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Pre-Publication Accepted Manuscript Dun Zhou Lifting properties of minimal sets for parabolic equation on S1 with reflection symmetry Proceedings of the American Mathematical Society DOI: 10.1090/proc/13283 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proof- read, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. Lifting Properties of Minimal Sets for Parabolic Equations on S1 with Reflection Symmetry Dun Zhou∗ Department of Mathematics University of Science and Technology of China Hefei, Anhui, 230026, P. R. China Abstract We consider the skew-product semiflow generated by the following parabolic equation 1 ut = uxx + f(t; u; ux); t > 0; x 2 S = R=2πZ; where f(t; u; ux) = f(t; u; −ux). It is proved that the flow on uniquely ergodic minimal set M is topologically conjugate to a subflow on R × H(f) and M is uniquely ergodic if and only if the set consists by 1-cover points of H(f) has full measure. It is further proved that any minimal set M is almost automorphic provided that f is uniformly almost automorphic. Moreover, for any almost automorphic solution u(t; x) contained in M, the frequency module M(u(t; x)) is contained in the frequency module of f. 1 Introduction In the current paper, we study the lifting properties for a minimal set of the following scalar reaction-diffusion equations on the circle 1 ut = uxx + f(t; u; ux); t > 0; x 2 S = R=2πZ; (1.1) 2 where f : R × R × R ! R is C admissible (see definition 2.3 ) and f(t; u; p) = f(t; u; −p). In non-autonomous systems, lifting properties of minimal sets is an important and interesting topic. The lifting properties describe whether or how much certain structure (say, periodicity, almost periodicity) in an equation can be lifted to its solutions. And it is also closely related to the asymptotic behavior of bounded solutions of a differential equation (see Shen and Yi [16]). Let Π('; σ) be a skew-product flow on X ×Y where Y is a compact minimal set in flow σ and X is metrizable. Let K ⊂ X × Y be a compact invariant set under Π. It was proved in Sacker and Sell [11] that if Y is distal and K is an N-cover of Y , then the skew-product flow Π on K is distal; ∗Partially supported by NSF of China No.11371338, 11471305, Wu Wen-Tsun Key Laboratory and the Fun- damental Research Funds for the Central Universities. 1 This is a pre-publication version of this article, which may differ from the final published version. CopyrightMay 6 restrictions2016 22:57:48 may apply. EDT Version 2 - Submitted to PROC moreover, if Y is almost periodic then K is almost periodic. In scalar non-autonomous ODE, if K −1 is minimal, then K is uniquely ergodic if and only if Y0 = fy 2 Y jcardK\p (y) = 1g has full set (see Johnson [7]). Shen and Yi in [16,17] generalized this property to scalar parabolic equations with separated boundary condition, besides, they also proved that K is almost automorphic if and only if Y is almost automorphic. In the present paper, we are focusing on equation (1.1) with periodic boundary condition. Let fτ (t; u; p) = f(t+τ; u; p)(τ 2 R) be the time-translation of f. Then the function f generates a family ffτ jτ 2 Rg in the space of continuous functions C(R × R × R; R) equipped with the compact open topology. Denote H(f) (called the hull of f) be the closure of ffτ jτ 2 Rg in the compact open topology. Then, H(f) is a compact metric space and every g 2 H(f) has the same regularity as f. The action of time-translation g · t ≡ gt (g 2 H(f)) defines a compact flow on H(f). Thus, equation (1.1) gives rise to a family of equations associated to each g 2 H(f), 1 ut = uxx + g(t; u; ux); t > 0; x 2 S : (1.2) 2 1 Let X be the fractional power space associated with the operator u ! −uxx : H (S ) ! L2(S1) such that the embedding relation X,! C1(S1) is satisfied. For any u 2 X,(1.2) admits (locally) a unique solution '(t; ·; u; g) in X with '(0; ·; u; g) = u(·). This solution also continuously depends on g 2 H(f) and u 2 X. Therefore, (1.2) defines a (local) skew product semiflow Πt on X × H(f): Πt(u; g) = ('(t; ·; u; g); g · t); t > 0: (1.3) It follows from [6] that if '(t; ·; u; g)(u 2 X) is bounded in X in the existence interval of the solution, then u is a globally defined classical solution. Assume !(u; g) be the !-limit set of the bounded semi-orbit Πt(u; g) in X × H(f). Then, for any δ > 0, f'(t; ·; u; g): t ≥ δg is relatively compact in X. As a consequence, !(u; g) is a nonempty connected compact subset of X ×H(f). It is also known that Πt restricted on the !-limit set !(u; g) admits a flow extension, so that it defines a skew-product flow (see [5,6]). If (1.1) is an autonomous system(i.e., f(u; ux) = f(u; −ux)). Matano [10] showed that !-limit set of any bounded solution of (1.2) is an equilibrium. In the case that f is time-periodic with 1 period 1 (equivalently, H(f) is homeomorphic to the circle T = R=Z), Chen and Matano [3] proved that for f = f(t; u) independent of ux, the !-limit set !(u) of any bounded solution consists of a unique time-periodic orbit with period 1. As a matter of fact, such conclusion is also correct for f(t; u; ux) = f(t; u; −ux). Recently, Shen et al. [15] considered the equation (1.1) with time almost periodically external force. It was proved in [15] that any minimal set of (1.2) is almost 1-cover of H(f). Moreover, if dim V c(M) = 0 (where V c(M) is the center space of M associated with the Sacker-Sell spectrum), this minimal set is spatially homogeneous and 1-cover of H(f); if dim V c(M) = 1, M is either spatially homogeneous or spatially inhomogeneous and 1-cover of H(f). 2 This is a pre-publication version of this article, which may differ from the final published version. CopyrightMay 6 restrictions2016 22:57:48 may apply. EDT Version 2 - Submitted to PROC The present paper is devoted to further studying the lifting properties of the minimal set of (1.2). Throughout this paper we assume that H(f) is recurrent (which means that H(f) is minimal). This is satisfied, for instance, when f is a uniformly almost periodic or, more generally, a uniformly almost periodic function, i.e. when it is admissible and almost periodic or almost automorphic. Let M be a minimal set of (1.3). Then, we will prove that the almost automorphic property of M equivalents to the almost automorphic property of H(f). Moreover, if f is uniformly almost automorphic then the frequency module of any almost automorphic solution in M is contained in the frequency module of f. Finally, we will prove that, if M is uniquely ergodic then the flow on M is topologically conjugate to a flow on R × H(f). An outline of this paper is as follows. In section 2, we agree on some notations, give relevant definitions and preliminary results, including the almost periodic (automorphic) theory and zero number property. In section 3 and section 4, we discuss the lifting property of uniquely ergodicity and almost automorphicly of the minimal set H(f) of (1.3) separately. 2 Preliminaries In this section, we summarize some preliminary materials to be used in later sections. We start by introducing some definitions in dynamic systems. Then, we give a brief about the almost periodic(automorphic) functions (flows). Finally, we present some basic properties of zero numbers of solutions for linear parabolic equations. 2.1 Basic Conceptions of Dynamic Systems Let Y be a compact metric space with metric dY , and σ : Y ×R ! Y; (y; t) 7! y·t be a continuous flow on Y , denoted by (Y; σ) or (Y; R). A subset E ⊂ Y is invariant if σt(E) = E for every t 2 R. A subset E ⊂ Y is called minimal if it is compact, invariant and the only non-empty compact invariant subset of it is itself. Every compact and σ-invariant set contains a minimal subset and a subset E is minimal if and only if every trajectory is dense. The continuous flow (Y; σ) is called to be recurrent or minimal if Y is minimal. Definition 2.1. Let (Y; R), (Z; R) be two continuous compact flows.
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