The Long-time Behavior Of Solutions Of Non-autonomous Infinite Dimensional Dynamical System In Unbounded Domain

In this doctoral dissertation, first, we consider the asymptotic behavior of solutions of following nonautonomous reaction-diffusion equations in unbounded domainwhere a = (?) is a N×N real matrix with positive symmetric part (?)(a + a^*)≥βI,β> 0, a^* is the transpose of a, u = u(x,t) = (u₁,...,u_N),g=g(x,t)=(g₁,...,g_N),f=f(u,t)=(f₁,...f_N).We assume that g = g(x, t)∈L_b²(R; H), f = f(u, t)∈C(R^N×R;R^N) and the following conditions hold:where letter C denotes a positive constant which may be different in each occasion throughout this paper. We mainly consider the existence and structure of uniform attractor for Eq.(11) in unbounded domain, we prove the existence of uniform attractor for Eq.(11) in spaces L²(R^N) and L^p(R^N), p > 2, respectively, and at the same time, obtain it's structure. In order to prove the existence of uniform attractor in space L^p(R^N), we use the method that developed by C. Zhong, M. Yang, C. Sun in [42] called asymptotic a priori estimate. In order to describe the structure of uniform attractor in space L^p(R^N), we must need some continuity of associated processes in space L^p(R^N), if we don't restrict the order of p, then there isn't any continuity of processes in space L^p(R^N), even norm-to-weak continuity, since the spaces L^q(R^N) and L^p(R^N) are not nested, if p≠q. In this doctoral dissertation, we use the continuity of processes in L²(R^N) to instead of the continuity of processes in space L^q(R^N) to obtain the structure of uniform attractor in space L^p(R^N), for more details, please see chapter three.Then, we consider the asymptotic behavior of the positive solutions of the following nonlinear,nonautonomous reaction-diffusion equations in unbounded domainwhere u₀∈E = L^q(Ω), 1 < q <∞, E is a Banach space that defined order≤,Ω(?) R^N is an unbounded smooth domain, f : R×Ω×R→R is suitable smooth function, and satisfies f(t,x,u)≥0,Our goal is that based on the ideal in papers [1,5,65], and combining with the theory of infinite dimensional dynamical system in unbounded domain, prove the existence of pullback attractor and forward attractor for Eq.(15) in unbounded domain. Under other assumptions on nonlinear force and suppose that the processes {U(t, s)}_t≥s associated to Eq.(15) is order preserving. Using comparison principle, method of sub-super solutions, monotony of operators, the continuity of processes {U(t, s)}_t≥s in space E, the property of exponential stable of processes {U(t, s)}_t≥s in space E, we prove that there exist two extremal equilibria (?)_m(t)≥0 and (?)_m(t), minimal and maximal, respectively, and they are asymptotic stable. Furthermore, we obtain that the order interval [(?)_m(t),(?)_m(t)] is forward invariant respect to processes {U(t, s)}_t≥s. In order to prove the existence of non-degenerate minimal equilibria, we use the method of domain approximation, first we find non-degenerate minimal equilibria in bounded domain, then by domain approximation, we obtain the non-degenerate minimal equilibria in whole unbounded domain. At the end, we obtain the existence of pullback attractorΑ₁ and forward attractorΑ₂ in spaces L^q(Ω), 1 < q <∞, and H_D^(2α),q(Ω),α∈[-1, +1], respectively, and at the same time, we obtain thatΑ₁ (?) [(?)_m(t),(?)_m(t)],Α₂ (?) [(?)_m(t),(?)_m(t)].In order to prove the compactness of the processes {U(t, s)}_t≥s in space L^q(Ω), 1 < q <∞, we use the method of cutoff function, decomposing the whole space into two parts, one is bounded, another it's complement, in bounded domain we use compact sobolev embedding, in another part, let the L^q-norm of solution very small. In order to obtain the compactness of processes {U(t, s)}_t≥s in space H_D^(2α),q(Ω), we use the formula of constant variation and energy estimate, for more details and further talks, please see chapter four.As a concrete example, we consider the asymptotic behavior of positive solutions of following nonautonomous Logistic equationwhereΩ(?) R^N is unbounded smooth domain, p > 1, b(t)∈C¹(R),β,λ∈R. Also assume b(t) satisfies following conditions: there exist positive constant B₀, such that for all t∈RWhenβ≥λ, the asymptotic behavior of positive solutions of Eq.(17) is simple, there exists only one complete obit, it is 0, such that pullback attractorΑ₁ = {0} and forward attractorΑ₂ = {0}.Whenβ<λ, if the processes {U(t, s)}_t≥s is unstable at point 0, then the asymptotic behavior of positive solutions of Eq.(17) is complicate. We will show that the asymptotic behavior of positive solutions is affected by the velocity of b(t) go to 0 completely. There exists non-trivial complete obit u^*(t) for Eq.(17), attracts other positive solutions in pullback sense, the pullback attractorΑ₁ exists, andΑ₁ = {u^*(t)}_t∈R. But when t→∞, u^*(t) maybe unbounded, obviously, the forward attractor nonexist. However, in this situation, we can still describe the asymptotic behavior of positive solutions of Eq.(17). By compute absolute error and relative error between u^*(t) and other positive solutions, if the velocity of b(t) go to 0 is slowly, the relative error between u^*(t) and other positive solutions of Eq.(17) go to 0, then u^*(t) is the ''first order approximation" of forward attractor, in this situation, u^*(t) can be considered as a forward attractor. We also give a regain forλand b(t) and compute the absolute error between u^*(t) and other positive solutions of Eq.(17), we will see that in one situation u^*(t) is forward attractor, in another one it is not, for more details, please see chapter five.