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    THE EXISTENCE OF GLOBAL SOLUTIONS FOR THE FULL NAVIER-STOKES-KORTEWEG SYSTEM OF VAN DER WAALS GAS
    Hakho Hong
    Acta mathematica scientia,Series B    2023, 43 (2): 469-491.   DOI: 10.1007/s10473-023-0201-9
    Abstract95)      PDF       Save
    The aim of this work is to prove the existence for the global solution of a non-isothermal or non-isentropic model of capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), in the case of van der Waals gas. Under the small initial perturbation, the proof of the global existence is based on an elementary energy method using the continuation argument of local solution. Moreover, the uniqueness of global solutions and large time behavior of the density are given. It is one of the main difficulties that the pressure $p$ is not the increasing function of the density $\rho$.
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    CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE*
    Xin Wei, Zhi-Ying Wen
    Acta mathematica scientia,Series B    2023, 43 (3): 981-993.   DOI: 10.1007/s10473-023-0301-6
    Abstract93)      PDF       Save
    Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves.
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    THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION*
    Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang
    Acta mathematica scientia,Series B    2023, 43 (2): 492-504.   DOI: 10.1007/s10473-023-0202-8
    Abstract49)      PDF       Save
    The singular convergence of a chemotaxis-fluid system modeling coral fertilization is justified in spatial dimension three. More precisely, it is shown that a solution of parabolic-parabolic type chemotaxis-fluid system modeling coral fertilization $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\epsilon}+(u^{\epsilon}\cdot\nabla)u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}=-(s^{\epsilon}+e^{\epsilon})\nabla \phi,\\ \nabla\cdot u^{\epsilon}=0, \\ e_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )e^{\epsilon}-\Delta e^{\epsilon}=-s^{\epsilon}e^{\epsilon},\\ s_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )s^{\epsilon}-\Delta s^{\epsilon}=-\nabla\cdot(s^{\epsilon}\nabla c^{\epsilon})-s^{\epsilon}e^{\epsilon}, \\ \epsilon^{-1} \left(c_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )c^{\epsilon}\right)=\Delta c^{\epsilon}+e^{\epsilon},\\ (u^{\epsilon}, e^{\epsilon},s^{\epsilon},c^{\epsilon})|_{t=0}= (u_{0}, e_{0},s_{0},c_{0})\\ \end{array} \right. \end{eqnarray*}$ converges to that of the parabolic-elliptic type chemotaxis-fluid system modeling coral fertilization $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\infty}+(u^{\infty}\cdot\nabla)u^{\infty}-\Delta u^{\infty}+\nabla\mathbf{P}^{\infty}=-(s^{\infty}+e^{\infty})\nabla \phi, \\ \nabla\cdot u^{\infty}=0, \\ e_t^{\infty}+(u^{\infty}\cdot\nabla )e^{\infty}-\Delta e^{\infty}=-s^{\infty}e^{\infty}, \\ s_t^{\infty}+(u^{\infty}\cdot\nabla )s^{\infty}-\Delta s^{\infty}=-\nabla\cdot(s^{\infty}\nabla c^{\infty})-s^{\infty}e^{\infty}, \\ 0=\Delta c^{\infty}+e^{\infty}, \\ (u^{\infty}, e^{\infty},s^{\infty})|_{t=0}= (u_{0}, e_{0},s_{0})\\ \end{array} \right. \end{eqnarray*}$ in a certain Fourier-Herz space as $\epsilon^{-1}\rightarrow 0$.
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    RANDERS SPACES WITH SCALAR FLAG CURVATURE*
    Jintang LI
    Acta mathematica scientia,Series B    2023, 43 (3): 994-1006.   DOI: 10.1007/s10473-023-0302-5
    Abstract42)      PDF       Save
    Let $(M, F)$ be an $n$-dimensional Randers space with scalar flag curvature. In this paper, we will introduce the definition of a weak Einstein manifold. We can prove that if $(M, F)$ is a weak Einstein manifold, then the flag curvature is constant.
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    STABILITY CONDITIONS AND THE MIRROR SYMMETRY OF K3 SURFACES IN ATTRACTOR BACKGROUNDS*
    Wenxuan Lu
    Acta mathematica scientia,Series B    2023, 43 (3): 1007-1030.   DOI: 10.1007/s10473-023-0303-4
    Abstract42)      PDF       Save
    We study the space of stability conditions on $K3$ surfaces from the perspective of mirror symmetry. This is done in the attractor backgrounds (moduli). We find certain highly non-generic behaviors of marginal stability walls (a key notion in the study of wall crossings) in the space of stability conditions. These correspond via mirror symmetry to some non-generic behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror correspondence in a synthesis of the homological mirror conjecture and SYZ mirror conjecture.
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    SINGULAR DOUBLE PHASE EQUATIONS*
    Zhenhai Liu, Nikolaos S. Papageorgiou
    Acta mathematica scientia,Series B    2023, 43 (3): 1031-1044.   DOI: 10.1007/s10473-023-0304-3
    Abstract41)      PDF       Save
    We study a double phase Dirichlet problem with a reaction that has a parametric singular term. Using the Nehari manifold method, we show that for all small values of the parameter, the problem has at least two positive, energy minimizing solutions.
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    THE EXISTENCE AND STABILITY OF NORMALIZED SOLUTIONS FOR A BI-HARMONIC NONLINEAR SCHRÖDINGER EQUATION WITH MIXED DISPERSION*
    Tingjian Luo, Shijun Zheng, Shihui Zhu
    Acta mathematica scientia,Series B    2023, 43 (2): 539-563.   DOI: 10.1007/s10473-023-0205-5
    Abstract41)      PDF       Save
    In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr\"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denoting by $Q_p$ the ground state for the BNLS with $\mu=0$, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $\mu\in ( -\lambda_0, \infty)$ for some $\lambda_0=\lambda_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$, we prove the orbital stability on a certain mass level below $\|Q^*\|_{L^2}$, provided that $\mu\in (-\lambda_1,0)$, where $\lambda_1=\frac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac8d]$.
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    The Smoothing effect in sharp Gevrey space for  the spatially homogeneous non-cutoff  Boltzmann equations  with a hard potential
    Lvqiao Liu, Juan Zeng
    Acta mathematica scientia,Series B    DOI: 10.1007/s10473-024-0205-0
    Accepted: 16 October 2023
    Online available: 06 December 2023

    BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM*
    Danyang LI, Hua LIU, Haotian ZHANG, Ming MA, Yong YE, Yumei WEI
    Acta mathematica scientia,Series B    2023, 43 (3): 1415-1438.   DOI: 10.1007/s10473-023-0324-z
    Abstract32)      PDF       Save
    Regarding delay-induced predator-prey models, much research has been done on delayed destabilization, but whether delays are stabilizing or destabilizing is a subtle issue. In this study, we investigate predator-prey dynamics affected by both delays and the Allee effect. We analyze the consequences of delays in different feedback mechanisms. The existence of a Hopf bifurcation is studied, and we calculate the value of the delay that leads to the Hopf bifurcation. Furthermore, applying the normal form theory and a center manifold theorem, we consider the direction and stability of the Hopf bifurcation. Finally, we present numerical experiments that validate our theoretical analysis. Interestingly, depending on the chosen delay mechanism, we find that delays are not necessarily destabilizing. The Allee effect generally increases the stability of the equilibrium, and when the Allee effect involves a delay term, the stabilization effect is more pronounced.
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    A LARGE DEVIATION PRINCIPLE FOR THE STOCHASTIC GENERALIZED GINZBURG-LANDAU EQUATION DRIVEN BY JUMP NOISE*
    Ran Wang, Beibei Zhang
    Acta mathematica scientia,Series B    2023, 43 (2): 505-530.   DOI: 10.1007/s10473-023-0203-7
    Abstract32)      PDF       Save
    In this paper, we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise. The main difficulties come from the highly non-linear coefficient and the jump noise. Here, we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle, which was initially proposed by Matoussi, Sabbagh and Zhang (2021).
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    MINIMAL FOLIATIONS FOR THE HIGH-DIMENSIONAL FRENKEL-KONTOROVA MODEL*
    Xueqing Miao, Jianhua Ge, Wenxin Qin, Yanan Wang
    Acta mathematica scientia,Series B    2023, 43 (2): 564-582.   DOI: 10.1007/s10473-023-0207-3
    Abstract32)      PDF       Save
    For the high-dimensional Frenkel-Kontorova (FK) model on lattices, we study the existence of minimal foliations by depinning force. We introduce the tilted gradient flow and define the depinning force as the critical value of the external force under which the average velocity of the system is zero. Then, the depinning force can be used as the criterion for the existence of minimal foliations for the FK model on a $\mathbb{Z}^d$ lattice for $d>1$.
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    From wave functions to tau-functions for the Volterra lattice hierarchy
    Ang Fu, Mingjin Li, Di Yang
    Acta mathematica scientia,Series B    DOI: 10.1007/s10473-024-0201-4
    Accepted: 16 October 2023
    Online available: 06 December 2023

    THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS*
    Lili WEN, Engui FAN, Yong CHEN
    Acta mathematica scientia,Series B    2023, 43 (3): 1045-1080.   DOI: 10.1007/s10473-023-0305-2
    Abstract30)      PDF       Save
    We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter $k$ into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the $z$-complex plane is divided into four analytic regions of $D_j: \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}$, but in the upper-half or lower-half plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in $D_{j}$. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries; these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this $N$-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.
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    TWO GENERALIZATIONS OF BOHR RADIUS*
    Chengpeng Li, Mingxin Chen, Jianfei Wang
    Acta mathematica scientia,Series B    2023, 43 (2): 583-596.   DOI: 10.1007/s10473-023-0206-4
    Abstract30)      PDF       Save
    The purpose of this paper is twofold. First, by using the hyperbolic metric, we establish the Bohr radius for analytic functions from shifted disks containing the unit disk $D$ into convex proper domains of the complex plane. As a consequence, we generalize the Bohr radius of Evdoridis, Ponnusamy and Rasila based on geometric idea. By introducing an alternative multidimensional Bohr radius, the second purpose is to obtain the Bohr radius of higher dimensions for Carathéodory families in the unit ball $B$ of a complex Banach space $X$. Notice that when $B$ is the unit ball of the complex Hilbert space $X$, we show that the constant $ {1}/{3} $ is the Bohr radius for normalized convex mappings of $B$, which generalizes the result of convex functions on $D$.
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    DOUBLE INERTIAL PROXIMAL GRADIENT ALGORITHMS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS*
    Kunrada Kankam, Prasit Cholamjiak
    Acta mathematica scientia,Series B    2023, 43 (3): 1462-1476.   DOI: 10.1007/s10473-023-0326-x
    Abstract29)      PDF       Save
    In this paper, we propose double inertial forward-backward algorithms for solving unconstrained minimization problems and projected double inertial forward-backward algorithms for solving constrained minimization problems. We then prove convergence theorems under mild conditions. Finally, we provide numerical experiments on image restoration problem and image inpainting problem. The numerical results show that the proposed algorithms have more efficient than known algorithms introduced in the literature.
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    PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION*
    Bo Chen, Zhengmao Chen, Junhui Xie
    Acta mathematica scientia,Series B    2023, 43 (3): 1161-1174.   DOI: 10.1007/s10473-023-0310-5
    Abstract28)      PDF       Save
    In the present paper, we consider the problem
    $\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $ \ \ \ \ \ $ (0.1)
    where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega $ is a convex domain in $ \mathbb{R}^{n} $. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven.
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    Three kinds of dentabilities in Banach spaces and their applications
    zihou Zhang, Jing Zhou
    Acta mathematica scientia,Series B    DOI: 10.1007/s10473-024-0204-1
    Accepted: 16 October 2023
    Online available: 06 December 2023

    POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS*
    Qing Guo, Leiga Zhao
    Acta mathematica scientia,Series B    2023, 43 (3): 1116-1130.   DOI: 10.1007/s10473-023-0308-z
    Abstract27)      PDF       Save
    In this paper, we study the Schrödinger equations
    $ (-\Delta)^s u+ V(x)u= a(x)|u|^{p-2}u+b(x)|u|^{q-2}u,\ \ x\in\ {\mathbb{R}}^{N},$
    where $0<s<1$, $2<q<p<2^*_s$, $2^*_s$ is the fractional Sobolev critical exponent. Under suitable assumptions on $V$, $a$ and $b$ for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.
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    JOHN-NIRENBERG-Q SPACES VIA CONGRUENT CUBES*
    Jin Tao, Zhenyu Yang, Wen Yuan
    Acta mathematica scientia,Series B    2023, 43 (2): 686-718.   DOI: 10.1007/s10473-023-0214-4
    Abstract27)      PDF       Save
    To shed some light on the John-Nirenberg space, the authors of this article introduce the John-Nirenberg-$Q$ space via congruent cubes, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$, which, when $p=\infty$ and $q=2$, coincides with the space $Q_\alpha(\mathbb{R}^n)$ introduced by Essén, Janson, Peng and Xiao in [Indiana Univ Math J, 2000, 49(2): 575--615]. Moreover, the authors show that, for some particular indices, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ coincides with the congruent John-Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into $JNQ^\alpha_{p,q}(\mathbb{R}^n)$. Furthermore, the authors characterize $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of `almost increasing' set function that is here introduced, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.
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    FIXED/PREASSIGNED-TIME SYNCHRONIZATION OF QUATERNION-VALUED NEURAL NETWORKS INVOLVING DELAYS AND DISCONTINUOUS ACTIVATIONS: A DIRECT APPROACH*
    Wanlu WEI, Cheng HU, Juan YU, Haijun JIANG
    Acta mathematica scientia,Series B    2023, 43 (3): 1439-1461.   DOI: 10.1007/s10473-023-0325-y
    Abstract26)      PDF       Save
    The fixed-time synchronization and preassigned-time synchronization are investigated for a class of quaternion-valued neural networks with time-varying delays and discontinuous activation functions. Unlike previous efforts that employed separation analysis and the real-valued control design, based on the quaternion-valued signum function and several related properties, a direct analytical method is proposed here and the quaternion-valued controllers are designed in order to discuss the fixed-time synchronization for the relevant quaternion-valued neural networks. In addition, the preassigned-time synchronization is investigated based on a quaternion-valued control design, where the synchronization time is preassigned and the control gains are finite. Compared with existing results, the direct method without separation developed in this article is beneficial in terms of simplifying theoretical analysis, and the proposed quaternion-valued control schemes are simpler and more effective than the traditional design, which adds four real-valued controllers. Finally, two numerical examples are given in order to support the theoretical results.
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