In this paper, we study composition operators on weighted Bergman spaces of Dirichlet series. We first establish some Littlewood-type inequalities for generalized mean counting functions. Then we give sufficient conditions for a composition operator with zero characteristic to be bounded or compact on weighted Bergman spaces of Dirichlet series. The corresponding sufficient condition for compactness in the case of positive characteristics is also obtained.

Let $0<p\leq1<q<\infty$, and $\omega_{1},\omega_{2}\in A_{1}$ (Muckenhoupt-class). We study an oscillating multiplier operator $T_{\gamma ,\beta }$ and obtain that it is bounded on the homogeneous weighted Herz-type Hardy spaces $H\dot{K}_{q}^{\alpha ,p}(\mathbb{R}^{n};\omega _{1},\omega _{2})$ when $\gamma=\frac{n\beta }{2}, \alpha =n(1-1/q)$. Also, for the unweighted case, we obtain the $H\dot{K}_{q}^{\alpha ,p}(\mathbb{R}^{n})$ boundedness of $T_{\gamma ,\beta }$ under certain conditions on $\gamma$. These results are substantial improvements and extensions of the main results in the papers by Li and Lu and by Cao and Sun. As an application, we prove the $H\dot{K}_{q}^{\alpha ,p}(\mathbb{R}^{n})$ boundedness of the spherical average $S_{t}^{\delta}$ uniformly on $t>0$.

In this paper, $C^{1, 1}$ regularity for solutions to the degenerate dual Orlicz-Minkowski problem is considered. The dual Orlicz-Minkowski problem is a generalization of the $L_p$ dual Minkowski problem in convex geometry. The proof is adapted from Guan-Li [17] and Chen-Tu-Wu-Xiang [11].

The differential system $\dot x= ax -y z, \ \dot y = -b y + x z, \ \dot z= -c z + x^2,$ where $a$, $b$ and $c$ are positive real parameters, has been studied numerically due to the big variety of strange attractors that it can exhibit. This system has a Darboux invariant when $c=2b$. Using this invariant and the Poincaré compactification technique we describe analytically its global dynamics.

In this paper, the problem of brake orbits with minimal period estimates are considered for the first-order Hamiltonian systems with anisotropic growth, i.e., the Hamiltonian functions may have super-quadratic, sub-quadratic and quadratic behaviors simultaneously in different variable components.

We consider large-time behaviors of weak solutions to the evolutionary $p$-Laplacian with logarithmic source of time-dependent coefficient. We find that the weak solutions may neither decay nor blow up, provided that the initial data $u(\cdot,t_0)$ is on the Nehari manifold $\mathscr{N}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)=0, \|\nabla v\|_p^p\neq0 \big\}$. This is quite different from the known results that the weak solutions may blow up as $u(\cdot, t_0)\in \mathscr{N}^{-}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)<0\big\}$ and weak solutions may decay as $u(\cdot, t_0)\in\mathscr{N}^{+}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)>0\big\}$.

We are concerned with a nonlinear elliptic equation, involving a Kirchhoff type nonlocal term and a potential $V(x)$, on $\mathbb{R}^3$. As is well known that, even in $H^1_r(\mathbb{R}^3)$, the nonlinear term is a pure power form of $|u|^{p-1}u$ and $V(x)\equiv 1$, it seems very difficult to apply the mountain-pass theorem to get a solution (i.e., mountain-pass solution) to this kind of equation for all $p\in(1,5)$, due to the difficulty of verifying the boundedness of the Palais-Smale sequence obtained by the mountain-pass theorem when $p\in(1,3)$. In this paper, we find a new strategy to overcome this difficulty, and then get a mountain-pass solution to the equation for all $p\in(1,5)$ and for both $V(x)$ being constant and nonconstant. Also, we find a possibly optimal condition on $V(x)$.

Combining $TT^{\ast}$ argument and bilinear interpolation, this paper obtains the Strichartz and smoothing estimates of dispersive semi-group ${\rm e}^{-{\rm i} t P(D)}$ in weighted $L^{2}$ spaces. Among other things, we recover the results in [1]. Moreover, the application of these results to the well-posedness of some equations are shown in the last section.

We investigate the long time existence of strong solutions to the initial value problem for the three-dimensional non-isentropic compressible Navier-Stokes-Korteweg system. Under the conditions of slight density and temperature variations, we verify that the full compressible Navier-Stokes-Korteweg equations admit a unique strong solution as long as the solution of the limiting system exists, when the Mach number is sufficiently small. Furthermore, we deduce the uniform convergence of strong solutions for the compressible system toward those for the corresponding incompressible system on the time interval in which the solution exists.

In this paper, we study the Cauchy problem of three-dimensional incompressible magnetohydrodynamics with almost symmetrical initial values in the cylindrical coordinates. Here the almost axisymmetric means that $(\partial_\theta u^r_0,\partial_\theta u^\theta_0,\partial_\theta u^z_0)$ is small. With additional smallness assumption on $(u^\theta_0,b^\theta_0)$, we prove the global existence of a unique strong solution $(\boldsymbol{u},\boldsymbol{b})$, which keeps close to some axisymmetric vector fields. Moreover, we give the initial data with some special symmetric structures that will persist for all time.

Let ${\cal I}$ be the set of all infinitely divisible random variables with finite second moments, ${\cal I}_0=\{X\in{\cal I}:{\rm Var}(X)>0\}$, $P_{\cal I}=\inf\limits_{X\in{\cal I}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $P_{{\cal I}_0}=\inf\limits_{X\in{\cal I}_0} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$. Firstly, we prove that $P_{{\cal I}}\ge P_{{\cal I}_0}>0$. Secondly, we find the exact values of $\inf\limits_{X\in{\cal J}}P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}$ and $\inf\limits_{X\in\cal J} P\{|X-E[X]|< \sqrt{{\rm Var}(X)}\}$ for the cases that $\cal J$ is the set of all geometric random variables, symmetric geometric random variables, Poisson random variables and symmetric Poisson random variables, respectively. As a consequence, we obtain that $P_{\cal I}\le {\rm e}^{-1}\sum\limits_{k=0}^{\infty}\frac{1}{2^{2k}(k!)^2}\approx 0.46576$ and $P_{{\cal I}_0}\le {\rm e}^{-1}\approx 0.36788$.

In this paper, we consider the Fisher informations among three classical type $\beta$-ensembles when $\beta>0$ scales with $n$ satisfying $\lim\limits_{n\to\infty}\beta n=\infty.$ We offer the exact order of the corresponding two Fisher informations, which indicates that the $\beta$-Laguerre ensembles do not satisfy the logarithmic Sobolev inequality. We also give some limit theorems on the extremals of $\beta$-Jacobi ensembles for $\beta>0$ fixed.

In this paper, we obtain a vector bundle valued mixed hard Lefschetz theorem. The argument is mainly based on the works of Tien-Cuong Dinh and Viêt-Anh Nguyên.

In this paper, we prove that for certain fiber bundles there is a $k$-Futaki-Ono conformally Kähler metric related to a metric in any given Kähler class for any $k \geq 2$.

Two sharp Chernoff type inequalities are derived for star bodies in $\mathbb{R}^2$, one is an extension of the dual Chernoff-Ou-Pan inequality, and the other is the reverse Chernoff type inequality. Furthermore, we establish a generalized dual symmetric mixed Chernoff inequality for two planar star bodies. As a direct consequence, a new proof of the dual symmetric mixed isoperimetric inequality is presented.

This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces. Moreover, we establish the controllability of the considered system. To do so, first, we investigate the approximate controllability of the corresponding linear system. Subsequently, we prove the nonlinear system is approximately controllable if the corresponding linear system is approximately controllable. To reach the conclusions, the theory of resolvent operators, the Banach contraction mapping principle, and fixed point theorems are used. While concluding, some examples are given to demonstrate the efficacy of the proposed results.

In this paper we investigate the existence of solution for the following nonlocal problem with Stein-Weiss convolution term $\begin{equation*} -\Delta_{\Phi}u+V(x)\phi(|u|)u=\dfrac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^{\lambda}|y|^\alpha}{\rm d}y\right)K(x)f(u(x)),\;\;x\in \mathbb{R}^{N}, \end{equation*}$ where $\alpha\geq 0$, $N \geq 2$, $\lambda>0$ is a positive parameter, $V,K\in {C}(\mathbb R^N,[0,\infty))$ are nonne-gative functions that may vanish at infinity, the function $f\in C (\mathbb{R}, \mathbb R)$ is quasicritical and $F(t)=\int_{0}^{t}f(s){\rm d}s$. To establish our existence and regularity results, we use the Hardy-type inequalities for Orlicz-Sobolev Space and the Stein-Weiss inequality together with a variational technique based on the mountain pass theorem for a functional that is not necessarily in $C^1$. Furthermore, we also prove the existence of a ground state solution by the method of Nehari manifold in the case where the strict monotonicity condition on $f$ is not required. This work incorporates the case where the $\mathcal{N}$-function $\tilde{\Phi }$ does not verify the $\Delta_{2}$-condition.

In this paper, we study the elliptic system$\begin{equation*} \left\{ \begin{array}{ll} -\Delta u +V(x) u=|v|^{p-2}v-\lambda_2|v|^{s_2-2}v, \\ -\Delta v + V(x)v=|u|^{p-2}u-\lambda_1|u|^{s_1-2}u, \\ u,v \in H^1(\mathbb{R}^N) \end{array} \right.\end{equation*}$ with strongly indefinite structure and sign-changing nonlinearity. We overcome the absence of the upper semi-continuity assumption which is crucial in traditional variational methods for strongly indefinite problems. By some new tools and techniques we proved the existence of infinitely many geometrically distinct solutions if parameters $\lambda_1, \lambda_2>0$ small enough. To the best of our knowledge, our result seems to be the first result about infinitely many solutions for Hamiltonian system involving sign-changing nonlinearity.

In this paper, we study the self-similar solutions of the degenerate diffusion equation $u_t-\mathrm{div}\left({|\nabla u^m|}^{p-2} \nabla u^m\right)=0$ of polytropic filtration diffusion in $\mathbb{R}^N \times (0,\pm\infty)$ or $(\mathbb{R}^N\backslash\{0\})\times (0,\pm\infty)$ with $N\ge1$, $m>0, p>1$, such that $m(p-1)>1$. We give a clear classification of the self-similar solutions of the form $u(x,t)=(\beta t)^{-\frac{\alpha}{\beta}}w((\beta t)^{-\frac{1}{\beta}}|x|)$ with $\alpha\in\mathbb{R}$ and $\beta=\alpha \left[m\left(p-1\right)-1\right]+p$, regular or singular at the origin point. The existence and uniqueness of some solutions are established by the phase plane analysis method, and the asymptotic properties of the solutions near the origin and the infinity are also described. This paper extends the classical results of self-similar solutions for degenerate $p$-Laplace heat equation by Bidaut-Véron [Proc Royal Soc Edinburgh, 2009, 139: 1-43] to the doubly nonlinear degenerate diffusion equations.

The paper is concerned with a class of elliptic equation with critical exponent and Dipole potential. More precisely, we make use of the refined Sobolev inequality with Morrey norm to obtain the existence and decay properties of nonnegative radial ground state solutions.

This paper studies the global existence and large-time behaviors of weak solutions to the kinetic particle model coupled with the incompressible Navier-Stokes equations in $\mathbb{R}^3$. First, we obtain the global weak solution using the characteristic and energy methods. Then, under the small assumption of the mass of the particle, we show that the solutions decay at the algebraic time-decay rate. Finally, it is also proved that the above rate is optimal. It should be remarked that if the particle in the coupled system vanishes (i.e. $f=0$), our works coincide with the classical results by Schonbek [32] (J Amer Math Soc, 1991, 4: 423-449), which can be regarded as a generalization from a single fluid model to the two-phase fluid one.

We investigate the radial symmetry of minimizers on the Pohozaev-Nehari manifold to the Schrödinger Poisson equation with a general nonlinearity $f(u)$. Particularly, we allow that $f$ is $L^2$ supercritical. The main result shows that minimizers are radially symmetric modulo suitable translations.

A mathematically rigorous framework for singular limits of the magnetohydrodynamic rotating shallow water equations with ill-prepared data is developed when the Rossby and Froude numbers tend to zero at different rates. The reduced systems are derived, respectively, for the stratification-dominant and the rotation-dominant cases by means of the developed three-scale fast averaging method.

In this paper, a composite numerical scheme is proposed to solve the three-dimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions. A mixed finite element is used for the flow equation. The velocity and pressure are computed simultaneously. The accuracy of velocity is improved one order. The concentration equation is solved by using mixed finite element, multi-step difference and upwind approximation. A multi-step method is used to approximate time derivative for improving the accuracy. The upwind approximation and an expanded mixed finite element are adopted to solve the convection and diffusion, respectively. The composite method could compute the diffusion flux and its gradient. It possibly becomes an efficient tool for solving convection-dominated diffusion problems. Firstly, the conservation of mass holds. Secondly, the multi-step method has high accuracy. Thirdly, the upwind approximation could avoid numerical dispersion. Using numerical analysis of a priori estimates and special techniques of differential equations, we give an error estimates for a positive definite problem. Numerical experiments illustrate its computational efficiency and feasibility of application.

In this paper, we examine the functions $a(n)$ and $b(n)$, which respectively represent the number of cubic partitions and cubic partition pairs. Our work leads to the derivation of asymptotic formulas for both $a(n)$ and $b(n)$. Additionally, we establish the upper and lower bounds of these functions, factoring in the explicit error terms involved. Crucially, our findings reveal that $a(n)$ and $b(n)$ both satisfy several inequalities such as log-concavity, third-order Turán inequalities, and strict log-subadditivity.