Study On The Global Dynamical Behavior For Several Kinds Of Nonlinear Reaction Diffusion Equations

This thesis is devoted to the existence of global attractors for several classes of autonomous dissipative dynamical systems.In chapter 1, we introduce the background of the theory and its applications of infinite dimensional dynamical system and the development of the global attractor and then the method and theory of the existence of global attractor and two-grid mixed finite element method are summed up this chapter.TIn Chapter 2, some space and inequalities that we will used in this thesis are presented.TIn Chapter 3, we considerTTclassical reaction diffusion equation.Firstly we prove that there exist the bounded absorbing sets for the system. Secondly, we prove that the unit operator is Holder&& continuous in a bounded absorbing set,and that solution semigroups are asymptotic compactness. Finally the result that reaction diffusion equations have the global attractor in the unbounded domain is obtained.In Chapter 4, we consider non-classicalTTreaction diffusion equation.Firstly, we prove that there exist the bounded absorbing sets for the system.Secondly we get solution semigroups w-limit compactness by the decomposition technique of the solution. Finally, the fact that reaction diffusion equations exists the global attractor in the unbounded domain is discussed in this part.In Chapter 5, we consider non-classicalTTreaction diffusion equation with fading memory.Firstly, we prove that there exist bounded absorbing sets for the system. Secondly, we get asymptotic compactness of solution semigroups by the decomposition technique of solution and compactness transitivity theorem.Finally, the reaction diffusion equations have the global attractor in the weak topological space and the strong topological space.TIn Chapter 6, we solve the stationary nonlinear reaction diffusion equations by two-grid stability of mixed finite element method. Firstly, we prove the existence and uniqueness of solutions for the discrete system. Secondly, the lowest order finite element such as（NCP₁ -P₁） is used for the finite element discretization the format, and error estimate is given. Finally the results of theoretical analysis is verified by a numerical example.TIn Chapter 7, two-grid method with lowest equal order p₁-p₁ elementTis used for the unsteady reaction diffusion system in mixed variational form.Firstly, by introducing the new variable the original equation is rewritten into mixed variational form, by which not only the regularity of the solution is weaken but also the solutionpthe gradient ofpcan be obtained simultaneously. Secondly, the discrete approximate of variables u andpis doneby using conforming p₁ -p₁ element. To offset the LBB condition the stabilized term is introduced based on the local pressure projection. The proposed scheme does not the requirement of stabilization parameters.Compared with other stabilized methods the numerical results show the new method has better stability and can reduce the amount of calculation and save a lot of time.