Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise
Abstract
For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to (0,∞] and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on (0,∞] in both square and p-order Lebesgue spaces, where p>2. The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.
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JDDE manuscript No. (will be inserted by the editor) Higher-order continuity of pullback random attractors for random quasilinear equations with nonlinear colored noise Yangrong Li ?·Fengling Wang ·Tomás Caraballo Received: date / Accepted: date Abstract For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to (0,1]and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on (0,1] in both square and p-order Lebesgue spaces, where p>2. The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system. Keywords Nonautonomous random dynamical systems ·pullback random bi-spatial attractors ·residual dense continuity ·random quasilinear equations ·colored noise Mathematics Subject Classification (2000) MSC 37L55 ·60H15 ·35B41 ?Corresponding author. Yangrong Li School of Mathematics and Statistics, Southwest University, Chongqing 400715, China. E-mail: [email protected] Fengling Wang School of Mathematics and Statistics, Southwest University, Chongqing 400715, China. E-mail: [email protected] Tomás Caraballo Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/Tarfia s/n, 41012-Sevilla, Spain. E-mail: [email protected]
2 Y.Li &F. Wang &T. Caraballo 1 Introduction For many stochastic or deterministic partial differential equations, there is a common phenomenon that the solutions belong to a regular space Ywhen the initial datum belongs to an initial space X. In this case, we say that the corresponding dynamical system is regular. By a pullback random attractor (abbr. PRA, see [6],[33]) for a nonautonomous random dynamical system (shortly, a cocycle) on Xover a probability space ⌦, we mean a family {A(t, !):t2R,!2⌦}of compact sets with the invariance, pullback attraction and measurability in X. For a regular cocycle in (X,Y ), we introduce the concept of a pullback random bi-spatial attractor (abbr. PRBA), which is a PRA in Xsuch that it is random compact in Yand pullback attracts some subsets of Xunder the topology of Y. A PRBA generalizes a PRA to the bi-spatial case and a random bi-spatial attractor in [25] to the nonautonomous case. We will establish an abstract result (see Theorem 1) on the unique existence of a PRBA. The main difficulty is to prove the measurability of the attractor in Yby using four verifiable conditions. In particular, we assume that the cocycle operator from X\Yinto Yis strong-weakly continuous (rather than quasi-continuity used in [11,17,26]). We remark that the measurability in the regular space has not been discussed in previous papers (e.g., [14,25,31,43,44]). Our main purpose is to study the continuity (upper and lower semicontinuity) of a family {A:2⇤}of PRBAs not only in the initial space but also in the regular space. More precisely, we investigate the continuity of the following set-valued mapping: 7! A(t, !),⇤!(C(X\Y),⇢ X\Y),8t2R,!2⌦,(1.1) where C(X\Y)is the collection of all nonempty compact sets in X\Y,⇢X\Yis Hausdorff metric, and ⇤is a topological space (which may not be a metric space). As pointed out by Carvalho et al. [7, page 65], even in the initial space, the continuity (especially the lower semicontinuity [3]) of attractors is extremely hard to prove in PDEs, the full continuity of the mapping (1.1) cannot be obtained generally. Hence we focus on the residual dense continuity of the mapping (1.1), which means that the set of points of continuity of the mapping forms a residual dense set in ⇤, where a set is called residual if its complementary set is a union of countably nowhere dense sets. The residual dense continuity of attractors in the initial space has been studied by Hoang, Olson and Robinson [18,19] in the deterministic case and by Li et al [27,29, 30] in the random case. However, even for autonomous deterministic attractors, any lower semi-continuity result in the regular space has not been reported in literature. Recall that some criteria for the residual dense continuity in the initial space may be given by the strong continuity and convergence of the cocycles and the locally uniform compactness of the attractors. However, the strong continuity of the cocycle operator in the regular space is not available generally. To overcome the difficulty, we only use the strong-weak continuity of the cocycle operator from X\Yinto Y. This condition cannot ensure that the image set of the
Higher-order continuity of pullback random attractors 3 cocycle operator on a compact subset of X\Yis a compact subset of Y. But we can prove that the image set is a closed bounded set in Y. So, we may consider the larger metric space CB(Y)(than C(Y)) of all closed bounded set in Y, equipped with the Hausdorff metric. In summary, we establish an abstract result (see Theorem 2) on the residual dense continuity of PRBAs in X\Yby using six verifiable conditions (see C1-C6 in Section 3). In the proof of this theorem, we will use the abstract Baire residual theorem ([18, theorem 5.1]), and the Baire density theorem ([29, Prop. 2]). The abstract results are illustrated in the random p-Laplace equation (where p> 2) on Rm, driven by nonlinear colored noise utdiv(|ru|p2ru)+↵u=f(x, u)+g(t, x)+h(x, u)⇣(✓t!),(1.2) where ⇣(✓t·)is the Ornstein-Uhlenbeck process (which is called colored noise in [15]) for a scalar Wiener process W(t)given by ⇣(!)=1 2Z0 1 es dW(s, !),8>0,!2⌦, and {✓t}t2Ris a group of self-transformations on the probability space ⌦. We remark that the previous works in references [15,16,34,36,39] considered 2(0,1) and the limits when !0in order to approximate the stochastic equation with white noise by the random equation with colored noise, the upper semicontinuity of attractors was established in the linear noise case, i.e., h(·)is linear. In this paper, we consider the limits as !02(0,1)and as !1. Due to the limit lim!1 ⇣(✓t!)=0(see Lemma 15), we regard the equation (1.2) for =1as the corresponding deterministic equation (i.e., the noise term in (1.2) disappears). It is worth pointing out that the approximation from random (or stochastic) to deterministic equations (!1) is significative (see [8,35,42]). In particular, Flandoli et al. [13] pointed out that a deterministic transport equation is unsolved, while the stochastic (or random) version is solved. In [40], it is studied the limit when !1in order to approximate the deterministic equation by the random equation with Wong-Zakai noise, but only the upper semicontinuity was established. Therefore, we take the parameter space by ⇤:= (0,1], where 12⇤, and the infinite noise corresponds to the deterministic equation. Our main application result is to prove the residual dense continuity (including the existence) of PRBAs {A(t, !)}on both spaces X:= L2(Rm)and Y:= Lp(Rm) with respect to 2⇤. All results may be valid if the colored noise is replaced by another simpler Wong- Zakai noise, i.e. the difference of the Wiener white noise, see [1,4,9,32,37,38]. 2 Concept and existence of a pullback random bi-spatial attractor Throughout this paper, (X, Y )is a limit-identical pair of separable Banach spaces, where the following limit-identical property {xn}⇢X\Y, kxnxkX!0,kxnykY!0)x=y, (2.1)
4 Y.Li &F. Wang &T. Caraballo is equivalent to that (X\Y,k·kX+k·kY)is a Banach space (see [25, Lemma 2.1]). Hausdorff semimetric and Hausdorff metric are respectively defined by dX(A, B)=sup x2A inf y2BkxykX,⇢ X(A, B) = max{dX(A, B),d X(B,A)} for A, B ⇢X. It is standard to prove that dX\Y=dX+dYand ⇢X\Y=⇢X+⇢Y. Let (⌦,F,P)be a probability space equipped with a group {✓t:⌦!⌦;t2R} of measure-preserving transformations. A measurable mapping :R+⇥R⇥⌦⇥ X!Xis called a cocycle on X(see [33]) if, for t, s 0,⌧2R,!2⌦, (t+s, ⌧,!)=(t, ⌧+s, ✓s!)(s, ⌧,!),(0,⌧,!)=IX. Definition 1 A cocycle on Xis called regular in (X,Y )if (t, ⌧,!)X⇢Y(thus ⇢X\Y),8t>0,⌧2R,!2⌦.(2.2) A set-valued map D:R⇥⌦7! 2X\{;}is called a bi-parametric set in X, while a bi-parametric set Din Xis called random in Xif !!d(x, D(⌧,!)) is F-measurable for x2X, and compact (closed, bounded) if D(⌧,!)is compact (closed, bounded) for each ⌧2Rand !2⌦. Throughout this paper, Dis an inclusion-closed universe of some bi-parametric sets in X. Definition 2 Let be a regular cocycle in (X,Y ). A bi-parametric set Ais called a D-pullback random (X,Y )-attractor for if (i) A2D; (ii) Ais compact in X\Y; (iii) Ais random in Xand in Yrespectively; (iv) Ais invariant under , i.e., for t0,⌧2R,!2⌦, (t, ⌧t, ✓t!)A(⌧t, ✓t!)=A(⌧,!); (2.3) (v) Ais D-pullback attracting in X\Y, that is, for each D2D, lim t!+1dX\Y((t, ⌧t, ✓t!)D(⌧t, ✓t!),A(⌧,!)) = 0.(2.4) Remark 1 (a) The regularity of and invariance in (2.3) imply that A(⌧,!)⇢Y. (b) If X=Y, a PRBA reduces to a usual PRA in [20,33,41]. (c) Even if X6=Y, a PRBA in (X,Y )must be a PRA in X. Definition 3 [33] A bi-parametric set Kin Xis called D-pullback absorbing for a cocycle in Xif, for each D2D,⌧2Rand !2⌦, there is T>0such that (t, ⌧t, ✓t!)D(⌧t, ✓t!)⇢K(⌧,!),8tT. (2.5) Definition 4 A regular cocycle in (X,Y )is called D-pullback asymptotically compact in X(resp. Y,X\Y) if, for D2D,⌧2R,!2⌦and tn!+1, the sequence {(tn,⌧tn,✓ tn!)xn}with xn2D(⌧tn,✓ tn!)(2.6) has a convergent subsequence in X(resp. Y,X\Y).
Higher-order continuity of pullback random attractors 5 For a bi-parametric set Bin X, the omega-limit set in Xis defined by ⌥X(B)(⌧,!):=\T>0[tT(t, ⌧t, ✓t!)B(⌧t, ✓t!) for ⌧2Rand !2⌦, where the overline denotes the closure in X. Since is regular in (X,Y ), the following set ⌥Y(B)(⌧,!):=\T>0clY[tT(t, ⌧t, ✓t!)B(⌧t, ✓t!) is well-defined in Y(even if Bdoes not lie in Y), where clYdenotes the closure in Y, and thus ⌥Y(B)is a closed bi-parametric set in Y. It is similar to define the omega-limit set ⌥X\Y(B). We now state and prove an abstract result on the unique existence of a PRBA. Theorem 1 Let be a regular cocycle in (X, Y )and Da universe in X. (i) If we assume that for t>0,⌧2Rand !2⌦, (A1) (t, ⌧,!): X!Xis continuous; (A2) there is a D-pullback absorbing set K2Dand Kis random closed in X; (A3) is D-pullback asymptotically compact in Xand in Yrespectively, then has a unique D-pullback (X,Y )-attractor given by A=⌥X(K)which is random in X. (ii) If we further assume (A4) (t, ⌧,!):X\Y!Yis strong-weakly continuous; (A5) there is another D-pullback absorbing set M2Dand Mis random closed in X\Y; (A6) !!(t, ⌧,!)yis (F,B(Y))-measurable for each y2X\Y; (A7) (X\Y,k·k X+k·k Y)is separable, then Ais a random set in Yand thus it is a D-pullback random (X, Y )-attractor. Proof (i) By (A1),(A2) and the X-part of (A3), the existence theorem in [33] shows that has a unique D-pullback random attractor in X, given by ⌥X(K)=:A. In particular, A2Dand Ais random, invariant, compact, D-pullback attracting in X. As the autonomous case in [25], using (A3) and the limit-identical property of (X,Y ), one can prove that ⌥X(K)=⌥Y(K), which is compact and D-pullback attracting in Y. (ii) The proof of the measurability of Ain Yis similar (but slightly different) to the autonomous case in [11, theorem 20]. We give here a sketch of the proof. For the absorbing set Min (A5), by the similar methods as in [11, theorem 20], one can prove that A=⌥X(M)=⌥X\Y(M)=⌥Y(M)=⌥dis Y(M),(2.7) where the last term is the discrete omega-limit set defined by ⌥dis Y(M)(⌧,!)) := \1 m=1clY[1 n=m(n, ⌧n, ✓n!)M(⌧n, ✓n!). Next, we need to consider the following weak sequential closure (rather than the weak closure in [11, theorem 20]): clYws E={y2Y;9{yn}1 n=1 ⇢Esuch that yn*yweakly in Y}
6 Y.Li &F. Wang &T. Caraballo for E⇢Y. We claim that for all ⌧2Rand !2⌦, ⌥dis Y(M)(⌧,!)) = \1 m=1clYws [1 n=m(n, ⌧n, ✓n!)M(⌧n, ✓n!).(2.8) The ⇢relationship in (2.8) is obvious. We need to prove the relationship. Suppose ybelongs to the right-hand side of (2.8). Then, for each m2N, there are {mi}i2N with mimfor all i2N(where {mi}may not be a subsequence of {m}) and xmi2M(⌧mi,✓ mi!)such that (mi,⌧mi,✓ mi!)xmi*yweakly in Yas i!1. Hence the diagonal sequence {mm}m2Nis a subsequence of {m}(because mm m!1) such that (mm,⌧mm,✓ mm!)xmm*yweakly in Yas m!1. Since mm!1,M2Dand is D-pullback asymptotically compact in Y(see (A3)), it follows that {mm}has a subsequence (not relabeled) such that (mm,⌧mm,✓ mm!)xmm!zin Y, as m!1, and thus z2⌥dis Y(M)(⌧,!). By the uniqueness of the limits, we have y=z2 ⌥dis Y(M)(⌧,!). Therefore (2.8) holds true. By (2.7) and (2.8), it is direct to deduce that A(⌧,!)=\1 m=1clYws [1 n=mclYws (n, ⌧n, ✓n!)M(⌧n, ✓n!).(2.9) By (A5), the mapping !!M(⌧n, !)defines a random closed set in X\Y, while, by (A7),X\Yis a separable Banach space. Hence, [10, Lemma II. 4.1] implies that there is a sequence {znj}1 j=1 of random elements in X\Ysuch that M(⌧n, !)=clX\Y{znj(!); j2N},8!2⌦. From which, we deduce from (A4) that clYws (n, ⌧n, ✓n!)M(⌧n, ✓n!) =clYws (n, ⌧n, ✓n!){znj(✓n!); j2N},(2.10) From (2.9) and (2.10) we obtain that A(⌧,!)=\1 m=1clYws [1 n=mclYws (n, ⌧n, ✓n!){znj(✓n!); j2N}. Hence, the similar method as in (2.8) implies that A(⌧,!)=\1 m=1clY[1 n=m(n, ⌧n, ✓n!){znj(✓n!); j2N}.(2.11) By the same method as in [11, theorem 20], using (A3),(A4) and (A6), one can prove that the last set in (2.11) is random in Y, and thus Ais random in Yas desired. Remark 2 In Theorem 1 (ii), the conditions (A4)-(A5) are different from those in [11], while we have to assume (A7), which appears to be missed in [11, theorem 20]. In general, the separability of Xand Ycannot imply the separability of X\Y. But we will prove the separability of L2(Rm)\Lp(Rm)in Lemma 6 later.
Higher-order continuity of pullback random attractors 7 3 Residual dense continuity of pullback random bi-spatial attractors 3.1 Lifted spaces and Baire categories Recall that C(X)(resp. CB(X),B(X)) denotes the collection of all nonempty compact (resp. closed bounded, bounded) sets in X. It is known that Hausdorff metric ⇢X is a complete metric on C(X)(see [27, Prop. 2.1]) and a metric on CB(X)(see [18, 19]). Recall that (X,Y )be a limit-identical pair of separable Banach spaces. Lemma 1 (i) C(X\Y)=C(X)\C(Y)and B(X\Y)=B(X)\B(Y). (ii) CB(X\Y)B(X)\CB(Y)and thus CB(X\Y)CB(X)\CB(Y). (iii) A set-valued mapping 7! F()is continuous at 02⇤into (C(X\ Y),⇢ X\Y)iff so it is into (C(X),⇢ X)and (C(Y),⇢ Y)respectively. (iv) If Ais a pre-compact set in X\Y, then the closures of Ain X,Yand X\Y are the same, and thus this same closure is compact in X,Yand X\Yrespectively. Proof (i) By [25, Lemma 2.1], if (X,Y )is limit-identical, then Ais compact in X\Y iff Ais compact in Xand in Yrespectively. Hence the first equality in (i) holds true, while the second equality follows from k·k X\Y=k·k X+k·k Y. (ii) Suppose that D2B(X)\CB(Y). By (i), Dis bounded in X\Y. Suppose xn2Dand x2X\Ysuch that kxnxkX\Y!0. Then kxnxkY!0. Since Dis closed in Y, we have x2D, and thus Dis closed in X\Y. The assertion (iii) follows from (i) immediately. (iv) Let Abe a pre-compact set in X\Yand y2clYA. Then there is a sequence yn2Asuch that kynykY!0. Since Ais pre-compact in X\Y, passing to a subsequence, we have kynzkX\Y!0for some z2X\Y, which implies that y=z2clX\YAand thus clYA⇢clX\YA. Obviously, clX\YA⇢clYA, and thus they are equal to each other. It is similar to prove that A=clX\YA. We need the following result on the strong-weak continuity of an operator. Lemma 2 Let :X!X\Ysatisfy that :X!Xis continuous. (a) If :X\Y!Yis strong-weakly continuous, then (C(X\Y)) ⇢CB(Y). (b) On the contrary, if (C(X\Y)) ⇢B(Y)and the embedding X⇤,!(X\Y)⇤ is dense, then :X\Y!Yis strong-weakly continuous. Proof (a) Given A2C(X\Y), by the strong-weak continuity of :X\Y!Y, we deduce that (A)is weakly compact in Y. In particular, (A)is weakly closed and thus (strongly) closed in Y. We then prove that (A)is bounded in Y. Indeed, for each y⇤2Y, the mapping y!hy,y⇤iis (Yw,R)-continuous, where Ywdenotes the weak topology. Since (A) is weakly compact in Y, it follows that the set {hy,y⇤i:y2(A)}is compact and thus bounded in Rfor each y⇤2Y⇤. On the other hand, each y2(A)can be regarded as a continuous linear functional on Y⇤, that is, y⇤⇤ :Y⇤!Rwith y⇤⇤(y⇤)=hy,y⇤ifor all y⇤2Y⇤. By
8 Y.Li &F. Wang &T. Caraballo the above proof, the family {y⇤⇤ 2Y⇤⇤ :y2(A)}is pointwise bounded (acting on each y⇤2Y⇤). By the resonance theorem, the set {y⇤⇤ 2Y⇤⇤ :y2(A)}is bounded in Y⇤⇤. Since ky⇤⇤k=kykfor all y2Y, it follows that (A)is bounded in Y. (b) Let xn!x0in X\Y. By the continuity of :X!X, we have xn! x0in X. In particular, {xn}is bounded in Xand hxn,x ⇤i!hx0,x ⇤i,8x⇤2X⇤.(3.1) Since {xn;n2N}[{x0}2C(X\Y), it follows from the assumption that {xn} is bounded in Y, and thus {xn}is bounded in X\Y. By the assumption, we know that X⇤is dense in (X\Y)⇤. Then the boundedness of {xn}in X\Y, together with (3.1), implies that hxn,y⇤i!hx0,y⇤i,8y⇤2(X\Y)⇤. Note that Y⇤,!(X\Y)⇤, and thus xn*x 0weakly in Y. We also recall two Barie category theorems. A set Qin a topological space ⇤is called nowhere dense if its closure Qdoes not contain any nonempty open set, and residual if the complement Qcis a union of countably many nowhere dense sets. Lemma 3 (Abstract Baire residual theorem) Suppose that ⇤is a topological space, Zis a metric space and Fn:⇤!Zis a sequence of continuous mappings such that lim n!1 Fn()=F()in Z, 82⇤. Then the continuity-set (namely, the set of points of continuity) of the mapping ! F()is residual in ⇤. The above theorem was proved in [29, Prop. 1], while a special case when ⇤is a complete metric space was proved in [18, Theorem 5.1]. Recall that ⇤is called a Hausdorff space if for two points 16=2there are open sets O1,O2such that 12O1,22O2and O1\O2=;. Lemma 4 (Baire density theorem) If ⇤is a complete metric space or a compact Hausdorff space, then any residual set in ⇤is dense. The above lemma can be found in [18] when ⇤is a complete metric space and in [29, Prop. 2] when ⇤is a compact Hausdorff space. 3.2 Residual dense continuity of PRBAs Let ⇤be a topological space (termed as a parameter space). For each 2⇤, we assign a regular cocycle in (X,Y )and assume that each possesses a D-pullback random (X,Y )-attractor denoted by A. We need some conditions which hold for all 2⇤,t>0,⌧2Rand !2⌦. C1.(t, ⌧,!):X!Xis continuous.
Higher-order continuity of pullback random attractors 9 C2.(t, ⌧,!):X\Y!Yis strong-weakly continuous. C3. For each K2C(X\Y)and 02⇤, lim !0 sup x2K k(t, ⌧,!)x0(t, ⌧,!)xkX\Y=0.(3.2) C4.⇤=[k2N⇤kand each ⇤kis closed in ⇤. C5.The union A⇤k(s, !)is pre-compact in X\Yfor each s<0, where A⇤k(⌧,!):=[2⇤kA(⌧,!),8⌧2R,k2N,!2⌦.(3.3) C6.For each k2N, there is Bk2Dsuch that A⇤k(s, !)⇢B k(s, !),8s<0.(3.4) In order to study the dense continuity, we strengthen C4 by C4*.⇤=[k2N⇤k, each ⇤kis a compact Hausdorff space or a complete metric space, and ⇤in kis residual in ⇤k,where ⇤in kdenotes the interior of ⇤kin ⇤. Theorem 2 Suppose that a family {A;2⇤}of D-pullback random (X,Y )- attractors satisfies C1-C6. Then, for each ⌧2Rand !2⌦, the continuity-set (denoted by ⇤⇤(⌧,!)) of the mapping 7! A(⌧,!),⇤!(C(X\Y),⇢ X\Y)(3.5) is a residual set in ⇤. Moreover, ⇤⇤(⌧,!)is dense in ⇤if C4* replaces C4. Proof Step 1.We prove the continuity-set (denoted by ⇤⇤ X(⌧,!)) of the mapping 7! A(⌧,!),⇤!(C(X),⇢ X) is residual in ⇤. Since Ais an attractor in X, by using C1,C4,C6 and the X-part of C3,C5, the assertion follows from [27, Theorem 4], but we use Baire residual theorem (i.e., Lemma 3) when ⇤is a topology space, which generalizes the case of a complete metric space used in [18,19,27]. Step 2.We prove under C2-C6 that the continuity-set (denoted by ⇤⇤ Y(⌧,!)) of the mapping 7! A(⌧,!),⇤!(C(Y),⇢ Y)(3.6) is residual in ⇤. We fix ⇤kwith k2N,⌧2R,!2⌦and a sequence {tn}nwith |⌧|<t n!+1. By C5 and ⌧tn<0, we know that for each n2N, A⇤k(⌧tn,✓ tn!):=[2⇤kA(⌧tn,✓ tn!) is a pre-compact set in X\Y. By Lemma 1 (iv), Kn:= A⇤k(⌧tn,✓ tn!)=clX\YA⇤k(⌧tn,✓ tn!),(3.7) and thus Kn2C(X\Y)for each n2N. By C2, the following mapping (tn,⌧tn,✓ tn!):X\Y!Y
16 Y.Li &F. Wang &T. Caraballo as i!1. By Hölder inequality again, ukj 2L2(Ok). Hence, for each k2N, {ukj :j2N}is dense in (L2(Ok)\Lp(Ok),k·k L2(Ok)\Lp(Ok)). The null-extension of v:Ok!Ris denoted by ev, namely, ev=von Okand ev⌘0 on Oc k. We then obtain that {eukj :j2N}is dense in (L2 k\Lp k,k·k L2 k\Lp k),8k2N,(4.9) where L2 k:= {ev:v2L2(Ok)}⇢Xand Lp k:= {ev:v2Lp(Ok)}⇢Y. Given now w2L2(Rm)\Lp(Rm), the truncation of won Okis defined by wk(x)=w(x),8x2Ok,w k(x)=0,8x2Rn\Ok. Obviously, wk2L2 k\Lp kfor each k2N. Moreover, as k!1, kwkwk!0,kwkwkp!0and thus kwkwkX\Y!0, which implies that [k2N(L2 k\Lp k)is dense in X\Y. Therefore, by (4.9), {eukj :k,j 2N}is dense in (X\Y,k·k+k·k p), which means that X\Yis separable. Using the test functions, one can prove (see [25]) the limit-identical property (2.1) for (X,Y ). Then it follows from [25, Lemma 2.1] that X\Yis a Banach space. The last assertion on the dense embedding is obvious. 4.2 The cocycle on the initial space Consider the Sobolev space W1,p(Rm)equipped with the norm kukW1,p =(kukp p+krukp p)1/p,8u2W1,p(Rm). We have W1,p(Rm),!Y, but the embedding is not compact. Both (A1) and C1 in Theorems 1, 2 are verified as follows. Lemma 7 For each >0,!2⌦and u⌧2X=L2(Rm), the equation (4.1) has a unique weak solution such that u2C([⌧,1); X)\Lp loc(⌧,1;W1,p(Rm)).(4.10) Moreover, the solution operator u⌧!u(t;⌧,!,u⌧)is continuous on Xand !! u(t;⌧,!,u⌧)is (F,B(X)) measurable.
Higher-order continuity of pullback random attractors 17 Proof Existence. As done in [23], by the Faedo-Galerkin method, one can prove the existence of a solution satisfying (4.10) by using the following a priori estimate: ku(t;⌧,!,u⌧)k2c⌥(t, ⌧,!, u⌧), Zt ⌧ e↵(st)(kru(s)kp p+ku(s)kp p+ku(s)k2)ds c⌥(t, ⌧,!, u⌧),(4.11) where t>⌧,cis an intrinsic positive constant and ⌥(t, ⌧,!, u⌧)=e↵(t⌧)ku⌧k2+Zt ⌧ e↵(st)(kg(s)k2+|⇣(✓s!)|2p2 pq+ 1)ds. The proof of (4.11) is similar to the autonomous case as given in [30]. Uniqueness, continuity and measurability in X. This is standard or deduced from the same method as in Proposition 2. By Lemma 7, for each >0, one can define a continuous cocycle :R+⇥ R⇥⌦⇥X!Xby (t, ⌧,!)u⌧=u(⌧+t;⌧,✓⌧!,u⌧)(4.12) for all t0,⌧2R,!2⌦and u⌧2X=L2(Rm). 4.3 Regularity and strong-weak continuity of the cocycle We prove that the cocycle is regular and strong-weakly continuous in Y=Lp(Rm), namely, (A2) and C2 hold true. Proposition 1 Let >0,T>0,⌧2Rand !2⌦. (i) The cocycle is regular in (X,Y ), i.e., (T,⌧,!)X⇢Y. (ii) The mapping (T,⌧,!):X\Y!Yis strong-weakly continuous. Proof (i) Multiplying (4.1) by |u|p2uand integrating the products over Rmwe obtain that 1 p d dtkukp p+hApu, |u|p2ui+↵kukp p =hf(u),|u|p2ui+hg(t),|u|p2ui+⇣(✓t!)hh(u),|u|p2ui. Note that hApu, |u|p2ui0. By (4.2), hf(u),|u|p2uiZRm |1||u|p1dx 1kuk2p2 2p2ck1k23 41kuk2p2 2p2. By (4.5) and 2p2 pq2, ⇣(✓t!)hh(u),|u|p2ui|⇣(✓t!)|ZRm (| 1||u|q+p2+| 2||u|p1)dx 1 8kuk2p2 2p2+ck 1k 2p2 pq 2p2 pq |⇣(✓t!)|2p2 pq+ck 2k2|⇣(✓t!)|2 1 8kuk2p2 2p2+c(|⇣(✓t!)|2p2 pq+ 1).
18 Y.Li &F. Wang &T. Caraballo We also have hg(t),|u|p2ui1 8kuk2p2 2p2+ckg(t)k2. Therefore, d dtkukp p+p↵kukp p+p1 2kuk2p2 2p2c(kg(t)k2+|⇣(✓t!)|2p2 pq+ 1).(4.13) Integrating (4.13) in t2[r, s], where ⌧+✏r<s⌧+Tand ✏>0, we obtain that ku(s)kp pku(r)kp p+cZ⌧+T ⌧ (kg(t)k2+|⇣(✓t!)|2p2 pq+ 1)dt ku(r)kp p+CT(!), where we use g2L2 loc(R,X)and the continuity of t!⇣(✓t!). Integrating the above inequality in r2[⌧+✏, s]yields ku(s)kp p1 s⌧✏Z⌧+T ⌧ ku(r)kp pdr +CT(!) for all s2(⌧+✏,⌧+T]. By (4.11) (or (4.10)), the above integral is finite. Therefore, for any ✏2(0,T/2), sup t2[2✏,T ] ku(⌧+t, ⌧,!, u⌧)kp pC✏,T (!). (ii) We claim that (T,⌧,!)C(X\Y)⇢B(Y). Indeed, for A2C(X\Y)and u⌧2A, by (4.13), the solution u(·;⌧,✓⌧!,u⌧)satisfies that d dtkukp pckg(t)k2+CT(✓⌧!),8t2[⌧,⌧+T]. Integrating it from t=⌧to ⌧+Twe obtain that ku(⌧+T)kp pku⌧kp p+cZ⌧+T ⌧ kg(t)k2dt +TCT(!) kAkp X\Y+cg+CT(!)CT(!)<+1. Hence (T,⌧,!)A2B(Y). Note that X⇤is dense in (X\Y)⇤as in Lemma 6. By Lemma 2 (b), we know that (T,⌧,!):X\Y!Yis strong-weakly continuous. 4.4 Luzin continuity and measurability of the cocycle in Y In order to verify (A6), we set ⌦N={!2⌦:|!(t)|Ne 2s,8s2R},8N2N. By (4.7), ⌦=[1 N=1⌦Nand each ⌦Nis a closed set in (⌦,d⌦), where the Frechet metric is defined by d⌦(!1,! 2)= 1 X n=1 1 2n sups2[n,n]|!(s)| 1+sup s2[n,n]|!(s)|,8!1,! 22⌦. We use the convergence of colored noise in samples ([11, Lemma 22]) to verify (A6).
Higher-order continuity of pullback random attractors 19 Lemma 8 Let !n,! 02⌦N(N2N) such that d⌦(!n,! 0)!0. Then the OU process satisfies sup s2[⌧,⌧+T] |⇣(✓s!n)⇣(✓s!0)|!0as n!1;(4.14) sup n2N0 sup s2[⌧,⌧+T] |⇣(✓s!n)|C, 8>0,⌧2R,T >0.(4.15) Proposition 2 (Measurability in Y)For each y2X\Y,T>0and ⌧2R, the mapping !!(T,⌧,!)yis (F,B(Y))-measurable and (F,B(X\Y))-measurable. Proof Let !n!!0in ⌦Nwith fixed N2Nand un(·)=u(·,⌧,! n,y),n2N0, the solutions of (4.1) with the initial data un(⌧)⌘y2X\Y. Then Un:= unu0 satisfies that dUn dt +ApunApu0+↵Un =f(un)f(u0)+⇣(✓t!n)h(un)⇣(✓t!0)h(u0). Multiplying it by |Un|p2Unand noting the monotonicity of Ap, we have 1 p d dtkUnkp p+↵kUnkp p=hf(un)f(u0),|Un|p2Uni +h⇣(✓t!n)h(un)⇣(✓t!0)h(u0),|Un|p2Uni. By the integrated mean-value theorem, (4.4) yields If:=hf(un)f(u0),|Un|p2Uni =ZRmZ1 0 @f @s(x, aun+(1a)u0)da|Un|pdx 3ZRmZ1 0 |aun+(1a)u0|p2da|Un|pdx +k3k1kUnkp p. Denote by Ih:=h⇣(✓t!n)h(un)⇣(✓t!0)h(u0),|Un|p2Uni =⇣(✓t!n)hh(un)h(u0),|Un|p2Uni +(⇣(✓t!n)⇣(✓t!0))hh(u0),|Un|p2Uni. By the integration mean-value theorem again, (4.6) and (4.15) yield that I1 h:= ⇣(✓t!n)hh(un)h(u0),|Un|p2Uni =⇣(✓t!n)ZRmZ1 0 @h @s(x, aun+(1a)u0)da|U|pdx CZRmZ1 0 k 3k1|aun+(1a)u0|q2da|Un|pdx +Ck 4k1kUnkp p 3ZRmZ1 0 |aun+(1a)u0|p2da|Un|pdx +C(k 3k p2 pq 1+k 4k1)kUnkp p
20 Y.Li &F. Wang &T. Caraballo for all t2[⌧,⌧+T]. Hence If+I1 hCkUnkp p. By (4.5), I2 h:= (⇣(✓t!n)⇣(✓t!0))hh(u0),|Un|p2Uni |⇣(✓t!n)⇣(✓t!0)|C(ku0kp+q2 p+q2+kunkp+q2 p+q2) C|⇣(✓t!n)⇣(✓t!0)|(ku0k2+kunk2+ku0k2p2 2p2+kunk2p2 2p2), where we use q<p. Therefore, d dtkUn(t)kp pCkUn(t)kp p+C|⇣(✓t!n)⇣(✓t!0)|n(t),8t2[⌧,⌧+T], where n(t)=ku0(t)k2+kun(t)k2+ku0(t)k2p2 2p2+kun(t)k2p2 2p2. By Un(⌧)=0, the Gronwall lemma implies that kUn(⌧+T)kp pCZ⌧+T ⌧ eCs|⇣(✓s!n)⇣(✓s!0)|n(s)ds Csup s2[⌧,⌧+T] |⇣(✓s!n)⇣(✓s!0)|Z⌧+T ⌧ n(s)ds, (4.16) where C=C(⌧,T). We then claim that the last integrals are bounded. Indeed, by (4.13) and (4.14), we have d dtkunkp p+p↵kunkp p+p1 2kunk2p2 2p2 c(kg(t)k2+|⇣(✓t!n)|2p2 pq+|⇣n(✓t!)|2+ 1) C(kg(t)k2+ 1), for all n2N0and t2[⌧,⌧+T]. Then the Gronwall lemma implies that Z⌧+T ⌧ ep↵skun(s)k2p2 2p2ds cep↵⌧ kun(⌧)kp p+CZ⌧+T ⌧ ep↵s(kg(s)k2+ 1)ds. Note that un(⌧)=y2X\Yand g2L2 loc(R,X). We have Z⌧+T ⌧ kun(s)k2p2 2p2ds ep↵⌧ Z⌧+T ⌧ ep↵skun(s)k2p2 2p2ds Ckykp p+Cep↵TZ⌧+T ⌧ (kg(s)k2+ 1)ds C for all n2N0. Hence, by (4.15) and (4.16), we have kUn(⌧+T)kp p!0as n!1. Hence the mapping !!(T,⌧,!)y,⌦N!Y, is continuous for all N2N. Note that lim N!1 P(⌦N)=P([1 N=1⌦N)=P(⌦)=1. Therefore, the mapping !!(T,⌧,!)yis Luzin continuous from ⌦into Yand thus (F,B(Y))-measurable. By a similar method as above, it is relatively easy to prove that the mapping !! (T,⌧,!)yis continuous from ⌦Ninto X. Hence the mapping !!(T,⌧,!)y is Luzin continuous from ⌦into X\Yand thus (F,B(X\Y))-measurable.
Higher-order continuity of pullback random attractors 21 5 Existence of a pullback random bi-spatial attractor To obtain a PRBA, we supply a backward tempered assumption: g2L2 loc(R,X)and sup r⌧Zr 1 e↵(sr)kg(s)k2ds < +1,8⌧2R,(5.1) where ↵>0as in (4.1). Let Dbe the universe of all tempered bi-parametric sets in X, where a bi-parametric set Dis called tempered if D(⌧,!)is a bounded subset of Xsuch that lim t!+1e✏tkD(⌧t, ✓t!)k=0,8✏>0,⌧2R,!2⌦.(5.2) 5.1 Uniform random absorbing sets in X\Y We verify (A2) and (A5), namely, the random absorption in Xand in X\Y. Lemma 9 For D2D,⌧2R,!2⌦, there is T0>2such that for all tT0and u⌧t2D(⌧t, ✓t!), ku(⌧;⌧t, ✓⌧!,u⌧t)k2cX(G(⌧)+E(!)) =: K(⌧,!),(5.3) sup r2[⌧1,⌧] ku(r;⌧t, ✓⌧!,u⌧t)kX\YM(⌧,!),(5.4) sup 0 sup r2[⌧1,⌧] ku(r;⌧t, ✓⌧!,u⌧t)kX\Ysup 0 M(⌧,!)<1,(5.5) sup 0 sup r2[⌧1,⌧] ku(r;⌧t, ✓⌧!,u⌧t)k2 W1,p cWsup 0 K(⌧,!)<1,(5.6) where 0>0,M(⌧,!)=cY(K 1 2 (⌧,!)+K 1 p (⌧,!)), G(⌧)=1+Z0 1 e↵skg(s+⌧)k2ds, E(!)=Z0 1 e↵s|⇣(✓s!)|2p2 pqds. (5.7) Proof By (4.11) and (5.2), one can deduce (5.3) and Z⌧ ⌧t e↵(s⌧)(kru(s;⌧t)kp p+ku(s)kp p+ku(s)k2)ds K(⌧,!),(5.8) for all tT0. By the energy inequality (4.13), u(·;⌧t, ✓⌧!,u⌧t)satisfies d dsku(s)kp pc2(kg(s)k2+|⇣(✓s⌧!)|2p2 pq+ 1). Integrating it from s=ˆr2[⌧2,⌧1] to s=r2[⌧1,⌧], we have ku(r)kp pku(ˆr)kp p+c2Zr ˆr (kg(s)k2+|⇣(✓s⌧!)|2p2 pq+ 1)ds ku(ˆr)kp p+c2Z⌧ ⌧2 (kg(s)k2+|⇣(✓s⌧!)|2p2 pq+ 1)ds.
22 Y.Li &F. Wang &T. Caraballo Integrating it from ˆr=⌧2to ˆr=⌧1, we obtain that for all r2[⌧1,⌧], ku(r)kp pZ⌧1 ⌧2 kukp pds +c2Z⌧ ⌧2 (kg(s)k2+|⇣(✓s⌧!)|2p2 pq+ 1)ds. By (5.6), we know that for all tT0and u⌧t2D(⌧t, ✓t!), Z⌧1 ⌧2 ku(s)kp pds e2↵Z⌧ ⌧2 e↵(s⌧)ku(s)kp pds e2↵K(⌧,!), and thus sup r2[⌧1,⌧] ku(r;⌧t)kp p(e2↵+c2)K(⌧,!). The similar inequality for L2-norm holds too and thus (5.4) holds true. Note that T0is independent of . We obtain that for all 0>0and tT0, sup 0 sup r2[⌧1,⌧] ku(r;⌧t)kX\Ysup 0 M(⌧,!). By (6.3) and (5.1), we have sup 0 K(⌧,!)=cXG(⌧)+cXsup 0 E(!)<1 and thus sup{M(⌧,!):0}is finite. Hence (5.5) holds true. It is standard to prove (5.6) (see [30] for Wong-Zakai noise). As the autonomous case in [30], one can establish the asymptotic tail property and omega-compactness in X. Lemma 10 For 0>0,D2D,⌧2Rand !2⌦, we have lim t,R!1 sup 0 sup r2[⌧1,⌧]Z|x|R |u(r;⌧t, ✓⌧!,u⌧t)(x)|2dx =0, uniformly for u⌧t2D(⌧t, ✓t!). Lemma 11 For each 0>0, the family {:0}of cocycles is uniformly D-pullback omega-compact in X, namely, XB(T)!0as T!1, where B(T)=⇣[ tT[ 0[ r2[⌧1,⌧] u(r;⌧t, ✓⌧!,D(⌧t, ✓t!)⌘,(5.9) for D2D,⌧2R,!2⌦, and the Kuratowski measure of B⇢Xdefined by X(B)=inf{r>0:Bhas a finite r-net in X}. Proof It is standard from Lemma 10 and Lemma 9.
Higher-order continuity of pullback random attractors 23 5.2 Large-valued estimates of solutions in Y Let m(e)be the Lebegue measure of a measurable set e⇢Rm. Lemma 12 Let 0>0,D2D,⌧2Rand !2⌦. We have lim !1 sup tT0 sup 0 sup r2[⌧1,⌧] m[|u(r;⌧t, ✓⌧!,u0)|]=0,(5.10) lim ,t!1 sup 0 sup r2[⌧1,⌧]Z[|u|] |u(r;⌧t, ✓⌧!,u0)(x)|2dx =0,(5.11) uniformly for u02D(⌧t, ✓t!), where T02as given in Lemma 9 and [|u|]={x2Rm,|u(r;⌧t, ✓⌧!,u0)(x)|}. Proof By (5.5) in Lemma 9, for all r2[⌧1,⌧],tT0and u02D(⌧t, ✓t!), m(Rm[|u(r;⌧t, ✓⌧!,u0)|]) 1 2Z[|u|] |u(r;⌧t)(x)|2dx 1 2ku(r;⌧t)k21 2sup 0 M2 (⌧,!)!0 as !1in view of sup0M(⌧,!)<1. Hence (5.10) holds true. Given ">0. By the uniform omega compactness in Lemma 11, there is T"T0 such that X(B(T")) <"/4, where B(T")is given by (5.9), and thus B(T")has a finite "/2-net in L2(Rm)with centers {v1,v 2,··· ,v k}⇢B(T"). By the absolute continuity of the Lebesgue integral, there is >0such that e⇢Rm,m(e)<)sup 1ikZe v2 i(x)dx<"2/4. Note that B(T")⇢B(T0)(as T"T0). By (5.10), there is >0such that m{x2Rm;|v(x)|}<,for all v2B(T"). For any v2B(T"), there is vi0, where i02{1,2,··· ,k}, such that kvvi0k<"/2, and thus Z[|v|] v2(x)dx 2Z[|v|] |vi0(x)|2dx +2Z[|v|] |v(x)vi0(x)|2dx <" 2/2+2kvvi0k2<" 2. Hence (5.11) holds true. Lemma 13 Let 0>0,D2D,⌧2Rand !2⌦. Then lim t,!1 sup 0Z[|u|] |u(⌧;⌧t, ✓⌧!,u0)(x)|pdx =0 (5.12) uniformly for u02D(⌧t, ✓t!).
24 Y.Li &F. Wang &T. Caraballo Proof The solution u(r;⌧t, ✓⌧!,u0)satisfies d dru(r)+Apu+↵u=f(x, u)+g(r, x)+h(x, u)⇣(✓r⌧!),(5.13) For any >0, multiplying (5.13) by (u)+, where a+= max(a, 0), we have 1 2 d drk(u)+k2+hApu,(u)+i+↵k(u)+k2 =hf(x, u),(u)+i+hh(x, u),(u)+i⇣(✓r⌧!) +hg(r, x),(u)+i.(5.14) On Rm[u], we have u>0and (u)+/u 1, then (4.2) implies that hf(x, u),(u)+i=Z[u] f(x, u)u (u)+ u dx 1Z[u] |u|p(u)+ u dx +Z[u] |1|dx. It is similar from (4.5) and (6.3) to see that hh(x, u),(u)+i⇣(✓r⌧!) |⇣(✓r⌧!)|Z[u] |(| 1|uq +| 2|u)(u)+ u dx 1 2Z[u] |u|p(u)+ u dx +cZ[u] (|⇣(✓r⌧!) 1|p pq+|⇣(✓r⌧!) 2|p p1)dx. We also have hApu,(u)+i0and hg(r, x),(u)+icZ[u] g2(r, x)dx +↵k(u)+k2. Substituting all estimates into (5.14), we obtain that d drk(u)+k2+1Z[u] |u|p(u)+ u dx 2Z[u] |1|dx +cZ[u] g2(r, x)dx +cZ[u] (|⇣(✓r⌧!) 1|p pq+|⇣(✓r⌧!) 2|p p1)dx. (5.15)
Higher-order continuity of pullback random attractors 25 Integrating (5.15) from r=⌧1to r=⌧we have Z⌧ ⌧1Z[u] |u(r)|p(u)+ u dxdr 1 1 k(u(⌧1) )+k2 +2 1Z⌧ ⌧1Z[u(r)] |1|dxdr +c6Z⌧ ⌧1Z[u(r)] g2(r, x)dxdr +c7Z⌧ ⌧1Z[u(r)] (|⇣(✓r⌧!) 1|p pq+|⇣(✓r⌧!) 2|p p1)dxdr. (5.16) Given ">0, by (5.11), there are 1>0and T">T 0such that for all 1,tT" and 0, 1 1 k(u(⌧1; ⌧t, ✓⌧!))+k21 1Z[u] u2 (⌧1)(x)dx < " 8.(5.17) By the absolute continuity of Lebesgue integrals, there is 1>0such that e⇢Rm,m(e)< 1)Ze |1|dx < 1" 16 . By (5.10), there is 21such that for all 2,tT",0, sup r2[⌧1,⌧] m[u(r;⌧t, ✓⌧!)]< 1 and thus for all 2,tT"and 0, 2 1Z⌧ ⌧1Z[u(r)] |1|dxdr 2 1Z⌧ ⌧1 1" 16 dr =" 8.(5.18) Lebesgue theorem and the absolute continuity imply that lim m(e)!0Z⌧ ⌧1Ze g2(r, x)dxdr =Z⌧ ⌧1 lim m(e)!0Ze g2(r, x)dxdr =0, which further implies that there is 2>0such that e⇢Rm,m(e)< 2)Z⌧ ⌧1Ze g2(r, x)dxdr < " 8c6 . By (5.10) again, there is 32such that for all 3,tT"and 0, sup r2[⌧1,⌧] m[u(r;⌧t, ✓⌧!)]< 2, and thus for all 3,tT"and 0, c6Z⌧ ⌧1Z[u(r)] g2(r, x)dxdr < " 8.(5.19)
32 Y.Li &F. Wang &T. Caraballo Applying the Gronwall lemma over [⌧,⌧+T]and noting U(⌧)=0, we have kU(⌧+T)kp p csup t2[⌧,⌧+T] |⇣(✓t!)|Z⌧+T ⌧ (ku(t)k2p2 2p2+ku1(t)k2p2 2p2+ 1)dt. (6.7) The Gronwall lemma on (4.13) implies that Z⌧+T ⌧ ku(t)k2p2 2p2dt ckKkp Y+cZ⌧+T ⌧ e↵s(kg(s)k2+|⇣(✓s!)|2p2 pq+|⇣(✓s!)|2+ 1)ds which is bounded as !1in view of (6.3). Hence, by (6.2), we see from (6.7) that kU(⌧+T)kp p!0, i.e., sup y2K k(T,⌧,!)y1(T,⌧)ykY!0as !1. The above limit holds in Xtoo, and thus (6.4) holds true. By the same method as in the proof of Proposition 2, using the convergence of ⇣(✓s!)as !0(see Lemma 15), one can prove (6.5). We finally establish the residual dense continuity of Ain both X=L2(Rm) and Y=Lp(Rm), where Ais the D-pullback random (X,Y )-attractors and A1is the D-pullback attractor for (6.1). Theorem 5 The continuity-set of the following set-valued mapping 7! A(⌧,!),(0,1]!C(X\Y),⇢ X\Y) is residual and dense in ⇤:= (0,1]for all ⌧2Rand !2⌦. continuous at 02R(⌧,!), For each ⌧2Rand !2⌦, there is a residual dense subset R(⌧,!) in ⇤=(0,1]such that the mapping 7! A(⌧,!),(0,1]!C(X\Y),⇢ X\Y) is continuous at 02R(⌧,!), where X=L2(Rm),Y=Lp(Rm)and ⇢X\Yis the Hausdorff metric. In addition, the mapping 7! A(⌧,!)is full upper semicontinuous from ⇤into X\Y. Proof We need to verify those conditions in Theorem 2. C1 and C2 (i.e. (A1) and (A4)) have been verified, while C3 follows from Lemmas 16. The parameter space ⇤=(0,1]can be divided into ⇤=[1 k=1⇤k, where ⇤1=[1,1],⇤ k=h1 k,1 k1i,k=2,3,··· . Obviously, each ⇤kis closed in ⇤and thus C4 holds. Moreover, ⇤1is a compact Hausdorff space (it is not a metric space) and ⇤in 1=(0,1], which is residual in ⇤1,
Higher-order continuity of pullback random attractors 33 while each ⇤k(k=2,3,···) is a complete metric space such that ⇤in kis residual in ⇤k. Hence C4* holds. By Lemma 9, for each k2N, we have the following uniform absorbing set Bk(⌧,!)={v2L2(Rm),kvk2cX(G(⌧)+ sup 1/k E(!)). Note that G(·)is tempered. For each ✏<↵, as t!1, e✏tsup 1/k E(✓t!)=e✏tsup 1/k Z0 1 e↵s|⇣(✓st!)|2p2 pqds e✏tsup 1/k Zt 1 e✏ 2(s+t)|⇣(✓s!)|2p2 pqds e✏ 2✏tCk(!)Zt 1 e3✏ 4sds !0 in view of Lemmas 5 and 15. Hence Bk2Dfor all k2N. Note that A⇢B kfor all 1/k, we know that for all k2, A⇤k(⌧,!):=[2⇤kA(⌧,!)⇢B k(⌧,!). For k=1, we have A⇤1(⌧,!):=[2⇤1A(⌧,!)⇢B 1(⌧,!)[A 1(⌧,!). Since A12D, we have B1[A 12D. Hence C6 hold true. It suffices to verify the condition C5 that A⇤k(⌧,!)is pre-compact in X\Y.We need to prove that for each k2Nthe set Ak(⌧,!):=[1/kA(⌧,!) is pre-compact in X\Y. Indeed, by the invariance, for any T>0and tT, Ak(⌧,!):=[1/k(t, ⌧t, ✓t!)A(⌧t, ✓t!) ⇢[ tT(t, ⌧t, ✓t!)Bk(⌧t, ✓t!). Since Bk2Das proved above, it follows from Propositions 5.9 and 14 that X\YAk(⌧,!)X\Y[tT(t, ⌧t, ✓t!)Bk(⌧t, ✓t!)!0 as T!1. Hence X\YAk(⌧,!)=0and thus Ak(⌧,!)is pre-compact in X\Y. Therefore, the residual dense continuity of 7! A(⌧,!)in X\Yfollows from the abstract Theorem 2. The full upper semi-continuity in Yfollows from the nonautonomous version of an abstract theorem as given by [25, Theorem 4.1].
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