A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.

The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations

In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space E0 to Ek, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.

A Family of Global Attractors for the Generalized Kirchhoff-Beam Equations

In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.

The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets Bk. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.

A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations

In this paper, the global dynamics of a class of higher order nonlinear Kirchhoff equations under n-dimensional conditions is studied. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup associated with the initial boundary value problem are proved, and the existence of a family of exponential attractors is obtained. Then, by constructing the corresponding graph norm, the condition of a spectral interval is established when N is sufficiently large. Finally, the existence of the family of inertial manifolds is obtained.

Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term

In this article, we prove the existence of exponential attractors of the nonclassical diffusion equation with critical nonlinearity and lower regular forcing term. As an additional product, we show that the fractal dimension of the global attractors of this problem is finite.

The Family of Global Attractors for Kirchhoff-Type Coupled Equations

This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the overall solution of the above problem by using a priori estimates in the spaces of E0 and Ek, and secondly, it proves that there is a family of global attractors for the above problem, and finally estimates the Hausdorff dimension and the Fractal dimension of the family of global attractors.

The Family of Global Attractors of Coupled Kirchhoff Equations

In this paper, we study the initial boundary value problem of coupled generalized Kirchhoff equations. Firstly, the rigid term and nonlinear term of Kirchhoff equation are assumed appropriately to obtain the prior estimates of the equation in E0 and Ek space, and then the existence and uniqueness of solution is verified by Galerkin’s method. Then, the solution semigroup S(t) is defined, and the bounded absorptive set Bk is obtained on the basis of prior estimation. Through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek. Finally, by linearizing the equation, it is proved that the solution semigroup S(t) is Frechet differentiable on Ek, and the family of global attractors Ak have finite Hausdroff dimension and Fractal dimension.

A Family of the Random Attractors for a Class of Generalized Kirchhoff-Type Equations

In this paper, we studied the existence of a family of the random attractor for a class of generalized Kirchhoff-type equations with a strong dissipation term. Firstly, according to Ornstein-Uhlenbeck process, we transformed the equation into a stochastic equation with random variables and multiplicative white noise. Secondly, we proved the existence of a bounded random absorbing set. Finally, by using the isomorphic mapping method and the compact embedding theorem, we get the stochastic dynamical system with a family of random attractors.

The Modi Exponentiated Exponential Distribution

In this study, a new four-parameter distribution called the Modi Exponentiated Exponential distribution was proposed and studied. The new distribution has three shape and one scale parameters. Its mathematical and statistical properties were investigated. The parameters of the new model were estimated using the method of Maximum Likelihood Estimation. Monte Carlo simulation was used to evaluate the performance of the MLEs through average bias and RMSE. The flexibility and goodness-of-fit of the proposed distribution were demonstrated by applying it to two real data sets and comparing it with some existing distributions.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

A Family of Global Attractors for a Class of Generalized Kirchhoff-Beam Equations

The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are proved by prior estimation and Galerkin finite element method. Then the bounded absorption set is obtained by prior estimation, and the family of global attractors for the high-order Kirchhoff-Beam equation is obtained. The Frechet differentiability of the solution semigroup is proved after the linearization of the equation, and the decay of the volume element of the linearization problem is further proved. Finally, the Hausdorff dimension and Fractal dimension of the family of global attractors are proved to be finite.

Asymptotically Optimal Empirical Bayes Estimation of Parameter for Scale-exponential Family under PA Samples

参数的 Bayes 评估者为规模被获得在情况中的指数的家庭相等分布式、确实联系(PA ) 在加权的方形的损失 function.We 下面的样品构造实验 Bayes (EB ) 评估者并且证明它是 asymptotic 最佳。

PULLBACK EXPONENTIAL ATTRACTORS FOR THE NON-AUTONOMOUS MICROPOLAR FLUID FLOWS

This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.

The Exponential Attractor for a Class of Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms

This paper studies the exponential attractor for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The exponential attractor is also called the inertial fractal set, which is an intermediate step between global attractors and inertial manifolds. Obtaining a set that attracts all the trajectories of the dynamical system at an exponential rate by the methods of Eden A. Under appropriate assumptions, we firstly construct an invariantly compact set. Secondly, showing the solution semigroups of the Kirchhoff-type equations is squeezing and Lipschitz continuous. Finally, the finite fractal dimension of the exponential attractor is obtained.

Random Attractor Family for the Kirchhoff Equation of Higher Order with White Noise

The existence of random attractor family for a class of nonlinear high-order Kirchhoff equation stochastic dynamical systems with white noise is studied. The Ornstein-Uhlenbeck process and the weak solution of the equation are used to deal with the stochastic terms. The equation is transformed into a general stochastic equation. The bounded stochastic absorption set is obtained by estimating the solution of the equation and the existence of the random attractor family is obtained by isomorphic mapping method. Temper random compact sets of random attractor family are obtained.

A Family of the Global Attractor for Higher Order Nonlinear Kirchhoff Equation

In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations with strong damping terms. We will properly assume the stress term M(s) and nonlinear term g(ut). First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.

ARTISTIC CHARM OF MANDELBROT FAMILY AND JULIA FAMILY——A FURTHER DISCUSSION CONCERNING MANDELBROT FAMILY AND JULIA FAMILY

Fractals generated by iterating Z←Z 2+c have been discussed in the literature. In this paper, we use the “Escape-time Algorithm” to do some further analysis of the graphics that are generated by iterating Z←Z i+c, Z←Z i+c, Z←∑Ni=0Z i+c, Z←∑10i=0a iZ i+c on the complex c -plane and z -plane, respectively, where i∈N. The graphics obtained on c -plane and z -plane are called Mandelbrot Family and Julia Family successively. Then we do some tentative discussions on graphics of these families. On the basis of object-oriented thought, a software has been developed by using Visual C++ under Windows 3.1. Not only the graphics can be generated interactively by dialog boxes and mouse, but also their subtle structures can be observed by the mechanism of local enlargement and the Julia graphics can be generated conveniently by collecting parameters from the corresponding Mandelbrot graphics. With a friendly interface and with the mouse as its main tool, the software can be used easily.

EXPONENTIAL ATTRACTOR FOR THE GENERALIZED SYMMETRIC REGULARIZED LONG WAVE EQUATION WITH DAMPING TERM

The global fast dynamics for the generalized symmetric regularized long wave equation with damping term is considered. The squeezing property of the nonlinear semi_group associated with this equation and the existence of exponential attractor are proved. The upper bounds of its fractal dimension are also estimated.

The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping

In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor Ak in space Ek. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on Ek via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor Ak.