Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces

In this paper, we first prove the well-posedness for the nonautonomous reaction-diffusion equations with fractional diffusion in the locally uniform spaces framework. Under very minimal assumptions, then we study the asymptotic behavior of solutions of such equation and show the existence of \begin{document}$(H^{2(\alpha -e),q}_U(\mathbb{R}^N),H^{2(\alpha -e),q}_φ(\mathbb{R}^N))(0 -uniform(w.r.t. \begin{document}$g∈\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)$\end{document} ) attractor \begin{document}$\mathcal{A}_{\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)}$\end{document} with locally uniform external forces being translation uniform bounded but not translation compact in \begin{document}$L_b^p(\mathbb{R};L^q_U(\mathbb{R}^N)).$\end{document} The key to that extensions is a new the space-time estimates in locally uniform spaces for the linear fractional power dissipative equation.

1. Introduction. In this paper, we consider the long-time dynamical behavior of solutions for the following non-autonomous reaction-diffusion equations with fractional diffusion on R N : with the initial condition where α > 0 is a parameter and the fractional Laplacian (−∆) α is defined through the Fourier transform f : R N × R → R is a suitable continuous function and g is given the external force. Definition (1.3)allows for a wider range of parameters α. The interval of interest for fractional diffusion is 0 < α < 1, and for 0 < α < 1, the fractional Laplacian operator can also use the integral representation where P.V. stands for principal value and C N,2α is a normalization constant, with precise value C N,2α = 2 2α αΓ((N + 2α)/2)/(π s → 1 it is possible to recover respectively the identity or the standard minus Laplacian, −∆, cf. [29,37]. The fractional Laplacian operators of the form (−∆) α , α ∈ (0, 1), are actually the infinitesimal generators of stable Lévy processes, see, e.g. [10,36].
Our goal is to prove that under suitable conditions there exist the uniform attractor of reaction-diffusion equation with the fractional Laplacian and forcing term g(x, t) being translation uniform bounded but not translation compact in an appropriate sense.
The nonlinear term f is assumed to satisfy the following assumptions: Assumption I. Assume that f satisfies f (x, 0) = 0 and there exist constants C and ρ with C > 0, ρ > 1 such that Assumption II. There exist two positive constants µ 0 , µ 2 such that −µ 0 s 2 − µ 2 |s| ≤ sf (x, s), for all s ∈ R, x ∈ R N .
We furthermore assume that the external force g(·, t) ∈L q U (R N ) with 1 < q < N for almost every t ∈ R and g has finite norm in the space L p b (R,L q U (R N )) with p > 2, i.e., The main contribution of the present paper is to investigate the long-time behavior of solutions to non-autonomous reaction-diffusion equations associated with the fractional Laplace operator and to extend the external force to the case where g(x, t) is considered in much larger space L p b (R,L q U (R N )) and with weaker assumptions.
The long-time behavior of the solutions of (1.1) is of great current interest. It is well-known that this behavior of solutions of such equations with α = 1 arise from mathematical physics can be described as the existence of the so-called attractors of the corresponding semigroups(or process). In particular, when the domain is bounded, the global attractor of such problems have been extensively studied by using many different methods in the literature(see, e.g., [6,14,21,25,33,35,39,45] and the references therein).
In contrast to this, the case of unbounded domain becomes much more difficulty and some of methods for bounded domains are on longer valid. The main difficulty lies in the absence of the standard Sobolev compact embedding.
On the other hand, the disadvantage of the standard Lebesgue spaces L p1 (R N )(1 ≤ p 1 < ∞) is that it does not consider the behavior of the solutions for large spatial values and the family of such spaces are not nested. In order to overcome these difficulties, one should introduce an appropriate functional spaces to study these partial differential equations. With respect to the use of weighted Sobolev spaces, one of the pioneer works is in [7], where Babin and Vishik for the first time introduced weighted Sobolev space as the phase space and showed, under appropriate assumptions, the existence of global attractors for parabolic type evolutionary equations on unbounded domain, however, which requires the initial data and forcing term belonging to the corresponding weighted spaces(see also [1]). Later on, in [5], the authors also proved the compactness of the nonlinear semigroup and the existence of global attractor in weighted Sobolev spaces. It is worth noting that the usual Sobolev spaces L p1 (R N )(1 ≤ p 1 < ∞) do not include the constant solutions and traveling wave connections between two equilibria. For reaction-diffusion equations u t = ∆u + u − u 3 , the solutions u = ±1 and the traveling wave connections between u = 0 and u = ±1 are no longer included the Sobolev spaces like L p1 (R N )(1 ≤ p 1 < ∞), for example, see [23]. Hence, in [3,4,8,20,[41][42][43], the authors introduced locally uniform spaces as the phase space to include these special solutions in the global attractors. The idea of the locally uniform spaces can be traced back to Kato [24]. In [3], J. Arrieta et al. systematically study some properties and embedding relationship of locally uniform spaces. These spaces enjoys suitable nesting properties(e.g., if p 1 ≤ q 1 , then L q1 U (R N ) → L p1 U (R N )) and have locally compact embeddings.
We should mention that the methods introduced by Arrieta, Moya and Rodríguez-Bernal in [5] will play a crucial role in our work. In [5], by using weighted Sobolev spaces, the authors systematically study the properties of these spaces, including Sobolev type embeddings and the weighted L p − L q estimates for heat kernel. Then the authors prove the global existence and regularity of the following reactiondiffusion equations in the space L q ρ (R N ) with 1 < q < ∞, and construct, under quite general assumptions with respect to f the existence of global attractor. In particular, the authors in [16] established a global attractor A and showed that A is contained in an ordered interval [φ m , φ M ], where φ m , φ M ∈ A is a pair of stationary solutions. We also refer to [17]. In [26], the authors studied asymptotic behavior of solutions of non-autonomous parabolic equations with singular initial data in the bounded domain of R N . In [38], by using the concept of comparison of concentrations, the authors establish symmetrization results for the solutions of the linear fractional diffusion equation and its elliptic counterpart and moreover, Those results extend to the nonlinear version. Recently, Dlotko, Kania, and Sun [19] first studied the parabolic type equations with the fractional Laplacian operator in locally uniform spaces by the semigroup method and they were able to establish the global solvability of such problem and prove the existence of a global attractor by proving a Maximum principle. The motivation of this paper is to give a detailed study of the uniform attractor of reaction-diffusion equation with the fractional Laplacian operator and forcing term g(x, t) being translation uniform bounded but not translation compact in ). The rest of this paper is organized as follows. In section 2 we include some preliminaries concerning locally uniform spaces and uniform Bessel spaces, their embeddings, basic notations of non-autonomous dynamical systems. In section 3 we consider the well-posedness of the linear non-autonomous fractional order reactiondiffusion equations, i.e., f (x, u) = −µ 0 u − µ 2 , and prove a comparison principle in locally uniform spaces which plays an important role in obtaining the existence of uniform attractor. In section 4 we prove the existence and uniqueness of solutions for problem (1.1)-(1.2). In section 5, we consider the global well-posedness of problem (1.1)-(1.2) in H 2(α− ),q U (R N ) by a comparison principle. Motivated by ideas of [15,20,41,43](see also, for instance, [40,42,44]), we obtain continuous property of a family of process in H 2(α− ),q φ (R N ) by extension solutions obtained in section 4. In section 6, we prove Theorem 1.1. By introducing the uniform normal forcing we show the asymptotic compactness properties of the family of processes and obtain the existence and structure of (H , in fact, we obtain the attracting property in the topology of H 2(α− ),q φ (R N ), 0 < < α < 1. This paper is complemented by some properties of uniform normal functions, stated in appendix A.
Throughout this paper we will use C to denote positive constants that are independent of the parameter α. They may change from line to line or/and depend on the weight function. In particular, when a constant C depends on some particular parameters, say a, b, c, we shall denote it by C a,b,c . Now we are in a position to formulate the main result of our paper.
Theorem 1.1. Assume that f (x, u) satisfies assumptions I and II and the external force iv) the following formula holds: ∀τ ∈ R and B α is the bounded uniformly(w.r.t.g ∈ H L q U (R N ) (g 0 )) absorbing set defined as in (5.16).

2.
Preliminaries. For completeness, we will recall some functional spaces which will be used throughout of the paper. For more details, see, e.g., [3,4,8,20,43]. Definition 2.1. A function φ : R N → (0, +∞) is said to be strictly positive integrable weighted function of class C 2 (R N ) if for two positive constants φ 0 , φ 1 , and x ∈ R N , j, k = 1, · · · , N, the following three conditions are satisfied: In particular, according to the above definition, we can choose as a strictly positive integrable weighted function of class C 2 (R N ) satisfying the estimates |∇φ| ≤ C √ 0 φ and |∆φ| ≤ C 0 φ, and the integral property holds obviously. This weight function has the following properties: Proposition 2.2. There exist positive constants C 1 , C 2 such that Thus, similar to Proposition 1.2 of [44], it follows Proposition 2.3. There exist positive constants C 1 , C 2 such that for all u ∈ L p φ (R N ) with 1 ≤ p < ∞, For a weight function φ(x) defined as like (2.1), we define the weighted Sobolev spaces as follows, see also [43].
with the norm ii) For p = ∞, define the weighted Sobolev space with weight φ(x) by Analogously, the weighted Sobolev space W k,p φ (R N ), k ∈ N is defined as the space of distribution whose derivatives up to the order k inclusively belong to L p φ (R N ), with the norm We define also, for 1 ≤ p < ∞, the locally uniform space L p U (R N ) as the set of functions u ∈ L p loc (R N ) such that where B(x, 1) denote the unit ball centered at x, with norm In a similar way, for p = ∞, we have L ∞ From the definition we can easily know that L p U (R N ) contains L ∞ (R N ), L r (R N ) and L r U (R N ) for any r ≥ p. In order to consider the non-autonomous reaction diffusion equations in much larger spaces with initial data in the uniform space L q U (R N ) for 1 ≤ q ≤ ∞, we denote byL q U (R N ) the closed subspace of L q U (R N ) consisting of all elements which are translation continuous with respect to · L q U (R N ) under the action of the group of translations {τ y , y ∈ R N } by τ y φ(·) := φ(· − y), that is → 0 as |y| → 0, and the uniform spaces W m,p U (R N ) andẆ m,p U (R N ) can be defined, respectively, by L p U (R N ) andL p U (R N ) in a similar way as the definition of W k,p φ (R N ). We recall the locally uniform spaces. For 1 ≤ p < ∞, define the weighted Sobolev space with weight φ(x) as and its the closed subspacė , and the definition of W k,p lu (R N ) andẆ k,p lu (R N ) can be carry out by using the standard way.
If the weight function φ satisfies (2.1), then we know from [3] there exist positive constants C 1 , C 2 such that the following norms are equivalent: This equivalent norms imply that for k ∈ N ∪ {0}, the spaces W k,p U (R N ) and W k,p lu (R N ) coincide algebraically and topologically when the weight function φ satisfies (2.1), and we recall some well-known embeddings of these spaces: is continuous. Moreover, assume that 1 < p, p 1 , p 2 < ∞, 1 then there is a C > 0 such that one has . For the fractional power operators (−∆ + I) α , one has from [3] Lemma 2.6. Let 0 ≤ α ≤ 1. Then there exist positive constants C 1,α and C 2,α such that ChoosingL q U (R N ) with 1 < q < ∞ as a base space, the unbounded linear opera- for k ∈ N and then denote byḢ k,p the Bessel potentials spaces which coincide with the uniform spacesẆ s,q U (R N ) for integer s if 1 < q < ∞ or for all s if q = 2. By using the complex interpolation-extrapolation procedure, one can construct the locally uniform spaces associated to the fractional power operator −(−∆) α , which will be denotedḢ 2α,q Note that there is no results on the negative part of the scale of uniform Bessel spaces when α < 0. Moreover, the locally uniform spaces are given byḢ 2α,q U (R N ) which satisfy the following sharp embedding of Bessel spaces: Thus, for some constant λ, the operator −(−∆) α − λI generates an analytic semigroup e (−(−∆) α −λI)t in each uniform spacesḢ 2α,q U (R N ) with 0 < α < 1. We now state the singular Gronwall lemma used below(cf Lemma 7.1.1 in [23]).
where a and c are positive constants, M ≥ 0, 0 < τ ≤ ∞ and r > 0. Then there exists a positive constant C r,c such that Here, for our purpose, we only recall a special case of the Gronwall-Henry inequality, a general case and detailed proof of this inequality can be found in Henry [23], Lemma 7.1.1.
To proceed with our investigation we introduce the concept of attractor of our interest. The definitions we state below are taken from [9,12,13].
We now consider a family of equations of the following abstract form with symbols σ(t) from the hull H(σ 0 ) of the symbol σ 0 (t): For simplicity, we assume that the set H(σ 0 ) is a complete metric space. We suppose that, for every symbol σ ∈ H(σ 0 ), the Cauchy problem (2.5) has a unique solution for any τ ∈ R and for every initial condition u τ ∈ E. Thus, we have the family of processes {U σ (t, τ )}, σ ∈ H(σ 0 ) acting in the Banach space E. Then, the solution u(t) ∈ E of the problem (2.5) can be represented as By the unique solvability of problem (2.5), the operators {U σ (t, τ )} posses the following multiplicative properties: We note that the following translation identity holds for the family of processes {U σ (t, τ )}, σ ∈ H(σ 0 ) corresponding to (2.5): Further straightforward properties can be find in [21]. The family of processes {U σ (t, τ )}, σ ∈ H(σ 0 ) is said to be uniformly(w.r.t. σ ∈ H(σ 0 ))ω-limit compact if for any τ ∈ R and B ∈ B(E), B t := ∪ σ∈H(σ0) ∪ s≥t U σ (s, τ )B is bounded for every t and lim t→∞ κ(B t ) = 0. A set B 0 ⊂ E is said to be uniformly(w.r.t. σ ∈ H(σ 0 )) absorbing for the family of processes {U σ (t, τ )}, σ ∈ H(σ 0 ) if for any τ ∈ R and B ∈ B(E), A set P ⊂ E is said to be uniformly(w.r.t. σ ∈ H(σ 0 )) attracting for the family of processes {U σ (t, τ )}, σ ∈ H(σ 0 ) if for every set B ∈ B(E) and an arbitrary fixed We now recall the notion of the uniform attractor A H(σ0) .
To describe the general structure of the uniform global attractor of the family of processes, we need the notion of the kernel of the process that generalizes the notion of a kernel of a semigroup. A function u(s), s ∈ R with values in E is said to be a complete trajectory of the process Definition 2.9. The kernel K σ of the process {U σ (t, τ )} is the family of all bounded complete trajectories of this process: is called the kernel section at time t.
The next lemma gives a space-time estimate for the linear heat equation with the fractional Laplacian in the L p (R N ) space.
dξ be a kernel function. Then we have the following point-wise estimate, with 0 < t < ∞ and 1 ≤ p ≤ ∞.
Similar to the above lemma, let we also have . We first consider the linear fractional power dissipative equations in the space L p U (R N ). Observe that the norm of the solution is estimated in locally uniform space.
We have, for a certain constant C and t > 0, Proof. We give the proof of the first one only. By the Fourier transform, e −t(−∆) α ϕ(x) can be written as: where, in the last equality we used scaling. Thus, it follows from the Young inequality and Lemma 2.10 with 1 + 1 p = 1 q + 1 r , from which and the definition of the locally uniform space, we complete the proof of this lemma.
3. Linear problem. This section is devoted to study the following linear nonautonomous reaction diffusion equations of the type (1.1)-(1.2), ). Our main result in this section is the following well-posedness theorem concerning the linear problem in the locally uniform spaces.
Then for any interval [τ, T ], problem (3.1) with the initial data u τ is well defined and there exists a unique mild solution u(t), which is given by the variations of constants formula where uτ converges to u τ in the norm ofḢ 2(α− ),q U (R N ) asτ → τ, and τ <τ ≤ t ≤t, and satisfies

3)
and for every τ < t < T, there exist positive constant c 1 and c 2 such that

4)
where the constants c 1 depends only on T and c 2 depends on T, τ and p.
Proof. The existence of solutions can be obtained in a standard way(Theorem 3.3.3 in [23], page 54). We deduce only a priori estimate (3.4) for the solutions of the problem (3.1). By using (4.2) and Hölder inequality we obtain from (3.2) which gives the estimate (3.4). Based on this result, we will prove a comparison principle for the non-autonomous reaction diffusion equations in the locally uniform spaces that allows us to obtain the global existence of solutions to (1.1)-(1.2).
) and let the following equality be hold a.e. in R N × [τ, T ], Assume that the nonlinear terms f i (x, u i ), i = 1, 2, satisfy assumptions I and II and let f 1 (x, u 1 ) ≤ f 2 (x, u 2 ). Then for any two initial values u τ1 , u τ2 ∈Ḣ Proof. The existence and uniqueness of solutions of problem (3.8) can be obtained from the standard Theorem 3.3.3 in [23], page 54 in the base space X =L q U (R N ), for the detail of proof we refer to the proof of Theorem 4.2 in the next section. The corresponding solution can be given by the variation of constants formula Since the operator e −(−∆) α t is order preserving inL q U (R N ), which follows from Proposition 5.3 in [3], then the right hand side of (3.7) preserve the ordering. Hence we can obtain u 1 (t, τ ; u τ1 ) ≤ u 2 (t, τ ; u τ2 ), t ∈ [τ, T ].
Similar to the arguments in [5], from the above lemma we know that which is a solution of the following equations it follows from the property of order preserving of the semigroup e (−(−∆) α +µ0I)t in locally uniform spacesL q U (R N ), which implies U (t, τ ; |u τ |) is a supersolution and it follows from (3.8) A similar argument can be applied to −U (t, τ ; |u τ |), which shows that −U (t, τ ; |u τ |) ≤ u(t, τ ; u τ ).
Hence it follows immediately that (3.9) 4. Well-posedness. In order to initiate the discussion let us recall the definition of a mild solution.
Definition 4.1. We say that u : is a mild solution to the problem (1.1) with the initial data u τ ∈Ḣ where uτ converges to u τ in the norm ofḢ It is well known from [19] that the operator (−∆) α generates a strongly continuous analytic semigroup with the domainḢ 2α,q U (R N ) and by lemma 2.11, e −(−∆) α t is uniformly bounded, that is, The main properties of the Nemitski operator associated with f are included in the following lemma, which will be used to obtain the well-posedness of solutions. Lemma 4.3. Assume that f (x, u) satisfies Assumption I. Then there exists a positive constant C such that the following estimates hold: ), Proof. For any u 1 , u 2 ∈Ḣ 2(α− ),q U (R N ) and y ∈ R N , we have In the ball B(z, 1), by using the sharp embeddings of Bessel spaces: if 1 < q < N 2(α− ) and 1 q (B(z,1)) . Inserting this estimate into (4.6), it follows from the definition of the locally uniform spaces that the conclusions of this lemma holds true.
Proof of Theorem 4.2. We will use Theorem 3.3.3 in [23], page 54 in the base space X =L q U (R N ). It is well known from [19] that the operator (−∆) α with domainḢ 2α,q U (R N ) is a sectorial operator inL q U (R N ) for 1 < q < ∞. Since g 0 ∈ L p b (R,L q U (R N )) with 1 < q < ∞, it is enough to prove that f :Ḣ . Thus, we obtain the local existence of solutions to problem (1.1)-(1.2), that is, there exists a timet > 0 such that and satisfies

GAOCHENG YUE
Hence we can obtain from (3.9) and Theorem 3.1, where the constants c 1 depends only on T and c 2 depends on T, τ and p. Now, it follows from the uniform boundedness of e (−(−∆) α +µ0)t in (4.2) and the formula (4.4) of Lemma 4.3 that, for some ξ 0 > 0, , 1)), we can especially choose r = q and r = ρq in (4.8), and obtain Thus, according to Theorem 3.1 and plugging (4.10) into (4.9) it yields, for τ ≤ t ≤ T < ∞, Hence, the solution is global inḢ 2(α− ),q U (R N ). Last, the proof of (4.3), that is, continuous dependence of solutions with respect to the initial data and external force, is completely similar to the proof of (5.5) in Theorem 5.4. So, we here omit details. For the symbol of the original equation (1.1) g 0 (t) := g 0 (x, t) ∈ L p b (R, L q U (R N )), consider the translation group {T (h), h ∈ R} acting by the formula T (h)g 0 (t) = g 0 (t + h), h ∈ R and denote by The resulting family of symbols σ 0 forms the hull of the original symbol g 0 (t) that is the closure of σ 0 in the space L p,w loc (R; L q U (R N )), which is the subspace of L p loc (R; L q U (R N )) equipped with the local weak convergence topology. Similar to Proposition 2.3 of [12], for any g(t) belongs to the hull  In the sequel we will require the following known results, which can be found in [3,31] and [40], respectively.
. In the following theorem, we will prove the existence, uniqueness of global weak solutions and their continuous dependence with respect to the initial data in the H Theorem 5.4. Suppose that the nonlinearity f satisfies assumptions I and II and the external force g 0 ∈ L p b (R, L q U (R N )) with 1 < q < N 2(α− ) and ρ ≤  a solution of (1.1)-(1.2).
Proof. We will split the proof into two steps.
Step 1. (Continuity) We begin with continuous dependence on initial conditions in the space H 2(α− ),q φ (R N ) for the solution obtained by Theorem 4.2. To this end, let u i (t), i = 1, 2 be the solution of problem (1.1) with the initial data u i τ belongings toḢ 2(α− ),q U (R N ) and associated with the external force g i ∈ L p b (R,L q U (R N )), and for convenience, set w(t) = u 1 (t) − u 2 (t) and f i (s) = f (x, u i (s)), exploiting the variation of constant formula we get Thus, in a similar way as in the proof of Theorem 4.2, it follows for some θ 2 > 0 and t ∈ [τ, T ], Now, we estimate the difference of the nonlinearity f i (s), i = 1, 2, in the weighted space L q φ (R N ). Similar to (4.6), it fields where we used Sobolev type inclusion H 2(α− ),q φ (R N ) → L q φ (R N ) for q < N 2(α− ) and a weight function given as in Definition 2.1.
Then, by substituting (5.3) into the right hand side of inequality (5.2), one obtains from Lemma 2.7 that there exists a positive constantC = C M,C T −τ ,T, u 1 ,µ0 such that . The above two estimates indicate that if the initial conditions u τ belongs to a bounded set ofḢ 2(α− ),q U (R N ) and the external force g belongs to a bounded set of L p b (R,L q U (R N )), then the solution u(t) of problem (1.1) is continuous dependence on initial data in the topology of weighted(resp. uniform) spaces H with the corresponding external force belonging to L p b (R, L q φ (R N ))(resp. L p b (R, L q U (R N ))), uniformly with respect to t on any bounded subintervals of [0, ∞).
Step 2. (Existence) As a consequence of Theorem 4.2, one can represent smooth global solutions given by this theorem as U gn (t, τ )u n τ = u n (t), t ≥ τ, τ ∈ R, where the initial data u n τ ∈Ḣ 2(α− ),q U (R N ) and the external force g n ∈ HLq U (R N ) (g 0 ), n = 1, 2, · · · . In addition, from (4.3) we know that { U gn (t, τ )}, g n ∈ HLq On the other hand, from Lemma 5.3 we have that for each u τ ∈ H (R N )-norm and g n → g in the L p b (R; L q φ (R N ))-norm. Thus, for each u τ ∈ H 2(α− ),q U (R N ) and g ∈ H L q U (R N ) (g 0 ), we can define the following limit of the process { U gn (t, τ )} : (5.6) It follows from Theorem 4.2 and Lemma 5.2 that the process U g (t, τ )u τ belongs to H 2(α− ),q U (R N ). Since the estimates of continuity (5.4) and (5.5), U g (t, τ )u τ does not depend on the choice of {u n τ } ∞ n=1 and {g n } ∞ n=1 , and is the limit of U gn (t, τ )u n τ in the space C([τ, T ], H 2(α− ),q φ (R N )), where T > τ, τ ∈ R. {U g (t, τ )}, g ∈ H L q U (R N ) (g 0 ) forms the family of the process of the space H 2(α− ),q U (R N ) and is (H Therefore, U g (t, τ )u τ = u(t) is a unique global weak solution of (1.1)-(1.2) associated with the initial data u τ ∈ H 2(α− ),q U (R N ) and the corresponding external force g ∈ H L q U (R N ) (g 0 ). We are now able to state our result on dissipativity of the non-autonomous reaction-diffusion equations in the locally uniform space H 2(α− ),q U (R N ).
Suppose that the analytic semigroup e (−(−∆) α +µ0I)t generated by the operator −(−∆) α + µ 0 I with domain H 2α,q U (R N ) is exponential decay. In addition, let v ∈ H 2α,q U (R N ) be the unique solution of the corresponding elliptic equations Under the assumptions of Theorem 4.2, there is a positive constant C ṽ such that for any bounded(in H and τ, such that for each 0 < < α < 1 and t ≥ T 0 where C g,ṽ depends on ṽ(x) L q U (R N ) , g L p b (R;L q U (R N )) and the positive constants θ 0 , θ 1 , α, p, λ 0 .
Proof. For the solution U (x, t; |u τ |) of the linear problem (3.1) associated with the initial |u τ |, we now decompose U (x, t; |u τ |) as follows: whereṽ(x) is the solution of (5.7) and v(x, t) satisfies the following non-autonomous linear equation: By using the variation of constants formula, we have and then, for some θ 0 > 0, it follows from (4.2) v(x, t) L q U (R N ) i.e., for any τ ∈ R and bounded set B ⊂ L q U (R N ), there exists T = T (B, τ ) > τ such that On the other hand, we can choose λ 0 large enough such that the semigroup e −(−∆) α −λ0I generated by −(−∆) α − λ 0 I decays exponentially in L q U (R N ). Thus, for Eq. (1.1) we use the variation of constants formula from t to t + 1 and have and then, for some θ 1 > 0, For any τ ∈ R and bounded set B ⊂ H 2(α− ),q U (R N ), there exists T = T (B, τ ) > τ such that for all t ≥ T, similar to (4.10), we get . Now, substituting the above two estimates in (5.15) it follows for all t ≥ T, which implies that for all t ≥ T + 1, the estimate (5.8) holds true. It follows from (5.8) and (5.1) that the family of processes {U g (t, τ )}, g ∈ H L q U (R N ) (g 0 ) has a bounded uniformly(w.r.t. g ∈ H L q U (R N ) (g 0 )) absorbing set B α : for all 0 < < α < 1, that is, for any τ ∈ R and bounded B ⊂ H 6. Uniform asymptotic compactness and proof of Theorem 1.1. In order to prove the family of {U σ (t, τ )} possesses some compactness in the sense of the process, we will recall the concept of uniform asymptotic compactness that is proposed by Moise et.al. in [32] and is different from the one given by Haraux [22] and Chepyzhov and Vishik [13]. From example 2.2 of [31] we know that there is no compactness of the semigroup in the norm of the locally uniform spaces, but in the corresponding weighted spaces. Hence, we will prove the asymptotic compactness of the family of processes {U g (t, τ ), g ∈ H L q U (R N ) (g 0 )} in the weighted norm. First of all, we recall the following definition. Definition 6.1. Let E and H(σ 0 ) be two complete metric spaces. A family of processes {U σ (t, τ )}, σ ∈ Σ, on E with the symbol σ ∈ H(σ 0 ) is said to be uniformly (w.r.t. σ ∈ H(σ 0 )) asymptotically compact, if and only if for any fixed τ ∈ R, a bounded sequence In view of our scopes, the main tool is the following abstract result from [27]. For convenience, we denote by B(E) all of the bounded set of E. 1) has a bounded uniformly(w.r.t.σ ∈ H(σ 0 )) absorbing set B 0 , and 2) is uniformly(w.r.t.σ ∈ H(σ 0 )) asymptotically compact.
Next, we will derive some priori estimates about the difference of two solutions which will be used to obtain required uniform(w.r.t.σ ∈ Σ) asymptotic compactness. To this end, let u i (t), i = 1, 2 be the solution of problem (1.1) with the initial data u i τ belongings to B α ⊂ H 2(α− ),q U (R N ) and associated with the external force g i ∈ L p b (R, L q U (R N )), and for convenience, set w(t) = u 1 (t) − u 2 (t) and f i (s) = f (x, u i (s)). Thanks to the translation identity (2.6) and the invariant property of T (t), it follows that for any fixed τ ∈ R and g ∈ H L q Thus, without loss of generality, we start with τ = 0 and w(t) solves the following integral equations: Since w 0 (x) ∈ H 2(α− ),q U (R N ), by taking the norm H 2(α− ),q φ (R N ) on the two sides of the above equation, it follows for some θ 3 > 0 and t ∈ [0, T ], Now, we estimate the difference of the nonlinearity f i (s), i = 1, 2, in the weighted space L q φ (R N ). Similar to (4.6), it fields where we used Sobolev type inclusion H and a weight function given as in Definition 2.1. Then, we insert (6.2) into the right hand side of inequality (6.1) and use Lemma 2.7, it follows that there exists a positive constant ,µ0 such that 3) Before starting with the existence of compact uniform attractor, we first introduce a new class of external forces which is an extension of the definition 3.1 of [27] in the setting of locally uniform spaces. More properties will be given in Appendix A. Definition 6.3. A function g ∈ L p loc (R; L q U (R N )) with p > 2 is said to be uniform normal if for any ε > 0, there exists η > 0 such that The set of all the uniform normal functions is denoted by L p un (R; L q U (R N )). It is obvious that L p un (R; L q U (R N )) ⊂ L p b (R; L q U (R N )). Theorem 6.4. Let g ∈ L p b (R; L q U (R N )) with p > 2 is the uniform normal. Under the assumptions in Theorem 5.5, the family of processes {U g (t, τ )}, g ∈ H L q U (R N ) (g 0 ), generated by solutions of problem (1.1)-(1.2) is uniformly(w.r.t. g ∈ H L q U (R N ) (g 0 )) asymptotically compact in H 2(α− ),q φ (R N ).
Proof. Let u n be the solutions corresponding to initial data u n 0 ∈ B α ⊂ H 2(α− ),q U (R N ) w.r.t. the symbol g n ∈ H L q U (R N ) (g 0 )), n = 1, 2, · · · , and let τ * > 0 be fixed. Then, from the term ii) of Lemma 2.5 we know that the embedding H , where u n = U gn (t n − τ * , 0)u n 0 satisfying t n τ * , without loss of generality we still denote by {u n } ∞ n=1 , which is a Cauchy sequence in L q φ (R N ). Since the symbol space H L q U (R N ) (g 0 ) is invariant w.r.t. the translation semigroup {T (s)}, s ≥ 0, then it follows from (5. Note that g 0 is the uniform normal, then according to the definition 6.3, for any ε > 0, there exists η(ε) > 0 such that sup t∈R t+kη t+(k−1)η , k ∈ Z + , which implies for 1 η < k ≤ 1 + Thus the proof of Theorem 6.4 is complete.
In particular, if the symbol g 0 (t) belonging to the hull H L q U (R N ) (g 0 ) is a tr.c. function, then according to Theorem 5.4 and 6.4, we have from Corollary 5.2 of [12], On the other hand, for any η > 0 there exist k 2 such that 1 k2 ≤ η, set t = k 2 , then sup t∈R t+η t m(s)n(x) p L q U (R N )) ds ≥ k2+ 1 k 2 k2 m(s) p ds = 1, which implies g(x, t) belonging to L p b (R; L q U (R N )) but not L p un (R; L q U (R N )). Example A.5. For k 1 , k 2 = 1, 2, · · · , we take n(x) = k 1 , |x| ∈ [k 1 , k 1 + 1 k Thus, for any 0 < ε < 1  )2ds = 4, which implies from Proposition A.2 that g(x, t) = m(t)n(x) does not belong to L p c (R; L q U (R N )) but in L p un (R; L q U (R N )).