Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion

In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a \begin{document}$(L^2,L^2)$\end{document} pullback \begin{document}$\mathscr{D}_μ$\end{document} -attractor of this model. Then we show that the pullback \begin{document}$\mathscr{D}_μ$\end{document} -attractor attract the \begin{document}$\mathscr{D}_μ$\end{document} class (especially all \begin{document}$L^2$\end{document} -bounded set) in $L^{2+δ}$-norm for any \begin{document}$δ∈[0,∞)$\end{document} . Moreover, the solution of the model is shown to be continuous in \begin{document}$H^s$\end{document} with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the \begin{document}$(L^2,L^2)$\end{document} pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of \begin{document}$\mathscr{D}_{μ}$\end{document} in \begin{document}$H^s$\end{document} -norm, and thus the existence of a \begin{document}$(L^2, H^s)$\end{document} pullback \begin{document}$\mathscr{D}_μ$\end{document} -attractor is obtained.


1.
Introduction. Reaction diffusion equations, arising from heat diffusion and other numerous areas of applied sciences, have been extensively studied. The dynamics of these equations are also quite well understood, see e.g. [13,16,17,32,34,40] and references therein. It is known that the heat equation is a special reaction diffusion equation. If −∆ in the heat equation is replaced by a nonlocal operator (−∆) s (s ∈ (0, 1)), then we have a fractional diffusion equation. This class of equations are arising in many contexts. For instance, the critical quasi-geostrophic equation appearing in hydrodynamics (see e.g., [12,26]), fractional porous medium models (see e.g., [7,11,31]), the context of stable processes (see e.g., [3,23]) and many others. These fractional diffusion equations have been studied a lot in recent years, such as the existence of global solutions and the global regularity of smooth solutions.
However, the understanding of the dynamics of fractional diffusion equations is rather limited, especially the theory of attractors of these equations is far less developed. The main goal of this paper is to study the existence and properties of pullback attractors of the following non-autonomous fractional diffusion equation: where λ > 0, 0 < s < 1, τ ∈ R is the initial time, and where u τ ∈ L 2 (R N ), (2) g ∈ L 2 loc (R; L 2 (R N )).
Different from the standard Laplacian −∆ which is defined by pointwise differentiation, the fractional Laplacian (−∆) s is defined by a global integration with respect to a singular kernel (see e.g., [22]). The difference between these two types of operators is best seen in the theory of diffusion. The operator −∆ is often adopted to model the Brownian motion, whose feature is that the space scale of the propagation of the probability distribution is proportional to t 1/2 . When the space scale of the propagation of a diffusion is proportional to another power of time, (−∆) s is more proper to model the diffusion. This type of diffusion is often known as anomalous diffusion (see e.g., [6,18,25]). The fractional Laplacian operators are also the infinitesimal generators of stable Lévy processes (see e.g., [3]). Define Then by (4), there exist positive constants C i , k i for i = 1, 2 such that The first result of this paper provides the existence of pullback D µ -attractor in L 2 , see Section 3. In order to do this, we need the positive constants λ, C 1 , C 2 , k 1 , k 1 , k 2 satisfying the following conditions: and λ + 1 − 2 k 1 > 0. (8) It is obvious that if λ is large enough, then (7) and (8)  Assume that the conditions (2)-(8) hold, and let t −∞ e µs g(s) 2 L 2 (R N ) ds < +∞, for all t ∈ R.
In the proof of Theorem 1.1, we firstly provide the existence of (L 2 (R N ), H s (R N ) ∩L p (R N )) pullback D µ -absorbing sets; then establish a "Tail estimate" (see Lemma 3.8), which, together with Rellich-Kondrachov's theorem, implies that the process U generated by the equation (1) is pullback D µ -limit-set compact in L 2 . We emphasize that the pullback D µ -limit-set compactness of U cannot be deduced simply by the fact that U has a (L 2 (R N ), H s (R N )) pullback D µ -absorbing set, since the Sobolev embedding H s (R N ) → L 2 (R N ) is only continuous but not compact.
Note that g only satisfies the L 2 -integrability (3) and (9), and f grows as the polynomial of degree p − 1. The solutions of (1) at most belong to W 2s,2p−2 (R N ). On the other hand, the embedding W 2s,2p−2 (R N ) → L 2+δ (R N ) is not true when both the spatial dimension N and δ are large enough. Therefore, for any δ ∈ [0, ∞), the L 2+δ (R N )-norm pullback D µ -attraction obtained in Theorem 1.2 is not trivial.
The last result of this paper investigates the dynamics of equation (1) in H s (R N ). It states that for any δ ∈ [0, ∞), the (L 2 (R N ), L 2 (R N )) pullback D µ -attractor obtained in Theorem 1.1 attracts the D µ -class (especially for any L 2 (R N )-bounded set) in H s (R N )-norm. Theorem 1.3. Let θ be a positive constant defined in Theorem 1.1, µ ∈ (0, θ). Suppose that (3)-(5), (7)-(9) hold. Furthermore, we assume that f satisfies an additional condition f ∈ C 1 (R) and there exists a positive C f such that for any s ∈ R, Then the minimal (L 2 , L 2 ) pullback D µ -attractor A = {A(t) : t ∈ R} obtained in Theorem 1.1 pullback attracts the D µ -class in the topology of H s (R N ), i.e., for any The condition (11), which is natural and has been used in many contexts (see e.g. [28,32,34] and references therein), is adapted to prove the continuity of solutions of (1) in H s (R N ), see Lemma 5.2.
By Theorem 1.3, we find that the (L 2 (R N ), L 2 (R N )) pullback D µ -attractor is in fact a (L 2 (R N ), H s (R N )) pullback D µ -attractor, see Remark 5. In order to prove the existence of the (L 2 (R N ), H s (R N )) pullback D µ -attractor, generally we need to show that the process U associated with the equation (1) not only has a (L 2 (R N ), H s (R N )) pullback D µ -absorbing set but also is pullback D µ -asymptotically compact in the Sobolev space H s (R N ).
The existence of a (L 2 (R N ), H s (R N )) pullback D µ -absorbing set is proved in Theorem 1.1. However, it is more difficult to show that U is pullback D µ -asymptotically compact in H s (R N ). The reasons are listed as follows. Firstly, since g(t) only satisfies L 2 integrability (w.r.t. t), we cannot differentiate the equation (w.r.t. t) to get a nice apriori estimate of ∂ t u. Secondly, a sufficient condition, which is called the "Tail estimate" in H s (R N ), is difficult to establish because of the nonlocal operator (−∆) s . Thirdly, for a non-autonomous reaction-diffusion equation defined on a bounded domain Ω, Lukaszewicz [28] proved the existence of a unique minimal pullback attractor in H 1 0 (Ω) by the Kuratowski measure of non-compactness of a bounded set together with a new Gronwall lemma. However, because our model is a nonlocal equation and defined on the whole space R N . The method presented in [28] is inapplicable to prove that U is pullback D µ -asymptotically compact in H s (R N ). Based on the above arguments, new methods are needed. This paper is organized as follows. We begin with some preliminary results which will be used throughout this paper. Section 3 is devoted to the study of the existence of the (L 2 (R N ), L 2 (R N )) pullback D µ -attractor. In Section 4, first some higher-order integrability for the difference of solutions near the initial time (Lemma 4.1) is proved by using a bootstrap argument, and then Theorem 1.2 is proved. In Section 5, we first show the continuity of solutions in H s (R N ) with respect to the initial data (Lemma 5.2), and then we complete the proof of Theorem 1.3. Finally, as a corollary, the existence of a (L 2 (R N ), H s (R N )) pullback D µ -attractor (Remark 5) is obtained.

Preliminaries.
2.1. Abstract results on pullback attractors. In this section, we recall some abstract results on pullback attractors, which were originally introduced by Crauel and Flandoli [19] and generalized by Caraballo, Lukaszewicz and Real [14], see e.g., [15,19,27] for more details. Firstly, we recall the definition of a process. Definition 2.1 (Kloeden and Rasmussen [27]). Let (X, d) be a complete metric space with the metric d(·, ·). A process is a continuous mapping U (t, τ ) : X → X for t, τ ∈ R with t τ , which satisfies the initial value and evolution properties: (i) U (t, t)x = x for all t ∈ R and x ∈ X, (ii) U (t, τ )x = U (t, r)U (r, τ )x for all τ r t and x ∈ X.
A process is also called a two-parameter semigroup on X in contrast with the one parameter semigroup of an autonomous semi-dynamical system since it depends on both the initial time τ and the actual time t rather than just the elapsed time t − τ .
Suppose that D is a nonempty class of parameterized sets D = {D(t) ∈ P(X) : t ∈ R}, where P(X) denotes the family of all nonempty subsets of X.
Definition 2.2 (Carvalho et al. [15,27]). A process on X is said to be pullback D-asymptotically compact if for any t ∈ R, any D ∈ D and any sequence {τ n } ⊂ (−∞, t] and {x n } ⊂ X satisfying τ n → −∞ and x n ∈ D(τ n ) for all n, the sequence {U (t, τ n )x n } is relatively compact in X. [29]). Let B be a nonempty bounded set in X. The Kuratowski measure of noncompactness is defined by α(B) = inf {δ | B admits a finite cover by sets of diameter δ}.
Definition 2.4 ( Lukaszewicz [29]). A process U (t, τ ) on X is said to be a pullback D-limit-set compact if for any D ∈ D, Definition 2.5 (Anguiano et al. [2]). Let X, Y be two Banach spaces. We say that B = {B(t) : t ∈ R} ⊂ P(Y ) is (X, Y ) pullback D-absorbing for the process U on X if for any t ∈ R and any D ∈ D ⊂ P(X), there exists a τ 0 (t, D) t such that for any τ τ 0 (t, D).
Definition 2.6 (Anguiano et al. [2,15,27]). Let X, Y be two Banach spaces. The family A = {A(t) : A(t) ∈ P(Y ), t ∈ R} is called to be a (X, Y ) pullback D-attractor for the process U (·, ·) if (i) A(t) is closed in X and compact in Y for all t ∈ R; (ii) A is pullback D-attracting in the topology of Y , i.e. lim τ →−∞ dist Y (U (t, τ )D(τ ), A(t)) = 0, for all D ∈ D(⊂ P(X)) and all t ∈ R, We have the following result on the existence of a minimal pullback D-attractor.
Theorem 2.7 ( Lukaszewicz [29]). Let X be a Banach space. Consider a continuous process U : R 2 × X → X, a universe D in P(X), and a family B = {B(t) : t ∈ R} ⊂ P(X) which is (X, X) pullback D-absorbing for U , and assume also that U (t, τ ) is pullback D-limit-set compact in X. Then the family A = {A(t) : t ∈ R} defined by is a minimal (X, X) pullback D-attractor for U . The family A is minimal in the sense that if C = {C(t) : t ∈ R} ⊂ P(X) is a family of closed sets such that for any Let U (t, τ ) be a process on a Banach space X. Denote by K the collection of all complete trajectories of U (t, τ ), that is We also recall the following result of the structure of pullback D-attractors (see [15,27] for more details), which means that each pullback D-attractor contains at least one complete trajectory.
Lemma 2.8 (Carvalho et al. [15,27]). Let U (t, τ ) be a process on a Banach space and consequently, there exists at least one complete trajectoryv of U (t, τ ) that satisfiesv ∈ D.

2.2.
Fractional Sobolev spaces. The main aim of this subsection is to prove the following preliminary result: for any p 2.
At the beginning, we provide the definition of fractional Sobolev spaces and state some preliminary lemmas, we refer readers to [4,20,35] where details can be found. Let S be the usual Schwartz class on R N and S be the space of tempered distributions. The Fourier transformf of a L 1 -function f is given bŷ For f ∈ S , the Fourier transform of f is defined by (f , g) = (f,ĝ) for any g ∈ S.
(see e.g., [1]) (ii) For any s ∈ (0, ∞) ∩ Z c , i.e., s is not an integer, let s = k + σ, where k = max{k s : k ∈ Z} and σ ∈ (0, 1), we define the fractional Sobolev space W s,q (Ω) as follows (see e.g., [20]): endowed with the norm u W s,q (Ω) Note that for any s ∈ (0, ∞), (13) is equivalent to (14) for the case q = 2 and Ω = R N . In fact, we have the following result: Lemma 2.10 (Di Nezza et al. [20]). Let s ∈ (0, 1). Then the fractional Sobolev space H s (R N ) defined in (13) coincides with H s (R N ) defined in (14). In particular, there exists a constant c(N, s), such that, for any u ∈ H s (R N ) Thus we use the notation H s (R N ) to denote W s,q (R N ) for the case q = 2. The following result states that any function in the fractional Sobolev space W s,q (R N ) can be approximated by a sequence of smooth functions with compact support. However, C ∞ 0 (Ω) is not dense in W s,q (Ω) when Ω R N . Lemma 2.11 (Di Nezza et al. [20]). For any s ∈ (0, ∞) and any q ∈ [1, ∞), the space C ∞ 0 (R N ) of smooth functions with compact support is dense in W s,q (R N ). The following result is a Sobolev-type inequality involving the fractional norm · W s,q .
Lemma 2.12 (Di Nezza et al. [20]). Let s ∈ (0, ∞) and q ∈ [1, ∞) be such that sq < N . Let Ω ⊂ R N be a bounded subset with a smooth boundary. Then there exists a positive constant C = C(N, q, s, Ω) such that, for any u ∈ W s,q (Ω), That is, the space W s,q (Ω) is continuously embedded in L r (Ω) for any r ∈ [1, q * ].
If Ω = R N , then W s,q (R N ) is continuously embedded in L r (R N ) for any r ∈ [q, q * ].
If sq > N , then W s,q (Ω) is continuously embedded in C 0,α (Ω), α = s − N q , where Ω is a subset with a smooth boundary or the whole space R N .
Proof. Note that |u + (x) − u + (y)| |u(x) − u(y)| for any x, y ∈ R N . This, together with (14) and (15), implies that u + ∈ H s (R N ) and The following result states that the product of two functions in a fractional Sobolev space is still in this Sobolev space.
We usually consider the action of smooth functions on the space W s,q (R N ). More precisely, if f is a smooth function vanishing at 0, and u is a function of W s,q (R N ), then f • u also belongs to W s,q (R N ). This result reads as follows.
Lemma 2.16 (Bahouri et al. [4]). Let f : R → R be a smooth function vanishing at 0, s be a positive real number and q ∈ Now we are ready to prove the result which is formulated at the beginning of this subsection.
M for any x ∈ R N . Thus, by Lemma 2.16 we know that both (u + ) p−1 and |u| p−1 belong to L ∞ (R N ) ∩ H s (R N ) and satisfying corresponding norm estimates. Note that Finally, by the equality |u| p−2 u = |u| p−1 + 2|u| p−2 u − , we obtain that |u| p−2 u ∈ L ∞ (R N ) ∩ H s (R N ) for any p 2 and the proof is completed.

Useful inequalities.
The following generalized Gronwall's inequality, which is derived by Lukaszewicz [28], is crucial to prove that the process U associated with (1) has pullback D µ -absorbing sets in H s (R N ).
Lemma 2.18 ( Lukaszewicz [28]). Suppose that for some λ > 0, τ ∈ R and for s > τ , where the function y, y , h are assumed to be locally integrable with y, h nonnegetive on the interval t < s < t + r for some t τ . Then t+r t e λs h(s)ds.
We also need the following classical Strook-Varopoulos inequality.
3.1. The well-posedness. In this subsection, motivated by the results of Dlotko et al. in [21], we establish some elementary results of the equation (1), including the existence and uniqueness of both the strong solution and the weak solution, and a L ∞ apriori estimate for the weak solution.
The following result about the existence and uniqueness of the strong solution has been essentially proved by Dlotko et al. [21].
u also satisfies the initial condition u(τ ) = u τ and the mapping u τ → u(t; τ, u τ ) is continuous in L 2 (R N ). In addition, for all v ∈ H s (R N ) ∩ L p (R N ) and for almost every t ∈ (τ, T ), the following identity holds: The following result states the existence and uniqueness of the weak solution of the problem (1)-(2), the proof is based on the approximation.
τ . It is obvious that every strong solution of (1) is also a weak solution, thus and for any v ∈ H s (R N ) ∩ L p (R N ) and almost every t ∈ (τ, T ), By (20), we have Taking (21) and integrating on (τ, t), we obtain that for any t ∈ [τ, T ], This, together with (4) and Cauchy's inequality, yields that . Consequently there exists an element u(t, x) and a subsequence which is still denoted by , it is obvious that w is the solution of the following equation Multiplying (22) by w and integrating on R N × (τ, T ), we obtain which yields that {u (m) } is a Cauchy sequence in C([τ, T ]; L 2 (R N )). Consequently, by the uniqueness of the limit, it follows that Hence, extracting a subsequence if necessary, we deduce that f (u (m) ) → f (u) a.e. in R N × (τ, T ). This, together with the fact that f (u (m) ) is uniformly bounded in L p (τ, T ; L p (R N )), yields that f (u (m) ) → f (u) weakly in L p (τ, T ; L p (R N )). Therefore, by the uniqueness of the weak limit, Φ = f (u). Let m → ∞ in (21), we finally obtain that u is the unique weak solution of the problem (1)-(2) and the proof is completed.
By Lemma 3.4, we can define a continuous process where u(t) is the weak solution to the problem (1)-(2) with u(τ ) = u τ . We will frequently use the technique of approximation in the rest of this paper. Thus it is necessary to deduce the following L ∞ -type result about the weak solution of (1).
We deduce from (4) that there exists a constant s 0 > 0, which depends on M 1 , λ, C i and k i (i = 1, 2), such that f (s) + λs M 1 for any s s 0 and f (s) + λs −M 1 for any s −s 0 .
Then multiplying (1) by ϕ and integrating on (τ, T ) × R N , we obtain Adding both sides of the above equality by − T τ Now we estimate integrals of (24) one by one. Firstly, note that for any convex function φ, we have the point-wise inequality Λ 2s uφ (u) Λ 2s (φ(u)), see e.g. [22]. Let φ(u) = (u − M ) + . Thus φ is convex and the following inequality holds: Secondly, we use the definition of M to obtain and Thirdly, integrating by parts yields Hence, substituting (25)-(28) into (24), we deduce that This implies that u M for almost every (x, t) ∈ R N × [τ, T ]. On the other hand, taking ϕ = (t−T )(u+M ) − , t ∈ [τ, T ], and repeating the above argument, we obtain that u −M for almost every (x, t) ∈ R N × (τ, T ). Therefore u L ∞ (τ,T ;L ∞ (R N )) M and the proof is completed.

3.2.
Existence of a (L 2 , L 2 ) pullback D µ -attractor. In this subsection we prove the existence of (L 2 (R N ), L 2 (R N )) pullback D µ -attractor. Firstly, by using Lemma 2.18, we prove the existence of (L 2 (R N ), H s (R N ) ∩ L p (R N )) pullback D µ -absorbing sets. Then following the idea presented in [40], where autonomous reaction diffusion equations are studied, we establish in addition a more important technical lemma called "Tail estimate". This, together with Rellich-Kondrachov's theorem, implies that the process U (t, τ ) defined in (23) is pullback D µ -limit-set compact. Assume that (4) and (7) hold. Then define It follows that θ > 0. Now let µ ∈ (0, θ) and let g(t) satisfy the following assumption t −∞ e µs g(s) 2 L 2 (R N ) ds < +∞, for all t ∈ R.
Proof. Taking u as a test function in (19) and using (4), we obtain that for almost every t ∈ (τ, ∞), and this implies that for a.e. t ∈ (τ, ∞), Multiplying (32) by e µt , where µ comes from (30), it follows that Integrating the above inequality with respect to t on the interval (τ, t) and using µ < λ − k 1 , we have in particular On the other hand, taking ∂ t u as a test function in (19) and using (3), we deduce that Combining (32) and (34), it yields that This, together with the fact that implies that Let It follows from (6) and (8) that Note that µ ∈ (0, θ) and Then, by (36) and (37), we have that Consequently, by Lemma 2.18, we have H(u(t + r)) e − 1 2 µr 2 r Now it suffices to estimate the first term on the right side of the inequality (39), and we establish this as follows. Combining (32) and (35), we see that Integrating the above inequality with respect to time on (t, t + r 2 ) and observing that where (33) is used to prove the third inequality. Thus the estimate of the first term on the right hand side of (39) is obtained. Substituting (40) into (39), it follows that there exists a constant τ 0 = τ 0 (t, D) < t such that H(u(t + r)) for any u(τ ) ∈ D(τ ) and τ < τ 0 . Let t = t − 1, r = 1. Then the above inequality implies that Substituting (37) into the above inequality, we obtain that for any τ < τ 0 and u(τ ) ∈ D(τ ), We denote by B = {B(t) : t ∈ R} the closed ball in phase space H s (R N ) ∩ L p (R N ) with center 0 and radius (R(t)) Therefore B = {B(t) : t ∈ R} is a class of pullback D µ -absorbing sets in H s (R N ) ∩ L p (R N ) and the proof is completed.

Lemma 3.8 (Tail estimate). Assume that
where B k = {x ∈ R N : |x| < k}, (B k ) c is the complement of B k in R N and U (t, τ )u τ is the weak solution of (1).
Remark 1. The approach presented in the proof of Theorem 1.1 is rather classical. We point out that the pre-compactness of U (t, τ ) can be also obtained by the energy method. We refer the interested reader to J. M. Ball [5] for more details on the energy method.
4. The pullback attraction in L 2+δ (R N )-norm. In this section, we study the properties of the pullback attractor obtained in Theorem 1.1. In fact, motivated by the results in [13,33], we prove that for any δ > 0, the (L 2 , L 2 ) pullback D µattractor obtained in Theorem 1.1 indeed attract the D µ -class in L 2+δ (R N )-norm. At the beginning, we establish a apriori estimate, which is called the higher-order integrability, for the difference of solutions near the initial time.

4.1.
Higher-order integrability near the initial time.
Lemma 4.1. Let u i be the weak solution of the problem (1)-(5) corresponding to the initial data u i τ (i = 1, 2). We denote by w the difference of u i , that is, Then for any T > τ and any k = 1, 2, · · · , there exists a positive constant M k = M (T − τ, N, l, k, λ, s) w(τ ) Remark 2. This result states that for any k ∈ Z + , the difference of two solutions is decay in L 2( N N −2s ) k (R N )-norm near the initial time. Proof.
and for i = 1, 2, where u i is the weak solution of the problem (1)-(5) corresponding to the initial data u i τ and forcing term g(t).
Step 2. A k and B k are both true for approximate solutions.
We prove that A k and B k are both true for w m for all m ∈ Z + . Let w m (t) = u 1 m (t) − u 2 m (t). We deduce from (50) and Lemma 2.17 that for any θ 2, the element |w m (t)| θ−2 w m (t) belongs to L ∞ (R N ) ∩ H s (R N ) for almost every t ∈ (τ, T ). It is obvious that w m (t) is the solution of the following equation We prove the result by induction on k.
Case 1. k = 1. Firstly, multiplying (52) by w and integrating over R N , using (5) and definition of H s (R N ) in Section 2.3, we obtain that where c λ = min{1, λ} > 0. Integrating the above inequality on (τ, t), it follows that By Lemma 2.12, we have Secondly, we multiply (52) by |w m | 2N N −2s −2 · w m and then integrate on R N . By Lemma 2.19 and (5), it follows that for a.e. t ∈ (τ, T ) (here we let c i (i = 1, 2, 3, 4) denote positive constants depending on λ, s and the spatial dimension N . Note that these c i may be vary from line to line), . (54) Multiplying (54) by (t − τ ) b1 2N N −2s , and combining with the fact that d dt Multiplying by t − τ , for a.e. t ∈ (τ, T ), we see that .

2N
N −2s , and thus for a.e. t ∈ (τ, T ) where we use (57) to prove the last inequality. Integrating the above inequality on [τ, T ], it follows that where (57) is used again in the proof of the second inequality. Hence it follows from the Sobolev inequality that Take M 1m to be max{M 1m , c λ,s,N C T −τ,N,s M 1m }. Then the proof of (A 1 ) and (B 1 ) is completed by (57) and (58).
In order to prove (B k+1 ), multiplying (60) by (t − τ ) 1+ N N −2s , it follows that Substituting (62) into (63), we have that for a.e. t ∈ [τ, T ], Integrating the above inequality on [τ, T ] and using (62) again, we have that Then applying the Sobolev embedding once more, we obtain that Therefore, combining Case 1 and Case 2, we obtain that A k and B k are both true for approximate solutions. That is, for any T > τ and any k = 1, 2, · · · , there exists a positive constant M km = M (T − τ, N, l, k, λ, s) w m (τ ) where b k are given in (48).
Step 3. Estimates for weak solutions. By (49), we have Furthermore, it follows from (51) that there exists a subsequence {w mj } and still denoted by {w m } without lose of generality, such that Therefore, by passing to the limit for (A km ) and (B km ), and then using Fatou' lemma, (65) and (66), we obtain that Hence the proof of Lemma 4.1 is completed.
Remark 3. Note that the L ∞ -estimate of w(t) can be also obtained. In fact, the monotone increasing sequence {b k } convergent tob ∈ R. On the other hand, there exists a constant c = c(T − τ, N, l, λ, s) such that for every t ∈ (τ, T ).

4.2.
The (L 2 (R N ), L 2+δ (R N )) pullback attraction. In the following subsection, the attraction of the (L 2 , L 2 ) pullback D µ -attractor can be proved to be L 2+δ -norm for any δ ∈ [0, ∞). Note that generally we do not know whether the pullback D µattractor A or U (t, τ )D(τ ) belongs to L 2+δ (R N ) or not. However, because of the estimates about decay rate of solutions in L 2( N N −2s ) k+1 -norm given by Lemma 4.1, the difference of any two elements which derived from D(τ ) and A(τ ) respectively will belongs to L 2+δ (R N ). By interpolation, there exists a constant γ ∈ (0, 1) such that for any u 1 , u 2 ∈ L 2( N N −2s ) k 0 (R N ), Using the definition of pullback attractors, we deduce that for each t ∈ R the section A(t) is compact in L 2 (R N ), and so is bounded in L 2 (R N ). We denote by B(t) the 1-neighborhood of A(t) under the L 2 (R N )-norm, then B(t) is bounded in L 2 (R N ). For any fixed t < T , applying Lemma 4.1 to the case of τ = t − 1 and k = k 0 , it follows that there exists a positive constant M k0 , which depends only on k 0 , N , l, c λ,s,N and B(t − 1) L 2 (R N ) , such that For the above t, M k1 and for any ε > 0, by the definition of pullback attractors again, we deduce that for any D = {D(t) : t ∈ R} ∈ D µ , there is a time τ 0 (< t − 1) depending only on t, ε > 0 and D, such that and Then it is a immediate consequence of (67)-(70) that for each τ τ 0 , By the arbitration of ε, the proof is completed.

5.
Dynamics in H s (R N ). In this section, we study the dynamics of the equation (1) in H s (R N ). Firstly, without any restriction on spatial dimension N and growthindex p of nonlinear term f , we show that the solution of (1) is continuous in H s (R N ) with respect to the initial data (Lemma 5.2). Then we prove that for any δ > 0, the pullback D µ -attractor obtained in Theorem 1.1 attract the D µ -class in the norm of H s (R N ) (Theorem 1.3). As a corollary, we obtain the existence of a (L 2 (R N ), H s (R N )) pullback D µ -attractor (Remark 5).
Proof. First of all, multiplying (1) by u and integrating with respect to x on R N , it follows that for a.e. t ∈ (τ, T ), This, together with (4) and Cauchy's inequality, implies that for any t ∈ [τ, T ], Integrating with respect to time on (τ, t), thus we obtain that for any t ∈ [τ, T ], Claim. there exists a fixed time t 0 ∈ [τ, τ +T 2 ] such that u(t 0 ) ∈ L p (R N ) and We prove the Claim by contradiction. If such t 0 does not exist, then This yields that which contradicts the inequality (73). Thus the Claim is proved. Now multiplying (1) by |u| p−2 u and integrating with respect to x on R N , it follows that for a.e. t ∈ (τ, T ), Using (18), (4) and Cauchy's inequality, we deduce that for a.e. t ∈ (τ, T ), For any t ∈ [ τ +T 2 , T ], integrating (75) with respect to time on (t 0 , t), we deduce that where we use (74) to prove the last inequality. Therefore for all t ∈ [ τ +T 2 , T ] which implies (71)-(72) immediately.

5.2.
Continuity in H s (R N ) with respect to the initial data. At the beginning, we recall some previous results on the continuity of reaction-diffusion equations in H 1 . Robinson [32] proved the continuity of solutions in H 1 0 (Ω) only for the case of N 2 and the case of N = 3 with a additional growth restriction (p 4) on the nonlinear term f . In 2008 Trujillo and Wang [36] reported a positive answer for other situations, i.e. any spatial dimension N and any growth condition on f .
The proof of Trujillo and Wang [36] essentially depends on differentiating the equation with respect to t in order to get a nice apriori estimate for ∂ t u. However, the method of Trujillo and Wang [36] is inapplicable to our non-autonomous equations (1) because g(t) only satisfies some L 2 integrability ((3), (9)). We cannot differentiate (1) with respect to t to get nice apriori estimate for ∂ t u.
In this subsection, based on Lemma 4.1, we provide a method to prove the continuity of the solution of (1) in H s (R N ). To do this, we assume further that the nonlinear term f satisfies f ∈ C 1 (R) and there exists a positive C f such that for any s ∈ R, Lemma 5.2. Assume that (3)-(5), (78) hold. Then for any τ ∈ R and any t > τ , where U (t, τ )u nτ and U (t, τ )u τ are the weak solutions corresponding to initial data u nτ and u τ respectively. More precisely, the following estimate holds: where the constant ϑ ∈ (0, 1) is the exponent of the interpolation · L 2p−2 (R N ) · ϑ L 2 (R N ) with some α ∈ N satisfying 2( N N −2s ) α > 2p − 2; the constant L 1 depends only on t − τ, N, p, l, ϑ, λ, s, and the constant L 2 depends only on l, ϑ, λ, s, u nτ L 2 (R N ) , u τ L 2 (R N ) , g L 2 ((τ,T )×R N ) , t − τ, N , C 1 , k 1 , p.
Proof. Let u n (t) = U (t, τ )u nτ and u(t) = U (t, τ )u τ be the solutions to the problem (1)-(2) with initial data u nτ and u τ respectively. Define w n (t) = U (t, τ )u nτ − U (t, τ )u τ . Then w n (t) is a solution of the following equation Multiplying the equation (81) by ∂ t w n and integrating on R N , it follows that where C λ = min{1, λ} and ρ(t) ∈ (0, 1). Hence where c = c(C λ , C f ). Since the result is obvious when p = 2, we suppose that p > 2 in the rest of the proof. Let α be the unique integer in N ∩ (log N N −2s (p − 1), 1 + log N N −2s (p − 1)] such that 2( N N − 2s ) α > 2p − 2.
Remark 4. We point out that the continuity of (1) in H s (R N ) can be also obtained by the energy method. And the additional assumption (78) might be unnecessary.
5.3. The (L 2 (R N ), H s (R N )) pullback attraction. In this subsection, we prove that the (L 2 , L 2 ) pullback D µ -attractor A obtained in Theorems 1.1 attract the class of D µ in H s (R N )-norm. The proof is based on Lemma 5.2.
Proof of Theorem 1.3. Let t ∈ R. We denote by B(t) the 1-neighborhood of A(t) under the L 2 -norm, thus we have that B(t) is bounded in L 2 . Applying Lemma 5.2 (especially the estimate (80)) to the case of τ = t − 1 and u τ , v τ ∈ B(t − 1), we know that there exist two constants L 1 , L 2 satisfying Since A is the (L 2 , L 2 ) pullback D µ -attractor, for any ε > 0 and any D = {D(t) : t ∈ R} ∈ D µ , there is a time τ 1 (< t − 1) depending only on t, ε and D, such that and dist L 2 (R N ) (U (t − 1, τ )D(τ ), A(t − 1)) ε for all τ τ 1 .
Therefore the result follows from the arbitration of ε and D.
Remark 5. By Lemma 2.8, Lemma 5.2 and Theorem 1.3, we see that A(t) − y t is compact in H s (R N ), where y t be an arbitrary point in A(t). Furthermore, it follows from Lemma 3.7 that A(t) is also a (L 2 (R N ), H s (R N )) pullback D µ -absorbing set. Hence it is easy to verify that the (L 2 (R N ), L 2 (R N )) pullback D µ -attractor A is also a (L 2 (R N ), H s (R N )) pullback D µ -attractor.