto-Nash equilibria. And we prove their approximation theorem. They not only realistically weaken the condition of approximation theorem, but they also improve the theoretical support for the algorithm of population games. References | Related Articles | Metrics

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Generic Stability of Cooperative Equilibria for Leader-Follower Population Games Wu Wenjun,Yang Guanghui,Fang Caiya,Yang Hui Acta mathematica scientia,Series A. 2023, 43 (3):  921-929.  Abstract ( 100 )   RICH HTML PDF (390KB) ( 286 )   Save In this paper, we study the existence and generic stability of cooperative equilibria in leader-follower population games with a leader and multiple population followers. Firstly, a cooperative equilibrium of leader-follower population games is defined in consideration of cooperative behavior among population followers. Secondly, we prove the existence of cooperative equilibria, and an example is illustrated that cooperative equilibria exist and are different from the classical Nash equilibria. Finally, using Fort's theorem, we prove that in the sense of Baire category most cooperative equilibria of leader-follower population games are of generic stability under perturbations of payoff functions. References | Related Articles | Metrics

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Essential Stability and Hadamard Well-Posedness of Excess Demand Equilibrium Problems Zeng Jing,Peng Jiayu Acta mathematica scientia,Series A. 2023, 43 (3):  930-938.  Abstract ( 60 )   RICH HTML PDF (358KB) ( 269 )   Save In this paper, we considered the excess demand equilibrium problems in finite dimensional Euclidean spaces. Firstly, the relationship among the semicontinuity of solution mapping, the essential equilibrium solution and the essential set is established, when both the excess demand mapping and the price parameter set are disturbed. Secondly, it is proved that the excess demand equilibrium problems is stable in the sense of Baire classification. Finally, taking advantage of a set defined by a scalar function, sufficient conditions of Hadamard well-posedness of the excess demand equilibrium problems are obtained. References | Related Articles | Metrics

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Equilibrium Investment Strategy of DC Pension Plan with Mispricing and Return of Premiums Clauses Under the 4/2 Stochastic Volatility Model Lu Jiaxin,Dong Hua Acta mathematica scientia,Series A. 2023, 43 (3):  939-956.  Abstract ( 85 )   RICH HTML PDF (1020KB) ( 294 )   Save In this paper, we consider a time-consistent investment strategy for DC pension plan with a return of premiums clause and mispricing under the mean-variance criterion. We assume that the pension plan manager is allowed to invest the wealth in the pension account in a financial market consisting of a risk-free asset, a pair of mispriced stocks, and a market index following a $4/2$ stochastic volatility model. Under the framework of game theory, the explicit expressions of the time-consistent equilibrium investment strategy and the equilibrium efficient frontier are obtained by using stochastic control methods and solving the extended HJB system. Finally, the effects of risk aversion coefficient, mispricing and return of premiums clauses on equilibrium strategy and efficient frontier are illustrated by numerical simulations. Figures and Tables | References | Related Articles | Metrics

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Proportional Reinsurance and Investment Based on the Ornstein-Uhlenbeck Process in the Presence of Two Reinsurers Huang ,Liu Haiyan,Chen Mi Acta mathematica scientia,Series A. 2023, 43 (3):  957-969.  Abstract ( 95 )   RICH HTML PDF (922KB) ( 263 )   Save This paper studies the optimal reinsurance and investment problem with two reinsurance companies under two risk models. The insurance company purchases proportional reinsurance and invests in the financial market consisting of one risk-free asset and one risky asset, where the price of the risky asset is influenced by the Ornstein-Uhlenbeck process. Assuming that premiums for reinsurance are calculated according to the exponential premium principle, and the insurer's goal is to maximize the expected exponential utility of terminal wealth. Using stochastic control theory and HJB equation, the explicit expressions of the optimal strategy and value function are derived. Finally, the influence of model parameters on optimal strategy is verified by numerical analysis. Figures and Tables | References | Related Articles | Metrics

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Dynamical Analysis of an Age-Space Structured HIV/AIDS Model with Homogeneous Dirichlet Boundary Condition Wu Peng,Wang Xiunan,He Zerong Acta mathematica scientia,Series A. 2023, 43 (3):  970-984.  Abstract ( 122 )   RICH HTML PDF (705KB) ( 150 )   Save In order to explore the impact of human movement, infection age, and a hostile boundary environment on the HIV/AIDS spatiotemporal transmission dynamics, we construct an age-space structure model with homogeneous Dirichlet boundary condition. Applying the method of characteristics, we transform the model into a system of a reaction-diffusion equation and an integral equation. We derive the basic reproduction ratio $R_0$ and investigate the threshold dynamics in terms of $R_0$. Out theoretical results show that, under appropriate conditions, the disease can be eliminated when $R_0<1$ and the infection is uniformly persistent among the population when $R_0>1$. We verify the theoretical result by numerical simulations in a two-dimensional spatial domain. Figures and Tables | References | Related Articles | Metrics