The Existence And Asymptotic Property Of The Solutions For Some Nonlinear Evolution Equations

In this paper, we consider the global smooth solutions and long time be-haviors for some nonlinear evolution equations, such as KdV equation, BBM equation, GBBM equation, KdV-Burgers equation, coupled generalized nonlin-ear wave equation. By using a priori estimates, the existence of global smooth solution, the existence of global attractors and its fractal dimensions for this sys-tems are obtained.This paper is organized in six chapters.Chapter 1, give the physical background for the nonlinear evolution equa-tions, such as KdV equation, BBM equation, GBBM equation, KdV-Burgers equation, coupled generalized nonlinear wave equation. We look back to some important results, review our work for doing.Chapter 2, consider a class non-homogeneous BBM equation. In section 2.2, by a priori estimates and Fourier spectral method, we prove the existence and uniqueness of the global smooth solution for the periodic initial value problem and obtain the large time error estimate between spectral approximate solution and the exact solution. In sections 2.3 and 2.4, by a priori estimates and Galerkin method, we prove the existence of the global smooth solution and global attrac-tors for the initial-boundary value problem.Chapter 3, consider the initial-boundary value problem of the multidimen-sional non-homogeneous GBBM equations. The existence of global smooth so-lution and global attractors of this problem was proved by means of a uniform priori estimate for time.Chapter 4, consider the periodic initial value problem of a dissipative gen-eralized KdV equations. In section 4.2, the existence of global smooth solution of this problem is proved and the existence of the global attractors is obtained. Insection 4.3, semi-discrete and fully discrete Fourier spectral and pseudo-spectral schemes are constructed. The convergence and stability for the schemes are proved, and the error estimates are obtained.Chapter 5, consider the damped coupled generalized nonlinear wave equations. In section 5.2, by coupled a priori estimates and Galerkin method, prove the existence and uniqueness of the global smooth solution for the periodic initial value problem and obtain the existence of global attractors. We get the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractors. In section 5.3, the Cauchy problem is studied, by using the weighted function space and the interpolating inequality, the existence of the global attractors for the damped generalized coupled nonlinear wave equations in an unbounded domain is proved. In section 5.4, the time periodic solution problem of damped generalized coupled nonlinear wave equations with periodic boundary conditions is studied, the existence of time periodic soluation of this problem is proved by using the convergence of approximate time periodic solution sequences.Chapter 6, consider a coupled generalized KdV-Burgers equation. In section 6.2, we study the initial-boundary value problem in the semi-unbounded domain, the existence of global solutions and global attractors is proved by means of a uniform priori estimate for time. In section 6.3, the Cauchy problem by using the weighted space, the existence of the global attractors for a coupled generalized KdV-Burgers in an semi-unbounded domain is proved.In this paper, the main difficulties are from a priori estimates for studing the high dimension, nonlinear systems and unboundary domain, we meet many problems which are difficult to be overcomed by using the standed method. We solve these difficulties by using the complicated, meticulous a priori estimates.