In this paper, we investigate the well-posedness result of the three-dimensional incompressible hyper-dissipative Hall-Magnetohydrodynamic equations with small anisotropic derivative. Making using of anisotropic Littlewood-Paley theory, we conclude that the hyper-dissipative Hall-MHD system has a unique global solution provided that $$\begin{align*} \left(\|J_{0}\|_{\mathcal{B}^{1-2\alpha}_{2}}+\|(\Lambda_{h}^{-1}\partial_{3}u_{0}, B_{0}^{h})\|_{\mathcal{B}^{1-2\alpha}_{2}}\right) \cdot F(u_{0}, B_{0}) \end{align*}$$ is sufficiently small. Here, $F(u_{0}, B_{0})$ is a bounded function, which depends on $\|(u_{0}, B_{0})\|_{\mathcal{B}^{1-2\alpha}_{2}}$ and $\|u_{0}^{h}\|_{H^{1}}$.

In this paper, we study the nonlinear stability problem for the two-dimensional Boussinesq system around the Couette flow in a finite channel with Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature with small viscosity $\nu$ and small thermal diffusion $\mu$. We establish that if the initial perturbation velocity and initial perturbation temperature satisfy $$\|u_{0}\|_{H^2}\leq \epsilon_0\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{1}{2}},$$ and $$\quad \|\theta_{0}\|_{H^1}+\| |D_x|^{\frac{1}{3}}\theta_{0}\|_{H^1}\leq \epsilon_1\min\left\lbrace \mu,\nu\right\rbrace ^{\frac{5}{6}},$$ for some small $\epsilon_0$ and $\epsilon_1$ independent of $\mu,\nu$, then the solution of the two-dimensional Navier-Stokes Boussinesq system does not transition away from the Couette flow for any time.

It is well-known that the propagation phenomena of nonlocal dispersal equations have been extensively studied, and the known results on the interface dynamics of this equation are under the compactly supported initial value. Moreover, there was no explicit formula regarding the interface due to the peculiarity of nonlocal dispersal operators. A natural question is whether it is possible to provide a precise characterization of the interface with respect to small parameter for the general initial values (including exponentially bounded and unbounded). This paper is concerned with the interface dynamics of the nonlocal dispersal equation with scaling parameter. For the exponentially bounded initial value, by choosing the hyperbolic scaling, we show that at a very small time, the interface is confined within a generated layer whose thickness is at most ${O}(\sqrt{\varepsilon}\vert\ln \varepsilon\vert)$, and subsequently, the interface propagates at a linear speed determined by the decay rate of initial value. For a class of exponentially unbounded initial value,

This paper is devoted to demonstrating several multiplicity results of nontrivial weak solutions to double phase problems of Kirchhoff type with Hardy potentials. The main features of the paper are the appearance of non-local Kirchhoff coefficients and the Hardy potential, the absence of the compactness condition of Palais-Smale, and the $L^{\infty}$-bound for any possible weak solution. To establish multiplicity results, we utilize the fountain theorem and the dual fountain theorem as main tools. Also, we give the $L^{\infty}$-bound for any possible weak solution by exploiting the De Giorgi iteration method and a truncated energy technique. As an application, we give the existence of a sequence of infinitely many weak solutions converging to zero in $L^{\infty}$-norm. To derive this result, we employ the modified functional method and the dual fountain theorem.

This paper deals with numerical computation and analysis for the initial boundary problems of two dimensional (2D) Sobolev equations with piecewise continuous argument. Firstly, a two-level high-order compact difference method (HOCDM) with computational accuracy ${\cal O}(\tau^2\!+\!h_x^4\!+\!h_y^4)$ is suggested, where $\tau, h_x, h_y$ denote the temporal and spatial stepsizes of the method, respectively. In order to improve the temporal computational accuracy of this method, the Richardson extrapolation technique is used and thus a new two-level HOCDM is derived, which is proved to be convergent of order four both in time and space. Although the new two-level HOCDM has the higher computational accuracy in time than the previous one, it will bring a larger computational cost. To overcome this deficiency, a three-level HOCDM with computational accuracy ${\cal O}(\tau^4+h_x^4+h_y^4)$ is constructed. Finally, with a series of numerical experiments, the theoretical accuracy and computational efficiency of the above methods are further verified.

In this paper, we study the exponential non-uniform Berry-Esseen bound for the maximum likelihood estimator of some time inhomogeneous diffusion process. As applications, the optimal uniform Berry-Esseen bound and optimal Cramér-type moderate deviations of the Ornstein-Uhlenbeck process and $\alpha$-Brownian bridge can be obtained. The main methods are the change of measure method and asymptotic analysis technique.

In this paper, we study the properties of the scaled packing topological pressures for a topological dynamical system $(X,G)$, where $G$ is a countable discrete infinite amenable group. We show that the scaled packing topological pressures can be determined by the scaled Bowen topological pressures. We obtain Billingsley's Theorem for the scaled packing pressures with a $G$-action. Then we get a variational principle between the scaled packing pressures and the scaled measure-theoretic upper local pressures. Finally, we give some restrictions on the scaled sequence $\mathbf{b}$, then in the case of the set $X_{\mu}$ of generic points, we prove that $$P^{P}(X_{\mu},\left\{F_{n}\right\},f,\mathbf{b})=h_{\mu}(X)+\int_{X} f \mathrm{d}\mu,$$ if $\left\{F_{n}\right\}$ is tempered and $\mu$ is a $G$-invariant ergodic Borel probability measure.

In this paper, we study the weighted higher order semilinear equation in an exterior domain $$\begin{equation*} (-\Delta)^{m} u=|x|^{\alpha}g(u) \quad \quad \text{in} \ \mathbb{R}^{N}\setminus B_{R_{0}}, \end{equation*}$$ where $N\geq1$, $m\geq2$ are integers, $\alpha>-2m$, $g$ is a continuous and nondecreasing function in $\left[ 0,+\infty\right) $ and positive in $\left( 0,+\infty\right) $, $ B_{R_{0}}$ is the ball of the radius $R_{0}$ centered at the origin. We prove that a positive supersolution of the problem which verifies $ (-\Delta )^{i}u > 0 $ in $\ \mathbb{R}^{N}\setminus B_{R_{0}}$ $(i=0,\cdots, m-1)$ exists if and only if $N>2m$ and $$\begin{equation*} \int_{0}^{\delta}\frac{g(t)}{t^{\frac{2(N-m)+\alpha}{N-2m}}}{\rm d}t<\infty, \end{equation*}$$ for some $\delta>0$. We further obtain some existence and nonexistence results for the positive solution to the Dirichlet problem when $g(u)=u^p$ with $p>1 $, by using the Pohozaev identity and an embedding lemma of radial Sobolev spaces.

In this paper, we investigate a Dirichlet boundary value problem for a class of fractional degenerate elliptic equations on homogeneous Carnot groups $\mathbb{G}=(\mathbb{R}^n ,\circ)$, namely $$\begin{equation*} \left\{ \begin{array}{cc} (-\triangle_{\mathbb{G}})^s u=f(x,u)+g(x,u) & \mbox{in} \Omega; \\[2mm] u\in {\cal{H}}_0^s(\Omega), \end{array} \right. \end{equation*}$$ where $s\in(0,1)$, $\Omega\subset\mathbb{G}$ is a bounded open domain, $(-\Delta_{\mathbb{G}} )^s$ is the fractional sub-Laplacian, ${\cal{H}}_0^s (\Omega)$ denotes the fractional Sobolev space, $f(x,u)\in C(\overline{\Omega}\times\mathbb{R}), g(x,u)$ is a Carathéodory function on $\Omega\times\mathbb{R}$. Using perturbation methods and Morse index estimates in conjunction with fractional Dirichlet eigenvalue estimates, we establish the existence of multiple solutions to the problem.

In this paper, we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations $$\begin{equation*} \begin{cases} (-\Delta)^{s} u+\lambda u=|u|^{p-2}u-|u|^{q-2}u,\ \ x\in \mathbb{R}^{N},\\ \displaystyle \int_{\mathbb{R}^{N}}|u|^{2}{\rm d}x=c>0,\\ \end{cases}\tag{$P$} \end{equation*}$$ where $N\geq 2$, $s\in (0,1)$, $2+\frac{4s}{N}<p<q\leq 2_{s}^{*}=\frac{2N}{N-2s}$, $(-\Delta)^{s}$ represents the fractional Laplacian operator of order $s$, and the frequency $\lambda\in \mathbb{R}$ is unknown and appears as a Lagrange multiplier. Specifically, we show that there exists a $\hat{c}>0$ such that if $c>\hat{c}$, then the problem ($P$) has at least two normalized solutions, including a normalized ground state solution and a mountain pass type solution. We mainly extend the results in [Commun Pure Appl Anal, 2022, 21: 4113-4145], which dealt with the problem ($P$) for the case $2<p<q<2+\frac{4s}{N}$.

We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential. By exploring some delicate energy estimates and studying decay properties of solution sequences, we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when $a\nearrow a^{\ast}:=\|Q\|_{2}^{2s}$, where $Q$ is the unique positive radial solution of $(-\Delta)^{s}u+su-|u|^{2s}u=0$ in $\mathbb{R}^{2}$. Based on the discussion of the concentration phenomenon, we prove the local uniqueness of minimizers by establishing a local Pohožaev identity and studying the blow-up estimates to the nonlocal operator $(-\Delta)^{s}$.

In this paper, we investigate the generalized quasilinear Schrödinger equation: $$ -\operatorname{div}\left(g^2(u) \nabla u\right)+g(u) g^{\prime}(u)|\nabla u|^2 +u=P(\varepsilon x) |u|^{\\\alpha p-2}u, \quad x \in \mathbb{R}^N, $$ where $N>3$, $g\!\!:\mathbb{R} \rightarrow \mathbb{R}^{+}$ is a $C^1$ even function, $g(0)=1$, $g^{\prime}(s) \geq 0$ for all $s \geq 0$, $g(s)=\beta|s|^{\alpha-1}+O\left(|s|^{\gamma-1}\right)$ as $s \rightarrow \infty$ for some constants $\alpha \in[1,2]$, $\beta>0$, $\gamma<\alpha$ and $(\alpha-1) g(s) \geq g^{\prime}(s) s$ for all $s \geq 0$, $\varepsilon>0$ is a positive parameter, and $p \in\left(2,2^*\right)$. We will study the impact of the nonlinearity's coefficient $P(x)$ on the quantity of positive solutions.

This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schrödinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing and defocusing nonlinearities, with exponents of the nonlinearity $\sigma\in[0,2)$ and $\sigma\in[0,\infty)$, and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method. In the focusing case, our result generalizes the earlier results in [12], where $\sigma=1$.

We consider nonlinear parabolic problems in a variational framework. The leading part is a monotone operator whose growth is controlled by time- and space-dependent Musielak functions. On Musielak's controlling functions we impose regularity conditions which make it possible to extend certain classical results such as the density of smooth functions, a Poincaré-type inequality, an integration-by-parts formula and a trace result. Bringing together these results, we adapt the classical theory of monotone operators and prove the well-posedness of the variational problem.

Dengue is a mosquito-borne disease that is rampant worldwide, with up to 70\% of cases reported to be asymptomatic during epidemics. In this paper, a reaction-diffusion dengue model with asymptomatic carrier transmission is investigated. We aim to study the existence, nonexistence and minimum wave speed of traveling wave solutions to the model. The results show that the existence and nonexistence of traveling wave solutions are fully determined by the threshold values, which are, the basic reproduction number $R_0$ and critical wave speed $c^*>0$. Specifically, when $R_0>1$ and the wave speed $c\ge c^*$, the existence of the traveling wave solution is obtained by using Schauder's fixed point theorem and Lyapunov functional. It is proven that the model has no nontrivial traveling wave solutions for $R_0\le1$ or $R_0>1$ and $0<c<c^*$ by employing comparison principle and limit theory. As a consequence, we conclude that the critical wave speed $c^*$ is the minimum wave speed of the model. Finally, numerical simulations are carried out to illustrate the effects of several important parameters on the minimum wave speed.

In this paper, we investigate the minimization problem $$\begin{equation*} e_{s}(\rho)=\inf_{u\in W^{1,N}_{V}(\mathbb{R}^{N}),\|u\|^{N}_{N}=\rho>0}E(u), \end{equation*}$$ where $$E(u)=\frac{1}{N}\int_{\mathbb{R}^{N}}|\nabla u|^{N}{\rm d}x+\frac{1}{N}\int_{\mathbb{R}^{N}}V(x)|u|^{N}{\rm d}x-\frac{1}{s}\int_{\mathbb{R}^{N}}|u|^{s}{\rm d}x.$$ Here $s>N$, $V$ is a spherically symmetric increasing function satisfying $$V(0)=0, \lim_{|x|\rightarrow\infty}V(x)=+\infty.$$ We discuss the problem in three cases. First, for the case $s>2N$, $e_{s}(\rho)=-\infty$ for any $\rho>0$. Secondly, for the case $N<s<2N$, for any $\rho>0$, we prove that it admits a minimizer which is nonnegative, spherically symmetric and decreasing via the $N$-Laplacian Gagliardo-Nirenberg inequality. When $s=2N$, the existence and nonexistence of minimizers of $ e_{s}(\rho)$ will also be given. During the arguments, we provide the detailed proof of the $N$-Laplacian Gagliardo-Nirenberg inequality and $N$-Laplacian Pohozaev identity.

In this paper, we investigate the Sobolev spaces $W^{1,p}(V)$ and $W_{0}^{1,p}(V)$ on a locally finite graph $G=(V,E)$, which are fundamental tools when we apply the variational methods to partial differential equations on graphs. As a key contribution of this note, we show that in general, $W^{1,p}(V)\neq W_0^{1,p}(V)$ on locally finite graphs, which is different from the situation on Euclidean space $\mathbb{R}^N$.

In this paper, we study the Neumann boundary value problem of the Yang-Mills $\alpha$-flow over a 4-dimensional compact Riemannian manifold with boundary. We establish the short-time existence of the Yang-Mills $\alpha$-flow in the framework of functional analysis and derive long-time existence and convergence results of classical solutions to the Yang-Mills $\alpha$-flow, provided that the $\alpha$-energy of initial connection is below some threshold. We also prove the validity of the boundary version of small energy estimates, removal of isolated singularities, and energy lower bound result for non-flat Yang-Mills connections. These results lead to the bubbling convergence of a sequence of Yang-Mills $\alpha$-connections, and as an application, we demonstrate the existence of non-trivial Yang-Mills connections with Neumann boundary.

For $1<p<\infty$, Coifman-Rochberg-Weiss established $L^{p}$ boundedness of commutators of smooth kernels. Later, many works tried to weaken the smooth condition. In this paper, we extend these mentioned results to the case of non-homogeneous but with strong Hörmander condition. Our main skills lie in wavelet decomposition, wavelet commutators, Hardy-Littlewood maximal operator and Fefferman-Stein's vector-valued maximum function Theorem.

The aim of this paper is to prove another variation on the Heisenberg uncertainty principle, we generalize the quantitative uncertainty relations in $n$ different (time-frequency) domains and we will give an algorithm for the signal recovery related to the canonical Fourier-Bessel transform.

We study certain weighted Bergman and weighted Besov spaces of holomorphic functions in the polydisk and in the unit ball. We seek conditions on the weight functions to guarantee that the dilations of a given function converge to the same function in norm; in particular, we seek conditions on the weights to ensure that the analytic polynomials are dense in the space.

In this paper, we investigate the propagation of chaos for solutions to the Liouville equation derived from the Linear-Formation particle model. By imposing certain conditions, we derive the rate of convergence between the $k$-tensor product $f_{t}^{\otimes k}$ of the solution to be Linear-Formation kinetic equation and the $k$-marginal $f_{N,k}^{t}$ of the solution to the Liouville equation corresponding to the Linear-Formation particle model. Specifically, the following estimate holds in terms of $p$-Wasserstein ($1 \leqslant p <\infty$) distance $$ W^p_p(f_{t}^{\otimes k},f_{N,k}^{t}) \leqslant C_{1} \frac{k}{N^{\min(p/2,1)}}\left(1+t^{p}\right){\rm e}^{C_{2}t}, \quad 1\leqslant k\leqslant N. $$

Explicit asymptotic properties of the integrated density of states $N(\lambda)$ with respect to the spectrum for the random Schrödinger operator $H^{\omega}=(-\Delta)^{\alpha/2}+V^{\omega}$ are established, where $\alpha\in (0,2]$ and $V^\omega(x)=\sum_{i \in \mathbb{Z}^{d}} \xi_i(\omega) W(x-i)$ is a random potential term generated by a sequence of independent and identically distributed random variables $\{\xi_i\}_{i\in \mathbb{Z}^d}$ and a non-negative measurable function $W(x)$. In particular, the exact order of asymptotic properties of $N(\lambda)$ depends on the decay properties of the reference function $W(x)$ and the spectrum properties of the first Dirichlet eigenvalue of $(-\Delta)^{\alpha/2}$.

The existence and uniqueness of stationary solutions to anisotropic Navier-Stokes equations is investigated by a Galerkin technique in this work. Based on this conclusion, we further explore the exponential stability of weak solutions to stochastic anisotropic Navier-Stokes equations. We present a relationship among different growth exponents, which is sufficient to guarantee the existence, uniqueness and exponential stability of stationary solutions.

Background: Schistosomiasis is a parasitic disease. It is caused by a prevalent infection in tropical areas and is transmitted through contaminated water with larvae parasites. Schistosomiasis is the second most parasitic disease globally, so investigating its prevention and treatment is crucial.
Methods: This paper aims to suggest a time-fractional model of schistosomiasis disease (T-FMSD) in the sense of the Caputo operator. The T-FMSD considers the dynamics involving susceptible ones not infected with schistosomiasis $(S_{h}(t))$, those infected with the infection $(I_{h}(t))$, those recovering from the disease $(R(t))$, susceptible snails with and without schistosomiasis infection, respectively shown by $I_{v}(t)$ and $S_{v}(t)$. We use a new basis function, generalized Bernoulli polynomials, for the approximate solution of T-FMSD. The operational matrices are incorporated into the method of Lagrange multipliers so that the fractional problem can be transformed into an algebraic system of equations.
Results: The existence and uniqueness of the solution, and the convergence analysis of the model are established. The numerical computations are graphically presented to depict the variations of the compartments with time for varied fractional order derivatives.
Conclusions: The proposed method not only provides an accurate solution but also can accurately predict schistosomiasis transmission. The results of this study will assist medical scientists in taking necessary measures during screening and treatment processes.