The Research Of The Long Time Behavior Of Solution For The Benjamin-Bona-Mahony Equations

This paper considers long time behavior of solutions for the Benjamin-Bona-Mahony equations.Firstly,we provide an analysis of the two-dimensional general-ized Benjamin-Bona-Mahony equation with periodic boundary conditions.By the method of orthogonal decomposition,we show that the existence of asymptotic at-tractor which overcome difficulties come from the precision of approximate inertial manifolds.Moreover,the dimensions estimate of the asymptotic attractor is ob-tained.Secondly,we provide an analysis of the semi-discrete generalized Benjamin-Bona-Mahony equation on Rn.We semi-discrete this generalized equation in time Crank-Nicolson scheme,and prove that it provides a discrete infinite dimensional dynamical systems in H1(Rn).Then we prove this system possesses a global attrac-tor AT in H1(Rn).In addition,we show that the global attractor AT is regular.At last,we estimate the finite fractal dimensions of the global attractor AT.The full paper is divided into three parts:The first chapter,we mainly introduce the background of the Benjamin-Bona-Mahony equation with periodic boundary and the semi-discrete generalized Benjamin-Bona-Mahony equation on Rn,and basic theory and method of long time behavior of solutions for evolution equation.The second chapter,we discuss the existence of the asymptotic attractor for the two-dimensional generalized Benjamin-Bona-Mahony equation and the dimensions estimate of the asymptotic attractor.The third chapter,we discuss the existence of the global attractor for the semi-discrete generalized Benjamin-Bona-Mahony equation on Rn and the finite fractal dimensions estimation of the global attractor.