Global Attractor Of Fourth Order Semi-linear Parabolic Equation

This work carries out qualitative studies on the existence of global solutions and attractor for the initial boundary value problem of fourth order semi-linear parabolic equation, which has a close relationship with Fisher-Kolmogorov equation and Swift-Hohenberg equations. Firstly, this thesis studies the existence of global solutions for the problem by using potential wells method. The invariant sets and the sharp conditions for the existence of global solutions for the problem are given. The existence of global solutions is a necessary condition for the existence of global attractor. Secondly, this thesis proves the nonexistence of global solutions for the problem by potential wells convexity method. Finally, we get the global attractor for the problem in Hk(Ω) by using Sobolev embedding theorem and iterative method. The maximum principle acts an important ingredient in a series of papers. However, for higher order equations such as this note we encounter serious difficulties due to the lack of maximum principle and lack of compactness. So we get rid of the bondage of maximum principle and compactness. Through the estimation of nonlinear term, we get bounded of module, what's more, we get the existence of global attractor in Hk(Ω) space by using Sobolev embedding theorem and iterative method.