We provide some characterizations of the boundedness for commutators of $n$-dimensional Hardy-type operators $H_b$ and $H^*_b$ from the central Morrey space $\dot{M}^{p,\lambda}(\mathbb{R}^n)$ to the weak central Morrey space $W\dot{M}^{p,\lambda}(\mathbb{R}^n)$, which extends the corresponding results on Lebesgue spaces.

Let $Q=\begin{pmatrix} b & 0\\ 0 & b \end{pmatrix}$ be an integer expanding matrix and let $\mathcal{D}=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\begin{pmatrix} 1 \\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \end{pmatrix},\begin{pmatrix} -1 \\ -1 \end{pmatrix} \right\}$ be a four element digit set. We considered the spectral structure of self-similar measure $\mu_{Q,\mathcal{D}}$ which generated by an integer expanding matrix $Q$ and a four element digits $\mathcal{D}$. It is well known that $\mu_{Q,\mathcal{D}}$ is a spectral measure if and only if $b=2q$ for some $q\in\mathbb{N}$. The spectrum for this spectral measure is the following set

where $\mathcal{C}_{q}=q\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 0 \end{pmatrix},\,\,\begin{pmatrix} 0 \\ 1 \end{pmatrix},\,\,\begin{pmatrix} 1 \\ 1 \end{pmatrix} \right\}$. In this paper, we investigate the structure of the maximum orthogonal set of $\mu_{Q,\mathcal{D}} $ through the maximum tree mapping and based on this, the relevant issues of its spectral eigenmatrix were discussed.

This paper mainly considers a kind of nonlinear elliptic equation with a Kirchhoff type nonlocal term

where$ a,b>0 $are constants,$ p\in(1,5) $,$ V(x) $and$ Q(x) $are$ L^\infty(\mathbb{R}^3) $functions. It is known that if we apply the mountain pass lemma directly to obtain solution (i.e., mountain pass solution) to the equation (0.1), we must require$ 3\le p<5 $because of the appearance of nonlocal terms. When$ p\in(1,3) $, the fundamental difficulty in applying the mountain pass lemma is that we are unable to verify the boundedness of the (PS) sequence. To overcome this difficulty, when$ Q(x)\equiv 1 $, paper [Acta Math Sci, 2025, 45B(2): 385-400] introduced a new technique to demonstrate the equation (0.1) exists a mountain pass solution for all$ p\in(1,5) $, and discussed the relationship between the mountain pass solution and the ground state solution obtained. The purpose of this paper is to extend the results of [Acta Math Sci, 2025, 45B(2): 385-400] to the general case$ Q(x)\not\equiv 1 $.

We establish a blow-up criterion for the Cauchy problem of compressible non-isothermal nematic liquid crystal flows with vacuum in $\mathbb{R}^3$. It is shown that the strong solution exists globally if the density is bounded from above and the velocity and the gradient of orientation field satisfy the Serrin condition. In particular, our criterion is independent of the temperature and is just the same as that of the isentropic case (Math MethodsAppl Sci, 2013, 36: 1363-1375). This paper presents some delicate analysis to exploit the structural characteristic of the system under consideration due to strong coupling and interplay interaction.

The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $L^\infty$ norm can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together with the structure of the Boussinesq equations help us to obtain the global smooth solutions.

In this paper, we consider the nonlinear parabolic equation associated with the fractional $p$-Laplace operator on the upper half space

First, the narrow region principle and the maximum principle for antisymmetric functions are proved in both bounded and unbounded domains. Then, the Hopf lemma for antisymmetric functions is established. Finally, using the method of moving planes, the monotonicity of solutions to the parabolic equation associated with the fractional $p$-Laplace operator on the upper half space is demonstrated.

The research mainly focuses on the Cauchy problem of the parabolic-parabolic Keller-Segel equation with a Logistic source. The equation is as follows

where the constants $c \in \mathbb{R}, \tau>0, d \geq 2$, initial values $u_{0} \in \mathcal{P} \mathcal{M}^{d-2}\left(\mathbb{R}^{d}\right), \varphi_{0} \in \mathcal{S}\left(\mathbb{R}^{d}\right)$. When $c=0$, Biler-Boritchev-Brandolese proved that for any initial value equation of arbitrary size, a global solution exists in the case of the diffusion parameter $\tau \gg 1$. Using the fixed point lemma and the method of scale invariance to obtain the existence and uniqueness of mild solutions for the parabolic-parabolic Keller-Segel equation with a square Logistic source, under initial conditions $u_{0}$ and $\varphi_{0}$ have certain restrictive conditions (which depend on the parameter $\tau$).

The spreading speeds of partially degenerate reaction-diffusion systems with advection term and time-space periodic coefficients in multi-dimensional space has been studied in the current paper. In the direction of $\mathbf{e}\in S^{N-1}$, we use the the spreading properties of solution with front-like initial values to show that there is a finite spreading speed interval of such time-space periodic system in any direction and the interval admits a single spreading speed under certain special conditions. In the direction of $\mathbf{\eta}$, we introduce the concept of asymptotic spreading ray speed interval, and under the compact supported initial values, we show that such time-space periodic system exists an asymptotic spreading ray speed and an asymptotic spreading set. The results show that the Freidlin-G$\ddot{\rm a}$rtner's formula can be used to describe the asymptotic spreading ray speed for such partially degenerate systems. We also apply these results to some partially degenerate models in multi-dimensional time and space periodic media including the benthic-pelagic model, a dengue transmission model and man-environment-man epidemics model.

In this article, the authors study the pullback attractor and invariant measures for retarded lattice reaction-diffusion equations. They first prove the global well-posedness of the addressed problem, and then show that the solution mappings generates a continuous process possessing a pullback attractor. Afterwards, they construct a family of invariant Borel probability measures for the process via the pullback attractor and the notion of generalized Banach limit.

This paper discusses the shadowing properties of the semilinear nonautonomous evolution equation

on a Banach space $X$, where the linear operator $A(t) : D(A(t)) \subset X \rightarrow X$ may not be bounded and $u'(t)=A(t)u(t)$ admits exponential dichotomy. This paper first establishes shadowing properties under the classical Lipschitz condition and a weaker $BS^p $ type Lipschitz condition for $f$. Then we further introduce the concepts of $L^p$ pseudo orbits and $L^p$ shadowing property, establishing corresponding shadowing theorem. Finally, an example of a parabolic partial differential equation is provided as an application of the abstract results. Compared to existing literature, this paper not only weakens the Lipschitz condition for the nonlinear term and introduces and discusses the new $L^p$ shadowing property, but most importantly, it allows $A(t)$ to be an unbounded operator, thereby enabling the abstract results to be applied to partial differential equations.

This paper investigates the boundedness of a Toeplitz operator $Q_\mu$ acting on the Hardy space $H^1(\mathbb{B}_{n})$. Let $\mu$ be a positive Borel measure on $\mathbb{B}_{n}$ and $0. The main results are following

1. If $\mu$ is a $(1,1)$-logarithmic Carleson measure, then $Q_\mu: H^1(\mathbb{B}_{n})\to H^1(\mathbb{B}_{n})$ is bounded;

2. If $Q_\mu: H^1(\mathbb{B}_{n})\to H^1(\mathbb{B}_{n})$ is bounded, then $\mu$ is a Carleson measure;

3. $Q_\mu: H^p(\mathbb{B}_{n})\to H^q(\mathbb{B}_{n})$ is bounded if and only if $\mu$ is a $(1+\frac1p-\frac1q)$-Carleson measure.

The paper discusses the Neumann problem for 2-species reaction-diffusion system of spatially inhomogeneous models. When the reaction function of the species is non-monotonic with respect to its population density, firstly the existence of a unique positive equilibrium solution for both species is established by virtue of the sub-super solution technique. It is also proven that the positive equilibrium solution is globally asymptotically stable when the diffusion coefficients are sufficiently small. Finally, the correctness of the conclusion is verified through numerical solutions of a specific example.

In this paper, the existence, uniqueness and $ \lambda $ stability of the global solutions to the pantogragh stochastic functional differential equations with Markovian switching are investigated. By applying the vector Lyapunov function method, we obtain some theorems ensuring the existence, uniqueness, moment $ \lambda $ stability and almost sure $ \lambda $ stability of the global solutions to the equations. Furthermore, impulsive effects are introduced to the systems.According to the different characteristics of the impulses and different decay rates of the $ \lambda $ functions we prove that similar results still hold under suitable conditions.Our results generalize some existing results.Finally, an example and numerical simulation is given to illustrate the effectiveness of the results.

This paper considers a class of single population dynamics models for fish that incorporates seasonal switching and impulsive perturbations. It investigates the effects of these factors on fish population dynamics. By employing the discrete dynamical system theory and stroboscopic mapping, we derive the sufficient conditions for the permanence and extinction of the fish population system. Using rational difference equations and fixed point theory, we demonstrate the existence of a unique globally attractive positive periodic solution. Finally, numerical simulations are provided to validate the theoretical results.

By using a class of nonlinear scalarization functions, we construct the bounded rationality functions of Nash equilibrium and cooperation equilibrium of population games with set payoffs, respectively. Then, we show the lower semicontinuity of the bounded rationality functions and the equivalence relations between their level sets of zero and game equilibrium sets. Finally, employing the acquired properties of the bounded rationality functions, we prove the generalized well-posedness of Nash equilibrium and cooperative equilibrium for set payoffs population games.

Noise reduction in complex environments is of paramount importance for the accurate extraction and analysis of signals. Prevailing research predominantly centers on the application of single or combined methods in specific scenarios, but encounters challenges in effectively addressing the nonlinearity and non-stationarity of complex signals. Based on the measurement of non-stationarity, this study proposes a two-stage noise reduction strategy using CEEMDAN. The NS index is used to quantify the non-stationarity of modal components, achieving precise separation of high-frequency noise from low-frequency signals. A novel logarithmic threshold function is adopted to remove high-frequency noise, and the SG filtering method is combined to smooth low-frequency signals, significantly improving the noise reduction effect and signal reconstruction accuracy. The results indicate that the new method demonstrates outstanding modal discrimination and noise reduction performance under different signal-to-noise ratios and noise types.

In this paper, we first study the Harris recurrence of the continuous time Markov process on general state space, then we study some problems about the decomposition of Harris, and finally we study the method to determine the Harris recurrence.

In this paper, we study an evenly convex optimization problem with the objective function and constraint functions being proper evenly convex. By using the concept of c-subdifferential, we introduce some new notions of constraint qualifications. Under those new constraint qualifications, we provide necessary and sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for the evenly convex optimization problem to have total Lagrangian dualities and stable total Lagrangian dualities.

As is well known, the study of nonuniform grids can effectively solve the initial value singularity phenomenon of fractional Caputo -type derivatives. In the theoretical analysis of nonuniform grids, fractional discrete Grönwall inequality is often used for error analysis, but there is a lack of specific research on error structures. An error convolution structure (ECS) was designed on nonuniform grids for analyzing the time fractional diffusion wave equation. A quadratic interpolation approximation was applied Caputo -type derivatives, and the BDF2 -type finite element method on nonuniform grids was obtained by discretizing it using a reduction method and a discrete complementary convolution kernel. The discrete complementary convolution kernel is crucial in the convergence analysis of algorithms, as it simplify the process of finite element theory analysis and construct global consistency errors based on the properties of convolution kernels and interpolation estimates. The $L^2$-norm error and $H^1$-norm error of the BDF2 finite element scheme on nonuniform grids were estimated in detail, and verifies the consistency between the proposed finite element scheme and the theoretical convergence order through experiments.

In order to solve the PageRank problem, a new algorithm Hessenberg-Chebyshev is proposed by combining Hessenberg-type algorithm with Chebyshev acceleration technology. This algorithm improves the Hessenberg-type algorithm based on the acceleration technique of Chebyshev polynomial. The implementation and convergence analysis of the new algorithm are discussed in detail, and the effectiveness of the algorithm is verified by numerical experiments.

Based on Gauss-Legendre integral or Newton-Cotes integral method, the integral-Newton type iteration method and the improved integral-Newton type iteration method for solving the generalized absolute value equation are presented. The convergence conditions of the two methods are proved theoretically. Numerical experiments show that the proposed methods are feasible and effective.

In this paper, the Levenberg-Marquardt type method is proposed for solving the generalized complementarity problems. Firstly, by integrating a class of complementary functions, the generalized complementarity problem is equivalently reformulated as a system of nonlinear equations. An adaptive modified Levenberg-Marquardt algorithm with line search is then introduced to address this reformulated problem. Furthermore, the convergence of the proposed algorithm is analyzed under appropriate conditions. Finally, numerical experiments are conducted to verify the feasibility and effectiveness of the proposed algorithm.

This paper studies the optimal reinsurance-investment problem of an insurer and a reinsurer under Heston's stochastic volatility model in the framework of Stackelberg stochastic differential game. We assume that the insurer purchases proportional reinsurance to transfer part of the risk to the reinsurer, and the reinsurer accepts the transferred risk and chooses an appropriate reinsurance premium strategy. At the same time, the insurer and the reinsurer can also invest their surplus in risk-free assets and Heston's stochastic volatility risk model. Under the criterion of maximizing the expected mean-variance utility of terminal wealth, the corresponding extended Hamilton-Jacobi-Bellman equations are developed using stochastic control theory, and verification theorems are given. We obtain specific forms for the equilibrium strategy and the value function. Finally, we analyze the effects of the model parameters on the equilibrium strategies through numerical simulations.

This paper considers the optimal investment and optimal contribution-benefit adjustment strategies of a continuous time collective hybrid pension plan with model uncertainty and default risk. We assume that pension funds are invested in a risk-free asset, a defaultable bond and a stock satisfied 4/2 random volatility model. The objective is to maximize the discount value of surplus and contribution-benefit adjustment amount or minimize the discount value when surplus and contribution-benefit adjustment amount are negative based on the CARA utility function. Firstly, applying stochastic optimal control approach, we establish the Hamilton-Jacobi-Bellman equations for both the pre-default case and the post-default case, respectively. Then, we derive the closed-from solutions for robust optimal strategies and corresponding value functions. Finally, numerical analyses illustrate the influence of model parameters and financial market parameters on optimal control problems.