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    The Schödinger Uncertainty Relation in the Fock-Type Spaces
    Li Wenxin,Lian Pan,Liang Yuxia
    Acta mathematica scientia,Series A    2023, 43 (5): 1321-1332.  
    Abstract222)   HTML19)    PDF(pc) (660KB)(442)       Save

    In this paper, the Schödinger uncertainty relation for the unilateral weighted shift operators on Fock space is established, and the explicit expression when the equality attained is given, which further extends the Heisenberg uncertainty relation on Fock space established in [4] and overcomes the difficulty in [16]. In addition, we generalize the uncertainty relation to the multiple operators case. A new uncertainty inequality in the form of non-self adjoint operators is obtained as well.

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    Locally Minimizing Solutions of a Two-component Ginzburg-Landau System
    Xiong Chen, Gao Qi
    Acta mathematica scientia,Series A    2023, 43 (2): 321-340.  
    Abstract145)   HTML8)    PDF(pc) (446KB)(402)       Save

    In this paper, we consider a Ginzburg-Landau functional for a complex vector order parameter $\Psi=[\psi_+, \psi_-]$. In particular, we consider entire solutions in all ${\Bbb R}^2$, which are obtained by blowing up around vortices. Among the entire solutions we distinguish those which are locally minimizing solutions, and we show that locally minimizing solutions must have degrees $n_\pm \in \{0, \pm1\}$. By studying the local structure of these solutions, we also show that one component of the solution vanishes, but the other does not, which describes the coreless vortex phenomenon in physics.

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    Stability and Bifurcation of a Pathogen-Immune Model with Delay and Diffusion Effects
    Jingnan Wang,Dezhong Yang
    Acta mathematica scientia,Series A    2021, 41 (4): 1204-1217.  
    Abstract79)   HTML0)    PDF(pc) (903KB)(367)       Save

    In order to understand the effects of diffusion and time-delay factors on the dynamics between pathogens and immune cells, a delayed pathogen-immune reaction diffusion model with homogeneous Neumann boundary condition is established. By using the diffusion ratio of pathogen-immune cells and immune delay as two parameters, the characteristic root distribution of the linearized system at the positive steady state is analyzed and the necessary and sufficient conditions for the positive steady state to undergo Turing instability and Hopf bifurcation are obtained by using the bifurcation theory of functional differential equations. In addition, the dynamic behavior close to the critical value of Turing instability and Hopf bifurcation is intuitively shown by Matlab numerical simulation. The biological and medicinal significance of corresponding dynamic behaviors are discussed. Furthermore, the obtained results provide certain theoretical support for controlling the growth of pathogen.

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    Blow-Up of the Smooth Solutions to the Quantum Navier-Stokes-Landau-Lifshitz Equations
    Zhen Qiu,Guangwu Wang
    Acta mathematica scientia,Series A    2022, 42 (4): 1074-1088.  
    Abstract306)      PDF(pc) (346KB)(357)       Save

    In this paper, we investigate the blow-up of the smooth solutions to the quantum Navier-Stokes-Landau-Lifshitz systems(QNSLL) in the domains $\Omega \subseteq \mathbb{R} ^n(n =1, 2)$. We prove that the smooth solutions to the QNSLL will blow up in finite time in the domains half-space $\mathbb{R} _+^n$, whole-space $\mathbb{R} ^n$ and ball. The paper also shows that the blow-up time of the smooth solutions in half-space or whole-space only depends on boundary conditions, while the the blow-up time of the smooth solutions in the ball depends on initial data and boundary conditions. In particular, the above conclusions are also valid for NSLL systems.

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    An Extension of Minkowski Formulae for Free Boundary Hypersurfaces in a Ball
    Sheng Weimin, Wang Yinhang
    Acta mathematica scientia,Series A    2023, 43 (6): 1641-1648.  
    Abstract167)   HTML18)    PDF(pc) (508KB)(345)       Save

    In this article, we prove a generalization of Hsiung-Minkowski formula for free boundary hypersurfaces in a ball in space forms. As corollaries, we obtain some Alexandrov-type results.

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    Existence Results for von Kármán Equations Modeling Suspension Bridges
    Yongda Wang
    Acta mathematica scientia,Series A    2022, 42 (4): 1112-1121.  
    Abstract104)   HTML5)    PDF(pc) (371KB)(340)       Save

    A nonlinear von Kármán equation with partial free boundary is considered. The equation is viewed as a mathematical model for suspension bridges with large deformation. The buckling loads, which carry a nonlocal effect into the model, are introduced. Uniqueness and multiplicity results are obtained by analyzing the critical points of the energy functionals.

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    Time Decay Rate for Large-Solution About 3D Compressible MHD Equations
    Chen Fei,Wang Shuai,Zhao Yongye,Wang Chuanbao
    Acta mathematica scientia,Series A    2023, 43 (5): 1397-1408.  
    Abstract72)   HTML4)    PDF(pc) (697KB)(336)       Save

    This paper focus on time decay rate for large-solution about compressible magnetohydrodynamic equations in $\mathbb{R}^3$. Provided that $(\sigma_{0}-1,u_{0},M_{0})\in L^1\cap H^2$, based on the work of Chen et al.[1], $\|\nabla(\sigma-1,u,M)\|_{H^1}\leqslant C(1+t)^{-\frac{5}{4}}$ is obtained in reference [2], obviously, time decay rate of the 2nd-order derivative of the solution in [2] is not ideal. Here, we improve that of $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ to be $(1+t)^{-\frac{7}{4}}$ by the frequency decomposition method[3].

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    On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement
    Jian Hui, Gong Min, Wang Li
    Acta mathematica scientia,Series A    2023, 43 (5): 1350-1372.  
    Abstract97)   HTML5)    PDF(pc) (797KB)(334)       Save

    This paper is devoted to the Cauchy problem of inhomogeneous nonlinear Schrödinger equation in the presence of a partial confinement, which is an important model in Bose-Einstein condensates. Combining the variational characterization of the ground state of a nonlinear elliptic equation and the conservations of mass and energy, we first obtain a global solution and show the existence of blow-up solutions for some special initial data by scaling techniques. Then, we study the $L^2$-concentration phenomenon for the blow-up solutions. Finally, we apply the variational arguments connected to the above ground state to investigate the dynamics of $L^2$-minimal blow-up solutions, i.e., the limiting profile, mass-concentration and blow-up rate of the blow-up solutions with minimal mass. We extend the global existence and blow-up results of Zhang[34] to the case of inhomogeneous nonlinearities and improve partial results of Pan and Zhang[23] to space dimensions $N\geq2$ in the inhomogeneous case.

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    Existence, Multiplicity and Concentration of Positive Solutions for a Fractional Choquard Equation
    Weiqiang Zhang,Peihao Zhao
    Acta mathematica scientia,Series A    2022, 42 (2): 470-490.  
    Abstract95)   HTML4)    PDF(pc) (451KB)(325)       Save

    We are concerned with the existence, multiplicity and concentration of positive solutions for the following fractional Choquard equation with subcritical nonlinearity where $\varepsilon>0$ is a parameter, $s\in(0, 1)$, $(-\Delta)^{s}$ is the fractional Laplace operator, $V:\mathbb{R} ^{N}\rightarrow\mathbb{R} $ is a positive potential having global minimum, $0<\mu<\min\{4s, N\}$, and $F$ is the primitive of $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ which is subcritical growth. The main research methods of this article are variational method and the Ljusternik-Schnirelmann theory.

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    Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity
    Li Xiaodong, Gao Hongliang, Xu Jing
    Acta mathematica scientia,Series A    2023, 43 (5): 1341-1349.  
    Abstract66)   HTML8)    PDF(pc) (770KB)(316)       Save

    In this paper, we study the exact multiplicity and bifurcation diagrams of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of

    $ \left\{\begin{array}{ll} -\left(\frac{u'}{\sqrt{1-u'^{2}}}\right)'=\lambda f(u), x\in(-L,L),\\ u(-L)=0=u(L), \end{array} \right. $

    where $\lambda>0$ is a bifurcation parameter and $L>0$ is an evolution parameters, $f\in C^{2}([0,\infty), \mathbb{R})$ satisfies $f(0)<0$ and $f$ is concave for $0. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of $\lambda$. The arguments are based upon a detailed analysis of the time map.

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    Blow-Up Conditions of Porous Medium Systems with Gradient Source Terms and Nonlinear Boundary Conditions
    Shen Xuhui,Ding Juntang
    Acta mathematica scientia,Series A    2023, 43 (5): 1417-1426.  
    Abstract40)   HTML4)    PDF(pc) (602KB)(308)       Save

    In this paper, we consider the blow-up of solutions to the following porous medium systems:

    $ \left\{ \begin{array}{ll} u_{t} =\Delta u^l+f(u,v,|\nabla u|^2,t), & \\\displaystyle v_{t} =\Delta v^m+g(u,v,|\nabla v|^2,t),&x\in\Omega, \ t\in(0,t^*), \\\displaystyle \frac{\partial u}{\partial\nu}=p(u), \ \frac{\partial v}{\partial\nu}=q(v), &x\in\partial\Omega, \ t\in(0,t^*), \\\displaystyle u(x,0)=u_{0}(x), \ v(x,0)=v_{0}(x), &x\in\overline{\Omega}, \end{array} \right. $

    where $l,m>1, \ \Omega\subset\mathbb{R}^N \ (N\geq2)$ is a bounded domain with smooth boundary $\partial\Omega$. Using the differential inequality techniques and the maximum principles, we give a sufficient condition to ensure that the positive solution $(u,v)$ of the above problem is a blow-up solution that blows up at a certain finite time $t^*$. An upper estimate of $t^*$ and an upper estimate of the blow-up rate of $(u,v)$ are also obtained.

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    Existence of Positive Solutions for a Class of Schrödinger-Newton Systems with Critical Exponent
    Cheng Qingfang,Liao Jiafeng,Yuan Yanxiang
    Acta mathematica scientia,Series A    2023, 43 (5): 1373-1381.  
    Abstract75)   HTML5)    PDF(pc) (629KB)(306)       Save

    In this paper, we study the existence of positive solutions for a class of Schrödinger-Newton system with critical exponents on bounded domain, and obtain two positive solutions by the variational method.

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    Survival Analysis of an SVIR Epidemic Model with Media Coverage
    Li Dan,Wei Fengying,Mao Xuerong
    Acta mathematica scientia,Series A    2023, 43 (5): 1595-1606.  
    Abstract58)   HTML2)    PDF(pc) (1602KB)(300)       Save

    We consider the long-term properties of a stochastic SVIR epidemic model with media coverage and the logistic growth in this paper. We firstly derive the fitness of a unique global positive solution. Then we construct appropriate Lyapunov functions and obtain the existence of ergodic stationary distribution when ${R}_{0}^{s}>1$ is valid, and also derive sufficient conditions for persistence in the mean. Moreover, the exponential extinction to the density of the infected is figured out when ${R}_{0}^{e}<1$ holds.

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    Bonnesen-Style Inequalities for Star Bodies
    Zengle Zhang
    Acta mathematica scientia,Series A    2021, 41 (5): 1249-1262.  
    Abstract185)   HTML17)    PDF(pc) (351KB)(300)       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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    The Radial Symmetry and Monotonicity of Entire Solutions for Fractional Parabolic Equations
    Tang Yanjuan
    Acta mathematica scientia,Series A    2023, 43 (5): 1409-1416.  
    Abstract51)   HTML3)    PDF(pc) (581KB)(291)       Save

    This paper mainly develops the radial symmetry and monotonicity of entire solutions for fractional parabolic equations. To obtain the symmetry and monotonicity of entire solutions, the narrow region principle and maximum principle for antisymmetric functions in [9] are needed. Furthermore, to circumvent the difficulty from nonlocality for the fractional Laplacian, a fractional parabolic version of the method of moving planes will be adopted.

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    Normalized Ground States for the Quasi-linear Schrödinger Equation with Combined Nonlinearities
    Gui Kunming,Tao Hongshan,Yang Jun
    Acta mathematica scientia,Series A    2023, 43 (4): 1062-1072.  
    Abstract40)   HTML1)    PDF(pc) (345KB)(289)       Save

    In this paper, we mainly investigate the existence of normalized ground states for the Schrödinger equation with combined nonlinearities. Our results extend those reported in [1-2]. Compared with the case they studied, the structure of the energy function correspongding to the equation in this paper is more complex.

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    Second Main Theorem for Algebraic Curves on Compact Riemann Surfaces
    Lizhen Duan,Hongzhe Cao
    Acta mathematica scientia,Series A    2021, 41 (6): 1585-1597.  
    Abstract207)   HTML13)    PDF(pc) (357KB)(288)       Save

    In this paper, we first establish some second main theorems for algebraic curves from a compact Riemann surface into a complex projective subvariety of the complex projective space, which is ramified over hypersurfaces in subgeneral position. Then we use it to study the ramification for the generalized Gauss map of complete regular minimal surfaces in $\mathbb{R}^{m}$ with finite total curvature.

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    Asymptotic Finite-Time Ruin Probability for a Bidimensional Perturbed Risk Model with General Investment Returns and Time-Dependent Claim Sizes
    Cheng Ming,Wang Dingcheng
    Acta mathematica scientia,Series A    2023, 43 (5): 1529-1558.  
    Abstract48)   HTML3)    PDF(pc) (810KB)(287)       Save

    The paper considers a bi-dimensional perturbed insurance risk model with general investment returns. Assume that the investment return is described by a càdlàg process, and two classes of claims and the inter-arrival times follow the Sarmanov dependence structure. When the claim-size distribution has a regularly varying tail, the paper derives the asymptotic formula of the finite-time ruin probability. When the càdlàg process describing investment returns is chosen as the Lévy process, Vasicek interest rate model, Cox-Ingersoll-Ross (CIR) interest rate model, or Heston model, the paper derives the asymptotic estimates for ruin probabilities under the corresponding investment returns.

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    Symmetric and Periodic Solutions for a Class of Weakly Coupled Systems Composed of Two Particles with Obstacles
    Wang Zihuan,Wang Chao
    Acta mathematica scientia,Series A    2023, 43 (5): 1427-1439.  
    Abstract35)   HTML2)    PDF(pc) (686KB)(276)       Save

    The problems of the existence and multiplicity of symmetric periodic solutions with impact for a class of weakly coupled systems of two degrees of freedom with obstacles are concerned. Under some superlinear assumption on time-mapping, the existence of infinite symmetric harmonic solutions and symmetric subharmonic solutions with impacts of the equation are proved. Furthermore, a sufficient condition for the existence of even and periodic bouncing solution is given for the coupled symmetric impact equations of two degrees of freedom.

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    Similarity and Unitary Similarity of a Class of Upper Triangular Operator Matrices
    Liqiong Lin,Jiahua Que,Yunnan Zhang
    Acta mathematica scientia,Series A    2022, 42 (5): 1281-1293.  
    Abstract194)   HTML18)    PDF(pc) (268KB)(276)       Save

    This paper introduces a class of upper triangular operator matrices related to Cowen-Douglas operators, and studies its similarity on Banach spaces and its unitary similarity on Hilbert spaces.

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