The Long-time Behaviors Of Solutions For Non-autonomous Reaction-diffusion Equations On Non-cylinarlcal Domains

In this dissertation, we consider the long-time behaviors of solutions for the following reaction-diffusion equation defined on non-cylindrical domains: where the nonlinear function g(·) satisfying a polynomial growth of arbitrary order.Since the spatial domains varying with time, the system mentioned above is intrinsically non-autonomous even if the terms (e.g., the forcing term f(·)) in the equation do not depend explicit on time.The main works of this paper are to establish new methods (framework) and a priori estimates to prove, without any additional assumptions, especially, no any smoothness assumptions on the forcing term, that the known (L2,L2) pullback D-attractor indeed can attract the same class Din L2+δ-norm (δ∈ [0,∞) is arbitrary) and H1-norm.We point out that even for the cylindrical domains case, up to now, the best known results about the attraction associated to the corresponding non-autonomous system above is (L2,LP) (the power p comes from (1.2)) pullback D-attraction for f∈ Lloc2(R;L2(O)), and there is no any result about (L2,Lp+s) pullback D-attraction for s> 0; as f∈ Lloc2(R;]H-1((D)), the best attraction is (L2,L2) pullback D-attraction, and the (L2,Lq) pullback attraction remains open for any q> 2. Our main results (Theorems 3.4.3 and 5.3.4) established for non-cylindrical domains case solve (even for the non-autonomous system defined on cylindrical domains) the problems mentioned above. On the other hand, for non-autonomous system defined on cylindrical domains, there is no any result about the continuity of solutions w.r.t. initial data in H1. Here, for any space dimension and any p ∈ [2,∞), we obtain the continuity of solutions w.r.t. initial data in H1, see Theorem 4.2.1, which makes up the gap even for cylindrical domains case. Moreover, we emphasize that our method, results and proof scheme are applicable to other dissipative equations.The structure of this paper is as follows.In Chapter one, we recall briefly the backgrounds about attractors and its ap-plications on reaction-diffusion equations defined on cylindrical and non-cylindrical domains, and give a introduction about the main contribution of this paper.In Chapter two, we make the preliminaries that will be used throughout this paper.In the case that the spatial domains satisfying some differentiable homeomor-phism, in Chapter three, we first establish a new criterion about the higher-order attraction of non-autonomous system; then, to guarantee the test functions we used to make sense, a maximum principle about the strong solutions is proved; finally, we prove that the known (L2,L2) pullback Dλ-attractor can pullback Dλ-attract in Lq-norm for any q ∈ [2,∞) (Theorem 3.4.3). In Chapter four, we first establish a Nash-Moser-Alikakos type a priori estimates for the difference of solu-tions near the initial time; and then, for any space dimension N and any growth power p≥2, we obtain the continuity (w.r.t. initial data) of solutions in H1 (Theorem 4.2.1). Based on such continuity, we obtain the pullback Dλ-attraction in L2+δ (δ ∈ [0, ∞))-norm and H1-norm respectively (see Theorems 4.3.1 and 4.4.1).In the case that the spatial domains expanding with time, we establish, in Chapter five, the maximum principle about the corresponding variational solu-tions; moreover, we obtain that the known (L2, L2) pullback Dλ1-attractor can pullback Dλ1-attract in L2+δ-norm for any δ ∈ [0,∞) (see Theorem 5.3.4).Finally, in Chapter six, we end this paper by summarizing and some remarks and problems that we will consider in the future.