Asymptotic Behavior For A Class Of Non-autonomous Reaction-diffusion Equations On The Whole Space

We consider the asymptotic behavior of the solution for a class of non-autonomous reaction-diffusion equations on the whole space: here g∈Lloc2(R,L2(Rn)), u(x,t) is an unknown function,f satisfies the following assumption: whereμ0,α1,α2,κ1,κ2 are positive constants.For the non-autonomous reaction-diffusion equations on the whole space, there are two main difficulties in the process of proving the asymptotic behavior of the solution. One difficulty is the Sobolev embedding theorem is invalid on the whole space, the other difficulty is the time-dependent external forcing g(x, t) is only local translation bounded but not translation compact, which bring about more diffi-culties for us to verify the existence of uniform attractor and obtain its structure. Motivated by the idea of [15,16,17], we will define a new class of functions-spacial absolutely continuous functions and make use of truncated functions to overcome the difficulties, then the existence and the structure of the uniform attractor are obtained.In Chapter 3, we will define a new class of functions-spacial absolutely con-tinuous functions, which are more general than translation-compact functions, then discuss its properties and the associations with other class of functions.In Chapter 4, we will make use of truncated functions and the properties of spa-cial absolutely continuous functions to prove the uniformly(w.r.tσ∈∑) asymptotic compactness of the solution, then the existence and the structure of the uniform attractor are obtained.