The Global Attractor On A Class Of Viscous Nonlinear Dispersive Wave Equations

Nonlinearity is universal and important phenomenon in nature. In recent years, many nonlinear partial differential equations were derived from physics, mechanics, chemistry, biology, engineering, aeronautics, medicine, economy, finance and many other fields. Because of the non-linearity and complexity of themselves, it is a big challenge to deal with them. In the paper, we study two nonlinear partial differential dispersive equations, that is, viscous Fornberg-Whitham equation and a class of viscous nonlinear dispersive wave equations. In this paper, we introduce the concept of global attractor and get the existence of global solution and the global attractor of a viscosity Fornberg-Whitham equation and the viscous nonlinear dispersive wave equation on periodical boundary condition.In the third chapter, the Galerkin Procedure is applied to show the existence of the global solution of Fornberg-Whitham equation in L~2(R). The Sobolev interpolation inequality and prior estimate on time-t are applied to show the existence of attracting set. Moreover, we prove the semi-group of the solution operator is a compact operator. Finally, we get the existence of the global attractor of a viscous Fornberg-Whitham equation in H~2(R).In the forth chapter, we give a study on a class of viscous nonlinear dispersive wave equations. We use the same discussion method as the third chapter. First we show the existence of global solution of the viscous nonlinear dispersive wave equation, then the existence of attracting set is obtained in H~2(R). Finally we proof that the equation has a global attractor.