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CONTINUITY PROPERTIES FOR BORN-JORDAN OPERATORS WITH SYMBOLS IN HÖRMANDER CLASSES AND MODULATION SPACES Maurice de GOSSON, Joachim TOFT Acta mathematica scientia,Series B    2020, 40 (6): 1603-1626.   DOI: 10.1007/s10473-020-0601-z Abstract （61）      PDF       Save We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hörmander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity, nuclearity and Schatten-von Neumann properties to the Born-Jordan calculus. Reference | Related Articles | Metrics

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ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS Chunlei CAO, Yuefei WANG Acta mathematica scientia,Series B    2020, 40 (4): 903-909.   DOI: 10.1007/s10473-020-0401-5 Abstract （53）      PDF       Save In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of ${\bf C^*}$ can neither contain a free Jordan arc nor be contained in any finite set of straight lines. Reference | Related Articles | Metrics

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MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS Hanwu LI Acta mathematica scientia,Series B    2021, 41 (2): 349-360.   DOI: 10.1007/s10473-021-0201-6 Abstract （37）      PDF       Save In this paper, we study the martingale inequalities under $G$-expectation and their applications. To this end, we introduce a new kind of random time, called $G$-stopping time, and then investigate the properties of a $G$-martingale (supermartingale) such as the optional sampling theorem and upcrossing inequalities. With the help of these properties, we can show the martingale convergence property under $G$-expectation. Reference | Related Articles | Metrics

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CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION Xinyu TU, Chunlai MU, Shuyan QIU Acta mathematica scientia,Series B    2021, 41 (1): 1-18.   DOI: 10.1007/s10473-021-0101-9 Abstract （36）      PDF       Save In this article, we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation. Based on defining a Finsler-type norm on the tangent space for solutions, we first establish the Lipschitz metric for smooth solutions, then by proving the generic regularity result, we extend this metric to general weak solutions. Reference | Related Articles | Metrics

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ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS Xuan Truong LE, Thanh Nhan NGUYEN, Ngoc Trong NGUYEN Acta mathematica scientia,Series B    2020, 40 (5): 1171-1184.   DOI: 10.1007/s10473-020-0501-2 Abstract （34）      PDF       Save In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator ?=-△+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space. Reference | Related Articles | Metrics

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BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS Zongguang LI, Rui LIU Acta mathematica scientia,Series B    2020, 40 (4): 910-920.   DOI: 10.1007/s10473-020-0402-4 Abstract （33）      PDF       Save In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution. Reference | Related Articles | Metrics

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ASYMPTOTIC STABILITY OF A BOUNDARY LAYER AND RAREFACTION WAVE FOR THE OUTFLOW PROBLEM OF THE HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY Lili FAN, Meichen HOU Acta mathematica scientia,Series B    2020, 40 (6): 1627-1652.   DOI: 10.1007/s10473-020-0602-y Abstract （33）      PDF       Save This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions. New weighted energy estimates are introduced, and the trace of the density and velocity on the boundary are handled by some subtle analysis. The decay properties of the boundary layer and the smooth rarefaction wave also play an important role. Reference | Related Articles | Metrics

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EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT Fanfan CHEN, Yang YANG Acta mathematica scientia,Series B    2020, 40 (6): 1666-1678.   DOI: 10.1007/s10473-020-0604-9 Abstract （23）      PDF       Save In this article, we study the following fractional $(p,q)$-Laplacian equations involving the critical Sobolev exponent: \[ (P_{\mu, \lambda}) \begin{cases} (-\Delta)_{p}^{s_{1}}u+(-\Delta)_{q}^{s_{2}}u=\mu |u|^{q-2}u +\lambda|u|^{p-2}u + |u|^{p_{s_{1}}^{*}-2}u, & \text{in $\Omega$,} \\ u=0, & \text{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, $\lambda,\ \mu >0, \ 0 < s_{2} < s_{1} < 1,\ 1 < q < p < \frac{N}{s_{1}} $. We establish the existence of a non-negative nontrivial weak solution to $(P_{\mu, \lambda})$ by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers. Reference | Related Articles | Metrics

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GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM Qingyou SUN, Yunguang LU, Christian KLINGENBERG Acta mathematica scientia,Series B    2020, 40 (5): 1185-1194.   DOI: 10.1007/s10473-020-0502-1 Abstract （23）      PDF       Save In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable $s$, the system about the variables $\rho$ and $u$ is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function $P( \rho,s)= {\rm e}^{s} {\rm e}^{-\frac{1}{\rho }}$, which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants $(w_{1}(x,0),w_{2}(x,0))$ of system (1.1) is small, $(w_{1}(x,0),w_{2}(x,0))$ are nondecreasing and the third Riemann invariant $s(x,0)$ is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1]. Reference | Related Articles | Metrics

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UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION Wenqiang ZHAO, Yijin ZHANG Acta mathematica scientia,Series B    2020, 40 (4): 921-933.   DOI: 10.1007/s10473-020-0403-3 Abstract （22）      PDF       Save In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature. Reference | Related Articles | Metrics

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GROUND STATE SOLUTIONS FOR A SCHRÖDINGER-POISSON SYSTEM WITH UNCONVENTIONAL POTENTIAL Yao DU, Chunlei TANG Acta mathematica scientia,Series B    2020, 40 (4): 934-944.   DOI: 10.1007/s10473-020-0404-2 Abstract （22）      PDF       Save We consider the Schrödinger-Poisson system with nonlinear term $Q(x)|u|^{p-1}u$, where the value of $\displaystyle\lim_{|x|\rightarrow\infty} Q(x)$ may not exist and $Q$ may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks. Reference | Related Articles | Metrics

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ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES Wei CHEN, Qiong WANG Acta mathematica scientia,Series B    2020, 40 (4): 1035-1044.   DOI: 10.1007/s10473-020-0411-3 Abstract （21）      PDF       Save This article investigates the algebraic differential independence concerning the Euler $\Gamma$-function and the function $F$ in a certain class $\mathbb{F}$ which contains Dirichlet $\mathcal{L}$-functions, $\mathcal{L}$-functions in the extended Selberg class, or some periodic functions. We prove that the Euler $\Gamma$-function and the function $F$ cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions $\phi$ with $\rho(\phi)<1$. Reference | Related Articles | Metrics

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ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES Qinghua XU, Yuanping LAI Acta mathematica scientia,Series B    2020, 40 (6): 1653-1665.   DOI: 10.1007/s10473-020-0603-x Abstract （21）      PDF       Save Let $\mathcal{K}$ be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function $f(z)=z+\sum\limits_{k=2}^\infty a_kz^k$ in the class $\mathcal{K}$, \begin{align*} |a_3-\lambda a_2^2|\leq \max \left\{\frac{1}{3}, |\lambda-1|\right\},\ \ \lambda \in \mathbb{C}. \end{align*} The above estimate is sharp for each $\lambda$. In this article, we establish the corresponding inequality for a normalized convex function $f$ on $\mathbb{U}$ such that $z=0$ is a zero of order $k+1$ of $f(z)-z$, and then we extend this result to higher dimensions. These results generalize some known results. Reference | Related Articles | Metrics

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RETRACTION NOTE: “MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD” Editorial Office of Acta Mathematica Scientia Acta mathematica scientia,Series B    2020, 40 (5): 1602-1602.   DOI: 10.1007/s10473-020-0524-8 Abstract （19）      PDF       Save Reference | Related Articles | Metrics

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ASYMPTOTIC STABILITY OF A VISCOUS CONTACT WAVE FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE Lishuang PENG Acta mathematica scientia,Series B    2020, 40 (5): 1195-1214.   DOI: 10.1007/s10473-020-0503-0 Abstract （17）      PDF       Save We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2-energy methods. Reference | Related Articles | Metrics

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THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES Weifeng JIANG, Zhen WANG Acta mathematica scientia,Series B    2020, 40 (5): 1229-1239.   DOI: 10.1007/s10473-020-0505-y Abstract （17）      PDF       Save In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special state equation, a bounded invariant region for the non-isentropic process may exist. Reference | Related Articles | Metrics

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QUASI-NEUTRAL LIMIT AND THE INITIAL LAYER PROBLEM OF THE DRIFT-DIFFUSION MODEL Shu WANG, Limin JIANG Acta mathematica scientia,Series B    2020, 40 (4): 1152-1170.   DOI: 10.1007/s10473-020-0419-8 Abstract （17）      PDF       Save In this article we study quasi-neutral limit and the initial layer problem of the drift-diffusion model. Different from others studies, we consider the physical case that the mobilities of the charges are different. The quasi-neutral limit with an initial layer structure is rigorously proved by using the weighted energy method coupled with multi-scaling asymptotic expansions. Reference | Related Articles | Metrics

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WEAK SOLUTION TO THE INCOMPRESSIBLE VISCOUS FLUID AND A THERMOELASTIC PLATE INTERACTION PROBLEM IN 3D Srđan TRIFUNOVIĆ, Yaguang WANG Acta mathematica scientia,Series B    2021, 41 (1): 19-38.   DOI: 10.1007/s10473-021-0102-8 Abstract （17）      PDF       Save In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function. Reference | Related Articles | Metrics

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GENERALIZED ROPER-SUFFRIDGE OPERATOR FOR $\epsilon$ STARLIKE AND BOUNDARY STARLIKE MAPPINGS Jie WANG, Jianfei WANG Acta mathematica scientia,Series B    2020, 40 (6): 1753-1764.   DOI: 10.1007/s10473-020-0610-y Abstract （16）      PDF       Save This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves $\epsilon$ starlikeness on the open unit ball of a complex Banach space $\mathbb{C}\times X$, where $X$ is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order $\alpha$ on the unit ball $B^n$ of ${\mathbb{C}}^{n}$, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order $\alpha$ on $B^n$. Finally, we propose some problems. Reference | Related Articles | Metrics

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EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS Yaghoub JALILIAN Acta mathematica scientia,Series B    2020, 40 (6): 1831-1848.   DOI: 10.1007/s10473-020-0614-7 Abstract （15）      PDF       Save In this paper, we study the coupled system of Kirchhoff type equations \begin{equation*} \left\{ \begin{array}{ll} -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla u|^{2}{\rm d}x}\bigg)\Delta u+ u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in \mathbb{R}^3, \\[3mm] -\bigg(a+b\int_{\mathbb{R}^3}{|\nabla v|^{2}{\rm d}x}\bigg)\Delta v+ v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in \mathbb{R}^3, \\[2mm] u,v\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation*} where $a,b > 0$, $ \alpha, \beta > 1$ and $3 < \alpha+\beta < 6$. We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity. Also, using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method, we obtain the existence of infinitely many geometrically distinct solutions in the case when $ \alpha, \beta \geq 2$ and $4\leq\alpha+\beta < 6$. Reference | Related Articles | Metrics