数学物理学报(英文版)

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CONSTANT DISTANCE BOUNDARIES OF THE

$t$ -QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE^*

Xin Wei, Zhi-Ying Wen

数学物理学报(英文版). 2023 (3): 981-993. DOI: 10.1007/s10473-023-0301-6

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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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RANDERS SPACES WITH SCALAR FLAG CURVATURE^*

Jintang LI

数学物理学报(英文版). 2023 (3): 994-1006. DOI: 10.1007/s10473-023-0302-5

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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

数学物理学报(英文版). 2023 (3): 1007-1030. DOI: 10.1007/s10473-023-0303-4

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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

数学物理学报(英文版). 2023 (3): 1031-1044. DOI: 10.1007/s10473-023-0304-3

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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 SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS^*

Lili WEN, Engui FAN, Yong CHEN

数学物理学报(英文版). 2023 (3): 1045-1080. DOI: 10.1007/s10473-023-0305-2

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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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ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION^*

Xueke Pu, Wenli Zhou

数学物理学报(英文版). 2023 (3): 1081-1104. DOI: 10.1007/s10473-023-0306-1

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In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

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REAL PALEY-WIENER THEOREMS FOR THE SPACE-TIME FOURIER TRANSFORM^*

Youssef El Haoui, Mohra Zayed

数学物理学报(英文版). 2023 (3): 1105-1115. DOI: 10.1007/s10473-023-0307-0

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This paper presents an extension of certain forms of the real Paley-Wiener theorems to the Minkowski space-time algebra. Our emphasis is dedicated to determining the space-time valued functions whose space-time Fourier transforms (SFT) have compact support using the partial derivatives operator and the Dirac operator of higher order.

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POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRÖDINGER EQUATIONS^*

Qing Guo, Leiga Zhao

数学物理学报(英文版). 2023 (3): 1116-1130. DOI: 10.1007/s10473-023-0308-z

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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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THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*

Leilei CUI, Jiaxing GUO, Gongbao LI

数学物理学报(英文版). 2023 (3): 1131-1160. DOI: 10.1007/s10473-023-0309-y

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In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations

$\begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)\Delta u +V(x)u =u^{p}, u>0 \text{in} \mathbb{R}^3, \end{equation*}$
which concentrate at non-degenerate critical points of the potential function

$V(x)$ , where

$a,b>0$ ,

$1<p<5$ are constants, and

$\varepsilon>0$ is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at

$\{a_i\}_{1\leq i\leq k}$ , where

$\{a_{i}\}_{1\leq i\leq k}$ are non-degenerate critical points of

$V(x)$ as

$\varepsilon\to 0$ .

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PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*

Bo Chen, Zhengmao Chen, Junhui Xie

数学物理学报(英文版). 2023 (3): 1161-1174. DOI: 10.1007/s10473-023-0310-5

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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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SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*

Xuemei Zhang, Shikun Kan

数学物理学报(英文版). 2023 (3): 1175-1194. DOI: 10.1007/s10473-023-0311-4

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Let

$\Omega$ denote a smooth, bounded domain in

$\mathbb{R}^N (N\geq 2)$ . Suppose that

$g$ is a nondecreasing

$C^1$ positive function and assume that

$b(x)$ is continuous and nonnegative in

$\Omega$ , and that it may be singular on

$\partial\Omega$ . In this paper, we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the

$p$ -Laplacian problem

$\Delta_p u=b(x)g(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } { dist}(x,\partial \Omega)\rightarrow 0$ .
The estimates of such solutions are also investigated. Moreover, when

$b$ has strong singularity, the nonexistence of boundary blow-up (radial) solutions and infinitely many radial solutions are also considered.

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HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*

Qingye Zhang, Chungen Liu

数学物理学报(英文版). 2023 (3): 1195-1210. DOI: 10.1007/s10473-023-0312-3

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In this paper, we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems

$\dot{z}=JH_z(t,z)$ , where the Hamiltonian function

$H$ possesses the form

$H(t,z)=\frac{1}{2}L(t)z\cdot z +G(t,z)$ , and

$G(t,z)$ is only locally defined near the origin with respect to

$z$ . Under some mild conditions on

$L$ and

$G$ , we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense, which is essentially forced by the subquadraticity of

$G$ near the origin with respect to

$z$ .

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LOCAL STRUCTURE-PRESERVING ALGORITHMS FOR THE KLEIN-GORDON-ZAKHAROV EQUATION^*

Jialing Wang, Zhengting Zhou, Yushun Wang

数学物理学报(英文版). 2023 (3): 1211-1238. DOI: 10.1007/s10473-023-0313-2

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In this paper, using the concatenating method, a series of local structure-preserv-ing algorithms are obtained for the Klein-Gordon-Zakharov equation, including four multi-symplectic algorithms, four local energy-preserving algorithms, four local momentum-preserving algorithms; of these, local energy-preserving and momentum-preserving algorithms have not been studied before. The local structure-preserving algorithms mentioned above are more widely used than the global structure-preserving algorithms, since local preservation algorithms can be preserved in any time and space domains, which overcomes the defect that global preservation algorithms are limited to boundary conditions. In particular, under appropriate boundary conditions, local preservation laws are global preservation laws. Numerical experiments conducted can support the theoretical analysis well.

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THE LOW MACH NUMBER LIMIT FOR ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A REVISED MAXWELL'S LAW^*

Yuxi Hu, Zhao Wang

数学物理学报(英文版). 2023 (3): 1239-1250. DOI: 10.1007/s10473-023-0314-1

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We investigate the low Mach number limit for the isentropic compressible Navier-Stokes equations with a revised Maxwell's law (with Galilean invariance) in

$\mathbb R^3$ . By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell's law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.

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GLOBAL SOLUTIONS TO THE 2D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SOME LARGE INITIAL DATA^*

Xiaoping Zhai, Xin Zhong

数学物理学报(英文版). 2023 (3): 1251-1274. DOI: 10.1007/s10473-023-0315-0

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We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in

$\mathbb{R}^2$ . By exploiting the global-in-time estimate to the two-dimensional (2D for short) classical incompressible Navier-Stokes equations and using techniques developed in (SIAM J Math Anal, 2020, 52(2): 1806-1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part

$\mathbb{P} u_0$ and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.

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GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY^*

Hongxia Lin, Sen Liu, Heng Zhang, Ru Bai

数学物理学报(英文版). 2023 (3): 1275-1300. DOI: 10.1007/s10473-023-0316-z

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This paper studies the global regularity of 2D incompressible anisotropic magneto-micropolar fluid equations with partial viscosity. Ma [22] (Ma L. Nonlinear Anal: Real World Appl, 2018, 40: 95-129) examined the global regularity of the 2D incompressible magneto-micropolar fluid system for 21 anisotropic partial viscosity cases. He proved the global existence of a classical solution for some cases and established the conditional global regularity for some other cases. In this paper, we also investigate the global regularity of 12 cases in [22] and some other new partial viscosity cases. The global regularity is established by providing new regular conditions. Our work improves some results in [22] in this sense of weaker regular criteria.

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GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN

${\mathbb{R}}^{3}$ ^*

Yuhui Chen, Qinghe Yao, Minling Li, Zheng-an Yao

数学物理学报(英文版). 2023 (3): 1301-1322. DOI: 10.1007/s10473-023-0317-y

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In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner (GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions on smooth function

$f$ , we find a particular solution to the GPTT model. In dimension three, we establish the global existence and the optimal time decay rates of strong solutions provided that the initial data is close to the particular solution. The results which are presented here are generalizations of the network viscoelastic models.

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A REMARK ON LARGE TIME ASYMTOTICS FOR SOLUTIONS OF A NONHOMOGENEOUS VISCOUS BURGERS EQUATION^*

Manas Ranjan Sahoo, Satyanarayana Engu, Smriti Tiwari

数学物理学报(英文版). 2023 (3): 1323-1332. DOI: 10.1007/s10473-023-0318-x

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The existence, uniqueness and regularity of solutions to the Cauchy problem posed for a nonhomogeneous viscous Burger's equation were shown in Chung, Kim and Slemrod [1] by assuming suitable conditions on initial data. Moreover, they derived the asymptotic behaviour of solutions of the Cauchy problem by imposing additional conditions on initial data. In this article, we obtain the same asymptotic behaviour of solutions to the Cauchy problem without imposing additional condition on initial data.

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THE SINGULAR LIMIT OF SECOND-GRADE FLUID EQUATIONS IN A 2D EXTERIOR DOMAIN^*

Xiaoguang You, Aibin Zang

数学物理学报(英文版). 2023 (3): 1333-1346. DOI: 10.1007/s10473-023-0319-9

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In this paper, we consider the second-grade fluid equations in a 2D exterior domain satisfying the non-slip boundary conditions. The second-grade fluid model is a well-known non-Newtonian fluid model, with two parameters:

$\alpha$ , which represents the length-scale, while

$\nu > 0$ corresponds to the viscosity. We prove that, as

$\nu, \alpha$ tend to zero, the solution of the second-grade fluid equations with suitable initial data converges to the one of Euler equations, provided that

$\nu = {o}(\alpha^\frac{4}{3})$ . Moreover, the convergent rate is obtained.

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THE OPTIMAL DEDUCTIBLE AND COVERAGE IN INSURANCE CONTRACTS AND EQUILIBRIUM RISK SHARING POLICIES^*

Lingling Jian

数学物理学报(英文版). 2023 (3): 1347-1364. DOI: 10.1007/s10473-023-0320-3

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In this paper, we consider the optimal risk sharing problem between two parties in the insurance business: the insurer and the insured. The risk is allocated between the insurer and the insured by setting a deductible and coverage in the insurance contract. We obtain the optimal deductible and coverage by considering the expected product of the two parties' utilities of terminal wealth according to stochastic optimal control theory. An equilibrium policy is also derived for when there are both a deductible and coverage; this is done by modelling the problem as a stochastic game in a continuous-time framework. A numerical example is provided to illustrate the results of the paper.

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ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*

Liangliang Mia, Yanhong Chen, Xiao Xiao, Yijun Hu

数学物理学报(英文版). 2023 (3): 1365-1381. DOI: 10.1007/s10473-023-0321-2

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In this paper, we focus on anticipated backward stochastic Volterra integral equations (ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps. Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

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SOME RESULTS ON BUNDLE SYSTEMS FOR A COUNTABLE DISCRETE AMENABLE GROUP ACTION^*

Juan Pan, Yunhua Zhou

数学物理学报(英文版). 2023 (3): 1382-1402. DOI: 10.1007/s10473-023-0322-1

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We consider the style number, independence number and entropy for a frame bundle dynamical system. The base system of which is a countable discrete amenable group action on a compact metric space. We obtain the existence of cover measures, an ergodic theorem about mean linear independence and the style number, and a variational principle for style numbers and independence numbers. We also study the relationship between the entropy of base systems and that of their bundle systems.

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HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING^*

Wenyi Pei, Litan Yan, Zhenlong Chen

数学物理学报(英文版). 2023 (3): 1403-1414. DOI: 10.1007/s10473-023-0323-0

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In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter

$H$ is supposed to be in

$(1/2,1)$ . As a direct application, the strong Feller property is presented.

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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

数学物理学报(英文版). 2023 (3): 1415-1438. DOI: 10.1007/s10473-023-0324-z

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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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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

数学物理学报(英文版). 2023 (3): 1439-1461. DOI: 10.1007/s10473-023-0325-y

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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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DOUBLE INERTIAL PROXIMAL GRADIENT ALGORITHMS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS^*

Kunrada Kankam, Prasit Cholamjiak

数学物理学报(英文版). 2023 (3): 1462-1476. DOI: 10.1007/s10473-023-0326-x

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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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