In this paper, we apply the three circle type method and a Hardy type inequality to a nonlinear Dirac type equation on surfaces, and provide alternative proofs to the energy quantization results.

We deal with a large solution to the semilinear Poisson equation with double-power nonlinearity $\Delta u = u^p + \alpha u^q$ in a bounded smooth domain $D \subset \mathbb{R}^n$, where $p>1$, $-1<q<p$ and $\alpha \in \mathbb{R}$. We obtain the asymptotic behavior of a solution $u$ near the boundary $\partial D$ up to the third or higher term.

In this paper, I consider the Hölder continuity of the Lyapunov exponent for a quasi-periodic Szegö cocycle with weak Liouville frequency. I extend the existing results about the regularity of the Lyapunov exponent from the Schrödinger cocycle in [24] to a Szegö cocycle.

Conservation law plays a very important role in many geometric variational problems and related elliptic systems. In this note, we refine the conservation law obtained by Lamm-Rivière for fourth order systems and de Longueville-Gastel for general even order systems.

The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk. We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions. Then, using these results, we establish five Landau-type theorems for subclasses of polyharmonic mappings $F$ and $L(F)$, where $F$ has bounded length distortion and $L$ is a differential operator.

In recent years, the research focus in insurance risk theory has shifted towards multi-type mixed dividend strategies. However, the practical factors and constraints in financial market transactions, such as interest rates, tax rates, and transaction fees, inevitably impact these strategies. By incorporating appropriate constraints, a multi-type mixed strategy can better simulate real-world transactions. Following the approach of Liu et al. [28], we examine a classical compound Poisson risk model that incorporates the constraints of constant interest rates and a periodic-threshold mixed dividend strategy. In this model, the surplus process of insurance companies is influenced by several factors. These factors include constant interest rates, continuously distributed dividends within intervals (threshold dividend strategy), and dividends at discrete time points (periodic dividend strategy). We derive the piecewise integro-differential equations (IDEs) that describe the expected present value of dividends (EPVDs) until ruin time and the Gerber-Shiu expected discounted penalty function. Furthermore, we provide explicit solutions to these IDEs using an alternative method based on the inverse Laplace transform combined with the Dickson-Hipp operator. This enables us to obtain explicit expressions for the dividend and Gerber-Shiu functions. Additionally, we present examples to illustrate the application of our results.

This paper concentrates on the dynamics of a waterborne pathogen periodic PDE model with environmental pollution. For this model, we derive the basic reproduction number $\mathcal{R}_{0}$ and establish a threshold type result on its global dynamics in terms of $\mathcal{R}_{0}$, which predicts the extinction or persistence of diseases. More precisely, the disease-free steady state is globally attractive if $\mathcal{R}_{0}<1$, while the system admits at least one positive periodic solution and the disease is uniformly persistent if $\mathcal{R}_{0}>1$. Moreover, we carry out some numerical simulations to illustrate the long-term behaviors of solutions and explore the influence of environmental pollution and seasonality on the spread of waterborne diseases.

Economic development has caused a lot of environmental problems, in turn, environmental pollution restricts economic development. Considering the influence of wind direction and speed, temperature and humidity on pollutants, as well as the influence of epidemic, war and exchange rate on economic development. In this paper, we develop a stochastic economic-environment model with pollution control strategies. Furthermore, sufficient and necessary conditions for the near-optimality are established. Finally, we perform some numerical simulations to demonstrate the correctness of the theoretical results, which shows that some control strategies could decrease the environmental pollution, and therefore, could alleviate economic losses caused by environmental pollution.

We consider the persistence of affine periodic solutions for perturbed affine periodic systems. Such $(Q,T)$-affine periodic solutions have the form $x(t+T)=Qx(t)$ for all $t\in\mathbf{R}$, where $T>0$ is fixed and $Q$ is a nonsingular matrix. These are a kind of spatiotemporal symmetric solutions, e.g. spiral waves. We give the averaging method for the existence of affine periodic solutions in two situations: one in which the initial values of the affine periodic solutions of the unperturbed system form a manifold, and another that does not rely on the structure of the initial values of the unperturbed system's affine periodic solutions. The transversal condition is determined using the Brouwer degree. We also present a higher order averaging method for general degenerate systems by means of the Brouwer degree and a Lyapunov-Schmidt reduction.

In this paper, we study multiplication operators on weighted Dirichlet spaces $\mathcal{D}_{\beta}$ $(\beta\in \mathbb{R})$. Let $n$ be a positive integer and $\beta\in \mathbb{R}$, we show that the multiplication operator $M_{z^{n}}$ on $\mathcal{D}_{\beta}$ is similar to the operator $\oplus_{1}^{n}M_{z}$ on the space $\oplus_{1}^{n}\mathcal{D}_{\beta}$. Moreover, we prove that $M_{z^{n}}$ $(n\geq 2)$ on $\mathcal{D}_{\beta}$ is unitarily equivalent to $\oplus_{1}^{n}M_{z}$ on $\oplus_{1}^{n}\mathcal{D}_{\beta}$ if and only if $\beta=0$. In addition, we completely characterize the unitary equivalence of the restrictions of $M_{z^{n}}$ to different invariant subspaces $z^{k}\mathcal{D}_{\beta}$ $(k\geq 1)$, and the unitary equivalence of the restrictions of $M_{z^{n}}$ to different invariant subspaces $S_{j}$ $(0\leq j<n)$.
Abkar, Cao and Zhu [Complex Anal Oper Theory, 2020, 14: Art 58] pointed out that it is an important, natural, and difficult question in operator theory to identify the commutant of a bounded linear operator. They characterized the commutant $\mathcal{A}'(M_{z^{n}})$ of $M_{z^{n}}$ on a family of analytic function spaces $A_{\alpha}^{2}$ $(\alpha\in \mathbb{R})$ on $\mathbb{D}$ (in fact, the family of spaces $A_{\alpha}^{2}$ $(\alpha\in \mathbb{R})$ is the same with the family of spaces $\mathcal{D}_{\beta}$ $(\beta\in \mathbb{R})$) in terms of the multiplier algebra of the underlying function spaces. In this paper, we give a new characterization of the commutant $\mathcal{A}'(M_{z^{n}})$ of $M_{z^{n}}$ on $\mathcal{D}_{\beta}$, and characterize the self-adjoint operators and unitary operators in $\mathcal{A}'(M_{z^{n}})$. We find that the class of self-adjoint operators (unitary operators) in $\mathcal{A}'(M_{z^{n}})$ when $\beta \neq 0$ is different from the class of self-adjoint operators (unitary operators) in $\mathcal{A}'(M_{z^{n}})$ when $\beta =0$.

This paper studies the strong convergence of the quantum lattice Boltzmann (QLB) scheme for the nonlinear Dirac equations for Gross-Neveu model in $1+1$ dimensions. The initial data for the scheme are assumed to be convergent in $L^2$. Then for any $T\ge 0$ the corresponding solutions for the quantum lattice Boltzmann scheme are shown to be convergent in $C([0,T];L^2(\R^1))$ to the strong solution to the nonlinear Dirac equations as the mesh sizes converge to zero. In the proof, at first a Glimm type functional is introduced to establish the stability estimates for the difference between two solutions for the corresponding quantum lattice Boltzmann scheme, which leads to the compactness of the set of the solutions for the quantum lattice Boltzmann scheme. Finally the limit of any convergent subsequence of the solutions for the quantum lattice Boltzmann scheme is shown to coincide with the strong solution to a Cauchy problem for the nonlinear Dirac equations.

In this short paper, we remove the restriction $\gamma \in (1,3]$ that was used in the paper "The rate of convergence of the viscosity method for a nonlinear hyperbolic system" (Nonlinear Analysis, 1999, 38: 435-445) and obtain a global Hölder continuous solution and the convergent rate of the viscosity method for the Cauchy problem of the variant non-isentropic system of polytropic gas for any adiabatic exponent $\gamma>1$.

The $Q_K(p)$-Teichmüllerspace is introduced and studied in this paper. Various characterizations of the $Q_K(p)$-Teichmüllerspace and the $Q_{K,0}(p)$-Teichmüllerspace are given. Their Schwarzian derivative model and pre-logarithmic derivative model are also discussed.

In this paper, we prove some Liouville-type theorems for the stationary magneto-micropolar fluids under suitable conditions in three space dimensions. We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials. And then we show similar results assuming that the solutions are contained in $L^p(\R^3)$ with $p\in[2,9/2)$. Finally, we show the same result for lower values of $p\in[1,9/4)$ with the further assumption that the solutions vanish at infinity.

In this paper, we propose a second-order moving-water equilibria preserving nonstaggered central scheme to solve the Ripa model via flux globalization. To maintain the moving-water steady states, we use the discrete source terms proposed by Britton et al. (J Sci Comput, 2020, 82(2): Art 30) by incorporating the expression of the source terms as a whole into the flux gradient, which directly avoids the discrete complexity of the source terms in order to maintain the well-balanced properties of the scheme. In addition, since the nonstaggered central scheme requires re-projecting the updated values of the nonstaggered cells onto the staggered cells, we modify the calculation of the global variables by constructing ghost cells and alternating the values of the global variables with the water depths obtained from the solution through the nonlinear relationship between the global flux and the water depth. In order to maintain the second-order accuracy of the scheme on the time scale, we incorporate a new Runge-Kutta type time discretization in the evolution of the numerical solution for the nonstaggered cells. Meanwhile, we introduce the "draining" time step technique to ensure that the water depth is positive and that it satisfies mass conservation. Numerical experiments verify that the scheme is well-balanced, positivity-preserving and robust.

In this paper, we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in $\mathbb{R}^{m}$, which is the case where the generalized Gauss map $\Phi$ is ramified over a family of hypersurfaces $\{Q_{j}\}_{j=1}^{q}$ in $\mathbb{P}^{m-1}(\mathbb{C})$ located in the $N$-subgeneral position. In addition, we investigate the Gauss curvature estimate for the $K$-quasiconformal harmonic surfaces immersed in $\mathbb{R}^{3}$ whose Gauss maps are ramified over a family of hypersurfaces located in the $N$-subgeneral position.

In this manuscript, we consider two kinds of the Fokker-Planck-type systems in the whole space. The first part involves proving the global existence and the algebraic time decay rates of the mild solutions to the Fokker-Planck-Boltzmann equation near Maxwellians if initial data satisfies some smallness in the function space $L^1_kL^\infty_TL^2_v\cap L^p_kL^\infty_TL^2_v$. The second part proves the global existence of the mild solutions to the Vlasov-Poisson-Fokker-Planck system in the function space $L^1_kL^\infty_TL^2_v$, and we also obtain the exponential time decay rates, which are different from the algebraic time decay rates of the Fokker-Planck-Boltzmann equation. Our analysis is based on $L^1_kL^\infty_TL^2_v$ function space introduced by Duan $et~ al$. (Comm Pure Appl Math, 2021, 74: 932-1020), the $L^1_k\cap L^p_k$ approach developed by Duan $et~ al$. (SIAM J Math Anal, 2024, 56: 762-800), and the coercivity property of the Fokker-Planck operator. However, it is worth pointing out that the $L^1_k\cap L^p_k$ approach is not required for the Vlasov-Poisson-Fokker-Planck system, due to the influence of the electric field term, which is different from the Fokker-Planck-Boltzmann equation in this paper and in the work of Duan $et~ al$. (SIAM J Math Anal, 2024, 56: 762-800).

In this paper, we prove the existence of the scattering operator for the fractional magnetic Schrödinger operators. In order to do this, we construct the fractional distorted Fourier transforms with magnetic potentials. Applying the properties of the distorted Fourier transforms, the existence and the asymptotic completeness of the wave operators are obtained. Furthermore, we prove the absence of positive eigenvalues for fractional magnetic Schrödinger operators.

A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations. Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and co-state variables in two space dimensions. A Crank-Nicolson difference scheme is constructed for time discretization. The resulting numerical solutions belong to $C^2$ in space, and the order of the coefficient matrix is low. Moreover, the Bogner-Fox-Schmit element is considered for comparison. Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.

In this paper, we consider the nonlinear equations involving the fractional $p\&q$-Laplace operator with a sign-changing potential. This model is inspired by the De Giorgi Conjecture. There are two main results in this paper. First, in the bounded domain, we use the moving plane method to show that the solution is radially symmetric. Second, for the unbounded domain, in view of the idea of the sliding method, we find the existence of the maximizing sequence of the bounded solution, then obtain that the solution is strictly monotone increasing in some direction.

For given $\ell,s\in \mathbb{N},$ $\Lambda=\{\rho_j\}_{j=1,\cdots,s},\rho_j\in\mathbb{T}$, the $C^*$-algebra $\mathcal{B}:=\mathcal{E}(\{r_j\}_{j=1,\cdots,s},\Lambda,\\ \ell)$ is defined to be the universal $C^*$-algebra generated by $\ell$ unitaries $\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell}$ subject to the relations $r_{j}(\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell})-\rho_j=0$ for all $j=1,\cdots,s,$ where the $r_j$ is monomial in $\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell}$ and their inverses for $j=1,2,\cdots,s$. If $\mathcal{B}$ is a unital $AF$-algebra with a unique tracial state, and $K_0(\mathcal{B})$ is a finitely generated group, we say that the relations $(\{r_j\}_{j=1,\cdots,s},\Lambda,\ell)$ are $AF$-relations. If the relations $(\{r_j\}_{j=1,\cdots,s},\Lambda,\ell)$ are $AF$-relations, we prove that, for any $\varepsilon>0,$ there exists a $\delta>0$ satisfying the following: for any unital $C^*$-algebra $\mathcal{A}$ with the cancellation property, strict comparison, nonempty tracial state space, and any $\ell$ unitaries $u_1,u_2,\cdots,u_\ell\in\mathcal{A}$ satisfying $$\|r_j(u_1,u_2,\cdots,u_\ell)-\rho_j\|<\delta,\,\,j=1,2,\cdots,s,$$ and certain trace conditions, there exist $\ell$ unitaries $\tilde{u}_1,\tilde{u}_2,\cdots,\tilde{u}_{\ell}\in\mathcal{A}$ such that $$r_j(\tilde{u}_1,\tilde{u}_2,\cdots,\tilde{u}_\ell)=\rho_j\,\,{\rm for}\,\,j=1,2,\cdots,s, \,\,{\rm and}\,\,\|u_i-\tilde{u}_i\|<\varepsilon\,\,{\rm for}\,\,i=1,2,\cdots,\ell.$$ Finally, we give several applications of the above result.

The present article is devoted to nonlinear stochastic partial differential equations with double reflecting walls driven by possibly degenerate, multiplicative noise. We prove that the corresponding Markov semigroup possesses an exponentially attracting invariant measure through asymptotic coupling, in which Foias-Prodi estimation and the truncation technique are crucial for the realization of the Girsanov transform.

In this article, we introduce and study the class of approximately Artinian (Noetherian) C*-algebras, called AR-algebras (AN-algebras), which is a simultaneous generalization of Artinian (Noetherian) C*-algebras and AF-algebras. We study properties such as the ideal property and topological dimension zero for them. In particular, we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite. This extends the Kirchberg's $\mathcal{O}_\infty$-absorption theorem, and implies that a weakly purely infinite C*-algebra is Noetherian iff every its ideal has a full projection.

This paper aims to construct six-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation. Their Hamiltonian formulations are furnished by the trace identity, which guarantee the commuting property of infinitely many symmetries and conserved Hamiltonian functionals. Illustrative examples of the resulting integrable equations of second and third orders are explicitly computed.

In this paper, we prove a transversal $V$-Laplacian comparison theorem under a transversal Bakry-Emery Ricci condition. We establish a Schwarz type lemma for transversally $V$-harmonic maps of bounded generalized transversal dilatation between Riemannian foliated manifolds by using this comparison theorem, including for the case of $V = \nabla^{\mathcal{H}} h$.