We proved the existence and uniqueness of comporessible subsonic impinging jet flow in the work \cite{WX}. As a continuation, in this paper, we investigate the shape of free boundary to the impinging jet flow established in \cite{WX}. More specifically, if the nozzle wall is concave to the fluid, then the free boundary of flow will be convex to the fluid. On another hand, the higher regularity of free boundary at separation point is obtained, provided that the nozzle satisfies corresponding hypotheses.

In this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. (J. Math. Anal. Appl. 2015, 422, 84-98) and Han (J. Math. Anal. Appl. 2019, 474, 513-517). On the one hand, we not only prove the existence of doubly exponential decay solutions, but also find its threshold, and obtain the solutions with ∥ u 0 ∥ 2 2 → 0 + is always zero. On the other hand, we also prove the existence of doubly exponential growth solutions. The reseach results in this note extend previous results from both decay and growth.

In this paper, we consider the global solvability and energy conservation for initial value problem of nonlinear semi-discrete wave equation of Kirchhoff type, which is a discretized model of Kirchhoff equation.

In this paper, we investigate the Lipschitz stability of a perturbed impulsive differential system concerning the unperturbed system. We employ the variation of parameters or the constant of variation for impulsive differential systems with an initial time difference.

In this paper, we propose a new nonuniform mesh method to simulate acoustic scattering problems in two dimensional periodic structures with non-periodic incident fields numerically. As existing methods are difficult to extend to higher dimensions, we have designed the new method with such extensions in mind. With the help of the Floquet-Bloch transform, the solution to the original scattering problem is written as an integral of a family of quasi-periodic problems. These are defined in bounded domains for each value of the Floquet parameter which varies in a bounded interval. The key step in our method is the numerical approximation of the integral by a quadrature rule adapted to the regularity of the family of quasi-periodic solutions. We design a nonuniform mesh with a Gaussian quadrature rule applied on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.

This paper investigates a stochastic two predators-one prey system with ratio-dependent functional response under regime switching. The stochastic extinction of species and the existence of ergodic stationary distribution for the system are established, and the transition probability of the solution converging to the stationary distribution also is obtained. To illustrate our theoretical results, the numerical examples and simulations are presented. Our findings also demonstrate that the stationary distribution and extinction of species for the stochastic two predators-one prey system are affected by random perturbations, leading to an imbalance in ecology.

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second-order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti-damping. The hybrid system is first reduced to a first order port-Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central-difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter-Kato Theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

This paper deals with the existence of weak solutions of the system that describes the two-phase two-component fluid flow in porous media. Both two-phase and possible one-phase flow regions are taken into account. Our research is based on a global pressure, introduced in3, an artificial variable that allows us to decouple the original equations. We introduce another variable that does not have physical meaning in one-phase region and name it capillary pseudo-pressure. By using these two variables we rewrite system's equations in a fully equivalent form. This allows us to derive an existence result for weak solutions in more tractable way compared to15.

Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two ‘proper’ discrete Ablowitz-Kaup-Newell-Segur equations and two ‘unproper’ discrete Ablowitz-Kaup-Newell-Segur equations are derived. The ‘proper’ equations admit local reduction, while the ‘unproper’ equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.

Based on partial Fourier series analysis, we adapt on a model case a new approach to classical results obtained in the literature describing the singularities of a family a solutions of a second order elliptic problems on open non-convex planar sectors. The method allows the exhibition of singular and regular frequencies, explicit decomposition and description of coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter are obtained via this method. They are not uniform near π where corners have opening angle generating a jump of singularity in Sobolev exponent, contrarily to the results obtained in A. Tami (2016),(2019),(2021) for harmonic and/or biharmonic problems on a family of convex planar sectors.

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

Two types of short-time Fourier transforms involving respectively nonlinear modulation and nonlinear translation are designed in this note. Some important properties such as twin in time-frequency domain and orthogonality relations are investigated. Moreover, Lieb type inequalities and uncertainty principle are established.

In this paper, we consider the following 2D incompressible chemotaxis-Navier-Stokes equations with the fractional diffusion { ∂ t n + u · ∇ n − ∆ n = − ∇ · ( n ∇ c )+ n ( 1 − n )( n − a ) , ∂ t c + u · ∇ c − ∆ c = − cn , ∂ t u + u · ∇ u + ∧ 2 α u + ∇ P = − n ∇ ϕ , where ∧ : = ( − ∆ ) 1 2 and α ∈ [ 1 2 , 1 ] . By establishing the new priori estimates, we get the global well-posedness for the above system with large initial data.

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Linear growth of spherically symmetric solutions to Navier-Stokes equations in R N with degenerate viscosity and free boundary

This paper focuses on the three-dimensional(3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space H k ( R 3 ) ( k ≥ 3 ) of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena.

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov’s with sextic-power law refractive index and generalized non-local laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell-shaped soliton solutions and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model’s solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

We develop the Riemann-Hilbert (RH) method for the generalized inhomogeneous Hirota equation with zero boundary condition. The RH problem is related to two kinds of scattering data: N simple poles and one N-order pole. Here we consider that when the scattering data have one or more higher-order poles, the formulas of bound-state (BS) solitons and multiple BS solitons are obtained, and the interaction between solitons and BS solitons are shown.