The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation with nonlinear strongly damped terms

In this paper ,we study the long time behavior of solution to the initial boundary value problems for higher -orderkirchhoff-type equation with nonlinear strongly dissipation:At first ,we prove the existence and uniqueness of the solution by priori estimate and Galerkin methodthen we establish the existence of global attractors ,at last,we consider that estimation of upper bounds of Hausdorff and fractal dimensions for the global attractors are obtain.

The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*

We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:2( ) ( )m mt t tu    u    D u  ( ) ( ) ( )m  u  g u  f x . Under of the properassume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkinmethod, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of theglobal attractor.

The Estimation of the Hausdorff and Fractal Dimensions of the Global Attractor for 2D Autonomous g-Navier-Stokes Equations

In this paper, by using the energyequation method, the 2D g-Navier-Stokes equations with linear dampness on some unbounded domains wereinvestigated without the restriction of the forcing term belongingto some weighted Sobolev space. Moreover,the estimation of theHausdorff and Fractal dimensions of such attractors were alsoobtained.

ASYMPTOTIC DYNAMICS OF 2D FRACTIONAL COMPLEX GINZBURG–LANDAU EQUATION

In this paper, we consider the well-posedness and asymptotic behaviors of solutions of the fractional complex Ginzburg–Landau equation with the initial and periodic boundary conditions in two spatial dimensions. We explore the existence and uniqueness of global smooth solution by means of the Galerkin method and establish the existence of the global attractor. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor are presented.

Topological dimensions of global attractors for semilinear PDE's with delays

An estimate is obtained on the Hausdorff and fractal dimensions of global attractors of semilinear partial differential equations with delay: ẋ(t) = Ax(t) + f(xt). The method employed is to associate such an equation with a nonlinear semigroup on a product space and then appeal to the upper estimate due to Constantin, Foias and Teman on topological dimensions of global attractors for general nonlinear dynamical systems.