Weakly Damped Generalized Kdv Equations And The Generalized Kdv-burgers Equation For A Long Time State

With the infinite-dimensional dynamical system of study continued to deepen and development, a large number of researchers are growing concerned about nonlinear evolution equations of a long time study. The generalized KdV equation and generalize-d KdV-Burgers equation appears in many physical models, is non-linear science, one of the important models, the overall existence of solutions, solution stability, solution of the explosion, traveling wave solutions, solitary wave solutions, etc. in the quantum field theory, plasma physics and solid-state physics has a broad application. KdV equation and KdV-Burgers equation with the equation parameters, which reflect a different meaning in different physical equations, so the generalized KdV equation and generalized KdV-Burgers equations in-depth study is very necessary and important.This paper mainly studies the following general KdV equation of weakly damped periodic initial boundary value problem: Which p=1,p=2,3≤p<4 (p=n/m,n is an integer and m is odd),γ>0,Ω=[0,L],f has nothing to do with time.And studies the following general KdV-Burgers equation of periodic initial boundary value problem: Which1≤p<8,(p=n/m,n is an integer and m is odd),γ>0,Ω=[0,L],f has nothing to do with time.This mainly consists of the following four chapters of the weakly damped generalized KdV equation and generalized KdV-Burgers equations to study the cycle of initial boundary value problem:In the first chapter, we introduces the physical background and study situation of the weakly damped generalized KdV equation and generalized KdV-Burgers equations. In the second chapter, we use the Galerkin method to prove the weakly damped generalized KdV existence and uniqueness of global solution.In the third chapter, we mainly obtained weakly damped generalized KdV equation of the existence of global attractor, and the global attractor of the Hausdorff dimension and the fractal dimension of the upper bound estimate.In the fourth chapter, we use the Galerkin method to prove the generalized KdV-Burgers existence and uniqueness of global solution.