Uniform attractors for non-autonomous p -Laplacian equations

Nonlinear Analysis 68 (2008) 3349–3363 www.elsevier.com/locate/na

Uniform attractors for non-autonomous p-Laplacian equationsI Guang-xia Chen ∗ , Cheng-Kui Zhong School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China Received 6 February 2007; accepted 19 March 2007

Abstract In this paper, by the Galerkin method, we give the existence of solutions for the non-autonomous p-Laplacian equation u t − div(|∇u| p−2 ∇u) + f (u) = g(t). After that, we explore the asymptotic behavior of the equation. The existence and the 1, p structures of the (L 2 (Ω ), L 2 (Ω ))-uniform attractor and the (L 2 (Ω ), L q (Ω ) ∩ W0 (Ω ))-uniform attractor are proved under the conditions below: the nonlinear term f is supposed to satisfy the polynomial condition of arbitrary order c1 |u|q − k ≤ f (u)u ≤ c2 |u|q + k and f 0 (u) ≥ −l, where q ≥ 2 is arbitrary; and the external force g(t) in L 2loc (R, L s (Ω )), s ≥ 2, is translation bounded and uniformly bounded in L s (Ω ) with respect to t ∈ R. c 2007 Elsevier Ltd. All rights reserved.

MSC: 35B40; 35B41; 35L70 Keywords: Galerkin method; Uniform attractor; Non-autonomous p-Laplacian equation; Asymptotic a priori estimate

1. Introduction Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω . We consider the equation u t − div(|∇| p−2 ∇u) + f (u) = g(t)

in Ω × R+

(1.1)

with the Dirichlet boundary condition u|∂ Ω = 0,

(1.2)

and initial condition u(x, τ ) = u τ (x),

∀τ ∈ R

(1.3)

where p ≥ 2; f : R → R satisfies the following assumptions f 0 (u) ≥ −l

for some l > 0

I The project is supported by the NNSF of China (No. 10471056). ∗ Corresponding author.

E-mail address: [email protected] (G.-x. Chen). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.03.025

(1.4)

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and c1 |u|q − k ≤ f (u)u ≤ c2 |u|q + k,

q ≥ 2.

(1.5)

The external force g(t) and the weak differential of g(t) with respect to t, denoted by h(t), are in the space L 2b (R, L s (Ω )); here s ≥ 2 and L 2b (R, L s (Ω )) is the translation bounded subspace in L 2loc (R, L s (Ω )) (see [6,7] for more details), i.e., g(t) ∈ L 2b (R, L s (Ω )), kgk2L 2 b

=

kgk2L 2 (R,L (Ω )) s b

Z = sup t∈R t

t+1

kgk2L s (Ω ) ds < +∞.

(1.6)

Furthermore, g(t) is uniformly bounded in L s (Ω ) with respect to t ∈ R, i.e., there exists a positive constant K , such that, sup kg(t, x)k L s (Ω ) ≤ K .

(1.7)

t∈R

In fact, the autonomous p-Lapalacian equation has been studied extensively in many monographs and lectures; see, e.g., [1–5,9] and the references therein. In [1], Babin and Vishik provided a detailed discussion about this problem: 1, p they got the existence of weak solution and the corresponding semigroup {S(t)}t≥0 has a (L 2 (Ω ), W0 (Ω )∩ L q (Ω ))weakly global attractor. In [2], Carvalho et al. considered the existence of global attractors for problems with monotone operators, and as an application, they got the existence of a (L 2 (Ω ), L 2 (Ω ))-global attractor for p-Laplacian equation, in which the nonlinear term satisfies a condition similar to (1.5). In [4], Carvalho and Gentile, combining with their 1, p comparison results developed in [3], obtained that the corresponding semigroup has a (L 2 (Ω ), W0 (Ω ))-global 1, p attractor. Recently, in [9], the authors got the existence of a (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-global attractor, they used a new a priori estimate method initiated by [8] to testify the asymptotic compactness of the corresponding semigroup. However, all of them considered the autonomous case. In fact, few people consider the non-autonomous case. In [6], Chepyzhov and Vishik studied the non-autonomous reaction–diffusion equation, which is a special case of the non-autonomous p-Laplacian equation when p = 2. But we haven’t got any results for the general p. In this paper, we talk about the general non-autonomous p-Laplacian equation; by the Galerkin method, we get the existence of a weak solution, and then applying the theories of non-autonomous dynamical systems established in [6,7], we construct the associated family of processes {Uσ (t, τ )}σ ∈Σ , and get the existence and structure of the 1, p (L 2 (Ω ), L 2 (Ω ))-uniform attractor and (L 2 (Ω ), L q (Ω ) ∩ W0 (Ω ))-uniform attractor for the corresponding family of processes. It is well known that if we want to prove the existence of the compact uniform attractor, we have to verify the 1, p family of processes {Uσ (t, τ )}σ ∈Σ has some kind of compactness, especially the compactness in L q (Ω ) ∩ W0 (Ω ) 1, p here. From Section 2, under the conditions (1.4)–(1.6), the solution of Eq. (1.1) is in L q (Ω ) ∩ W0 (Ω ), and has no higher regularity, therefore, there are no embedding results for these cases. Moreover, as we do not impose any 1, p restriction on p and q, it seems difficult to verify the uniformly asymptotic compactness in L q (Ω ) ∩ W0 (Ω ) by the 1, p usual methods. Hence, in order to obtain the compact (L 2 (Ω ), L q (Ω ) ∩ W0 (Ω ))-uniform attractor, one must use new method instead of the usual method. 1, p In this paper, motivated by the ideas in [8,9], we give a new method to verify the (L 2 (Ω ), L q (Ω ) ∩ W0 (Ω ))uniformly asymptotic compactness of {Uσ (t, τ )}σ ∈Σ corresponding to problem (1.1)–(1.3). Furthermore, we give the abstract result for the structure of the bi-space uniform attractor, under the conditions that {Uσ (t, τ )}σ ∈Σ is (X, Y )-uniformly asymptotically compact and (X × Σ , Y )-weakly continuous (see Section 3), here the symbol space Σ is weakly compact; this result improves the corresponding result in [6]. In fact, in [7], the authors have talked about this problem and got the structure of a uniform attractor: they assume that {Uσ (t, τ )}σ ∈Σ is uniformly ω-limit compact (see [7]) rather than uniformly asymptotically compact. As an application, we get the structure of the uniform attractor for the p-Laplacian equation. Throughout this paper, we use the following notations for convenience: let X be a Banach space, X ∗ be the dual space of X , k · kq be the norm of L q (Ω )(q ≥ 1), |u| the modular of u, (·, ·) be the inner product in L 2 (Ω ), and h·, ·i be the duality between X and X ∗ , m(e) (sometimes we also write it as |e|) the Lebesgue measure of e ∈ Ω , Ω (u ≥ M) = {x ∈ Ω : u(x) ≥ M} and Ω (u ≤ −M) = {x ∈ Ω : u(x) ≤ −M}, and C an arbitrary positive constant,

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which may be different from line to line (and even in the same line). Since Ω ⊂ Rn is a bounded domain, We take the 1, p equivalent norm in W0 (Ω ) to be p ! 1p n Z X ∂u k∇uk p = ∂x i i=1 Ω

1, p

for any u ∈ W0 .

2. Existence of weak solution In this section, we give the existence and the uniqueness of a weak solution. At first, we give some useful lemmas. Lemma 2.1 (See [1,6]). Suppose that E1 ⊂ E ⊂ E0, where E is a Banach space, and the embedding E 1 ⊂ E is compact. We assume that q1 ≥ 1

and

q0 ≥ 1.

Then the following embedding is compact Wq1 ,q0 (0, t; E 1 , E 0 ) ⊂ L q1 (0, t; E) where Wq1 ,q0 (0, t; E 1 , E 0 ) = {ϕ(s), s ∈ [0, t]|ϕ(s) ∈ L q1 (0, t; E 1 ), ϕ 0 (s) ∈ L q0 (0, t; E 0 )}, with norm Z kϕkWq1 ,q0 =

0

t

q

kϕ(s)k E11

1/q1

t

Z + 0

q

kϕ 0 (s)k E00

1/q0

.

Lemma 2.2 (See [1,6]). Let O be a bounded domain in Rn ×R, and let a sequence {gn }, {gn } ∈ L q (O), let 1 < q < ∞ be given. Assume that kgn k L q (O) ≤ C, where C is independent of n, gn → g (n → ∞) almost everywhere in O, and g ∈ L q (O). Then gn + g (n → ∞) weakly in L q (O). Theorem 2.3. Assume that Ω is a bounded domain in Rn , f satisfies (1.4) and (1.5) and g(t) ∈ L 2b (R, L s (Ω )) (s ≥ 2). Then, for any initial data u τ ∈ L 2 (Ω ) and any t ≥ τ , there exists a unique solution u for Eqs. (1.1)–(1.3) which satisfies 1, p

u ∈ L ∞ (τ, t; L 2 (Ω )) ∩ L p (τ, t; W0 (Ω )) ∩ L q (τ, t; L q (Ω )).

(2.1)

Proof. We implement the N -dimensional Galerkin methods. At first, we introduce a base consisting of functions 1, p e j ∈ H0l (Ω ), where l is sufficiently large such that H0l (Ω ) ⊂ L q (Ω ), H0l (Ω ) ⊂ W0 (Ω ), H0l (Ω ) ⊂ C 2 (Ω ), and the embedding H0l (Ω ) ⊂ L 2 (Ω ) is compact. The scalar product in L 2 (Ω ) generates on H0l (Ω ) ⊂ L 2 (Ω ) the bilinear function hu, vi0 , which can be represented in the form hu, vi0 = hLu, vil , where h.; .i0 and h.; .il denote the scalar product in L 2 (Ω ) and H0l respectively. Then we know the operator L is compact. Hence, L has a complete system of eigenvectors {e j }, j ∈ N; these vectors are orthogonal in L 2 (Ω ) and in H0l . By E n we denote the subspace span{e1 , e2 , . . . , en }, and by Pn the orthogonal projection in L 2 into E n . The Galerkin system of order n to (1.1) is the ordinary differential equation for u n ∈ E n ∂t u n + Pn Au n + Pn f (u n ) = Pn g(t),

u n (τ ) = Pn u τ

(2.2)

where Au = −div(|∇u| p−2 ∇u). Evidently, PP g(t) * g(t) (n → ∞) n u τ → u τ strongly in L 2 (Ω ) as n → ∞, and PnP weakly in L 2loc (R, L s (Ω )). Note that Pn u = nj=1 hu, e j ie j . On the other hand, since u n = ϕ j (t)e j , e j ∈ H0l ⊂

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C 2 (Ω ), then from the conditions imposed on f (u) and g(t), we know that the ordinary differential equation (2.2) with initial condition has a solution u n (t), which is defined on the interval τ ≤ t ≤ tn 0 , tn 0 ≥ τ . Eq. (2.2) implies that Z Z 1 d p g(t) · u n (2.3) f (u n ) · u n = ku n k22 + k∇u n k p + 2 dt Ω Ω from (1.5) we obtain that 1 d 1 p q ku n k22 + k∇u n k p + c1 ku n kq ≤ Ckg(t)k2L s (Ω ) + ku n k22 + k|Ω |. 2 dt 2 When q > 2, z ≥ 0, the following inequality holds (see Theorem 2.3.1 in [1]):

(2.4)

−µz q + λz 2 ≤ Cµ−2/(q−2) λq/(q−2) = C1

(2.5)

where C1 depends only on q. Setting µ = 12 c1 and λ = 1, we deduce from (2.4) and (2.5) that 1 d 1 1 p q ku n k22 + k∇u n k p + c1 ku n kq + ku n k22 ≤ Ckg(t)k2L s (Ω ) + k|Ω | + C1 2 dt 2 2 (when q = 2, (2.6) easily holds). So we have

(2.6)

1 d 1 ku n k22 + ku n k22 ≤ Ckg(t)k2L s (Ω ) + k|Ω | + C1 . 2 dt 2 By Gronwall’s lemma, we obtain ku n (t)k22 ≤ ku n (τ )k22 e−(t−τ ) + C2 (1 − e−(t−τ ) ) + C ≤

ku n (τ )k22 e−(t−τ )

where C2 = 2(k|Ω | + C1 ), and Z t Z e−(t−s) kg(s)k2L s ds ≤ τ

+ C2 (1 − e

t

t−1 Z t

≤ t−1

−(t−τ )

(2.7)

t

Z τ

e−(t−s) kg(s)k2L s (Ω ) ds

) + Ckg(t)k2L 2

(2.8)

b

e−(t−s) kg(s)k2L s ds + kg(s)k2L s ds + e−1

Z

Z

t−1

t−2 t−1

t−2

e−(t−s) kg(s)k2L s ds + · · ·

kg(s)k2L s ds + e−2

Z

t−2 t−3

kg(s)k2L s ds + · · ·

≤ (1 + e−1 + e−2 + · · ·)kg(t)k2L 2 b

≤ (1 − e−1 )−1 kg(t)k2L 2 b

≤ Ckg(t)k2L 2 . b

From equality (2.8) we know that the solution u n (t) of (2.2) exists for any t ≥ τ . Integrating (2.6) in s from τ to t, t ≥ τ , we obtain Z t Z t Z t p q 2 2 ku n (t)k2 − ku n (τ )k2 + 2 k∇u n (s)k p + c1 ku n (s)kq + ku n (s)k22 τ

t

Z ≤C

τ

τ

τ

kg(s)k2L s (Ω ) + C2 (t − τ ).

(2.9) 1, p

From (2.8) and (2.9), we deduce that the sequence {u n } is a bounded set of L ∞ (τ, t; L 2 (Ω )) ∩ L p (τ, t; W0 (Ω )) ∩ 1, p L q (τ, t; L q (Ω )). By (1.5), we know that { f (u n )} and {Au n } are bounded in L q 0 (τ, t; L q 0 (Ω )) and L p0 (τ, t; (W0 )∗ ) respectively (see [1]). Therefore, by taking, if necessary, a subsequence, we can assume that there exists a 1, p function u(s) ∈ L ∞ (τ, t; L 2 (Ω )) ∩ L p (τ, t; W0 (Ω )) ∩ L q (τ, t; L q (Ω )) such that u n (s) + u(s) weakly in 1, p L p (τ, t; W0 (Ω )), weakly in L q (τ, t; L q (Ω )), and ∗-weakly in L ∞ (τ, t; L 2 (Ω )) as n → ∞. See, from Theorem 1, p 2.3.1 in [1], as n → ∞, Au n (s) + Au(s) weakly in L p0 (τ, t; (W0 )∗ ), that ∂t u n (s) + ∂t u(s) weakly in

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L r (τ, t; (H0l (Ω ))∗ ), where r = min{ p 0 , q 0 } ≥ 1. Assume f (u n (s)) + η(s) weakly in L q 0 (τ, t; L q 0 (Ω )) for some η(s) ∈ L q 0 (τ, t; L q 0 (Ω )). Passing to the limit in (2.2), we obtain the equality ∂t u + Au + η = g(t)

(2.10) 1, p

in the space L r (τ, t; (H0l (Ω ))∗ ). Note by Lemma 2.1, where E 0 = H −l , E = L p (Ω ), E 1 = W0 (Ω ); we extract a subsequence of {u n } (denote by {u n }), u n + u in L p (τ, t; L p (Ω )) strongly, and u n (x, s) → u(x, s) for almost every (x, s) ∈ Ω × [τ, t] as n → ∞. Since f (u) ∈ C 1 (R); then f (u n (x, s)) → f (u(x, s)) (n → ∞) for almost every (x, s) ∈ Ω × [τ, t]. On the other hand, the sequence f (u n ) is bounded in L q 0 (τ, t; L q 0 (Ω )). From Lemma 2.2, we conclude that f (u n (x, s)) + f (u) (n → ∞) weakly in L q 0 (τ, t; L q 0 (Ω )); hence η(s) = f (u(x, s)). 1, p So u(s) ∈ L ∞ (τ, t; L 2 (Ω )) ∩ L p (τ, t; W0 (Ω )) ∩ L q (τ, t; L q (Ω )) is a solution of (1.1)–(1.3). At last, we verify the uniqueness of the solution. Let u 1 , u 2 be two solution of (1.1) with the initial data u 1 |t=τ = u 1 (τ ), u 2 |t=τ = u 2 (τ ) respectively. Subtracting the corresponding equation (1.1), we obtain ∂t (u 1 − u 2 ) + Au 1 − Au 2 + f (u 1 ) − f (u 2 ) = 0.

(2.11)

Multiplying (2.11) by u 1 − u 2 , and integrating on Ω , we obtain 1 d ku 1 − u 2 k22 + hAu 1 − Au 2 , u 1 − u 2 i + h f (u 1 ) − f (u 2 ), u 1 − u 2 i = 0. 2 dt

(2.12)

When p ≥ 2, the p-Laplacian operator has the property: there exists a positive constant δ, such that for all 1, p u 1 , u 2 ∈ W0 (Ω ), hAu 1 − Au 2 , u 1 − u 2 i ≥ δku 1 − u 2 k

p 1, p

W0 (Ω )

.

(2.13)

By (1.4) and (2.13), we deduce that 1 d p ku 1 − u 2 k22 + δk∇(u 1 − u 2 )k p ≤ lku 1 − u 2 k22 . 2 dt

(2.14)

Using the Gronwall lemma, we obtain ku 1 (t) − u 2 (t)k2 ≤ e2l(t−τ ) ku 1 (τ ) − u 2 (τ )k22 . So the uniqueness of the solution is proved.

(2.15)



3. Preliminaries and abstract results 3.1. Preliminaries In this section, we recall some basic concepts of bi-space uniform attractors (see [1,6,7] for details). Let Σ be a parameter set, X Y are two Banach spaces, Y ⊂ X continuously. {Uσ (t, τ )}σ ∈Σ is a family of processes in Banach space X . Denote by B(X ) the set of all bounded subsets of X . Definition 3.1. A set B0 ∈ B(Y ) is said to be (X, Y )-uniformly (with respect to (w.r.t.) σ ∈ Σ ) absorbing for the family S of processes {Uσ (t, τ )}σ ∈Σ if, for any τ ∈ R and every B ∈ B(X ), there exists t0 = t0 (τ, B) ≥ τ such that σ ∈Σ Uσ (t, τ )B ⊂ B0 for all t ≥ t0 . A set P belonging to Y is said to be (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting for the family of processes {Uσ (t, τ )}σ ∈Σ if, for an arbitrary fixed τ ∈ R and B ∈ B(X ), limt→+∞ (supσ ∈Σ distY (Uσ (t, τ )B, P)) = 0. Definition 3.2. A closed set AΣ ⊂ Y is said to be (X, Y )-uniform (w.r.t. σ ∈ Σ ) attractor of the family of processes {Uσ (t, τ )}σ ∈Σ if it is (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting and it is contained in any closed (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting set A0 of the family of processes {Uσ (t, τ )}σ ∈Σ : AΣ ⊂ A0 .

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Definition 3.3. Define the uniform (w.r.t. σ ∈ Σ ) ω-limit set of B by ωτ,Σ (B) = can be characterized, by following:

T

t≥τ

S

σ ∈Σ

S

s≥t

Uσ (s, τ )B. This

y ∈ ωτ,Σ (B) ⇔ there are sequences {xn } ⊂ B, {σn } ⊂ Σ , {tn } ⊂ Rτ , tn → ∞ such that Uσn (tn , τ )xn → y(n → ∞).

(3.1)

Definition 3.4. A family of processes {Uσ (t, τ )}σ ∈Σ possessing a compact (X, Y )-uniformly (w.r.t. σ ∈ Σ ) absorbing set is called (X, Y )-uniformly compact. And a family of processes {Uσ (t, τ )}σ ∈Σ is called (X, Y )-uniformly asymptotically compact if it possess a compact (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting set. Lemma 3.5. If a family of processes {Uσ (t, τ )}σ ∈Σ is (X, Y )-uniformly asymptotically compact, then for any τ ∈ R, B ∈ B(X ), (1) for any sequences {xn } ⊂ B, {σn } ⊂ Σ , {tn } ⊂ Rτ , tn → +∞ as n → ∞, there is a convergent subsequence of {Uσn (tn , τ )xn } in Y . (2) ωτ,Σ (B) is nonempty and compact in Y . (3) ωτ,Σ (B) = ω0,Σ (B) (4) limt→∞ (supσ ∈Σ distY (Uσ (t, τ )B, ωτ,Σ (B))) = 0. (5) if A is a closed set and (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting B, then ωτ,Σ (B) ⊂ A. Proof. (1) Let P be a compact (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting set of {Uσ (t, τ )}σ ∈Σ ; then for any ε > 0, τ ∈ R, and B ∈ B(X ), there is a t0 = t0 (τ, ε, B), such that [ Uσ (t, τ )B ⊂ NY (P, ε/2) ∀t ≥ t0 , (3.2) σ ∈Σ

where NY (P, ε/2) is the ε/2 neighborhood of P in Y . Then, for the fixed ε > 0, we can find N0 = N0 (t0 ), such that when n ≥ N0 , tn ≥ t0 . By the compactness of P, we can deduce that NY (P, ε/2) has a finite ε-net. Hence, when n ≥ N0 , there is a convergent subset of {Uσn (tn , τ )xn } in Y . (2) From (1), we know that ωτ,Σ (B) is nonempty. Let {yn }∞ n=1 be an arbitrary sequence in ωτ,Σ (B). From (3.1), we know that there are sequences {σnk } ⊂ Σ , {xnk } ⊂ B, {tnk } ⊂ Rτ , (n, k ∈ N), such that Uσnk (tnk , τ )xnk → yn for every n ∈ N as k → ∞. By choosing the diagonal sequence of {Uσnk (tnk , τ )xnk }, we can prove that {yn }∞ n=1 is a Cauchy sequence in ωτ,Σ (B) ⊂ Y . (3) see [7] for the proof. (4) and (5) the proof is similar to Proposition VII.1.1 in [6].  Theorem 3.6. If a family of processes {Uσ (t, τ )}σ ∈Σ acting on X is (X, Y )-uniformly asymptotically (w.r.t. σ ∈ Σ ) compact, then it possesses a (X, Y )-uniform (w.r.t. σ ∈ Σ ) attractor AΣ , the set AΣ is compact in Y , and attracts the bounded subset of X in the topology of Y . Proof. Assume P is a compact (X, Y )-uniformly (w.r.t. σ ∈ Σ ) attracting set of {Uσ (t, τ )}σ ∈Σ , and let A be a bounded neighborhood of P in Y ; then A is a bounded (X, Y )-uniformly (w.r.t. σ ∈ Σ ) absorbing set, and from Lemma 3.5, we can deduce that ωτ,Σ (A) is the (X, Y )-uniform (w.r.t. σ ∈ Σ ) attractor for {Uσ (t, τ )}σ ∈Σ .  Assumption 1. Let {T (h)|h ≥ 0} be a family of operators acting on Σ and satisfying: (1) T (h)Σ = Σ , ∀h ∈ R+ , (2) translation identity: Uσ (t + h, τ + h) = UT (h)σ (t, τ ),

∀σ ∈ Σ , t ≥ τ, τ ∈ R, h ≥ 0.

Definition 3.7. The kernel K of the process {U (t, τ )} acting on X consists of all bounded complete trajectories of the process {U (t, τ )}: K = {u(·)|U (t, τ )u(τ ) = u(t), dist(u(t), u(0)) ≤ Cu , ∀t ≥ τ, τ ∈ R}. The set K(s) = {u(s)|u(·) ∈ K} is said to be kernel section at time t = s, s ∈ R.

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Definition 3.8. {Uσ (t, τ )}σ ∈Σ is said to be (X ×Σ , Y )-weakly continuous if, for any fixed t ≥ τ , τ ∈ R, the mapping (u, σ ) → Uσ (t, τ )u is weakly continuous from X × Σ to Y . Assumption 2. Let Σ be a weakly compact set and {Uσ (t, τ )}σ ∈Σ be (X × Σ , Y )-weakly continuous. Theorem 3.9. Under Assumptions 1 and 2 with {T (h)}h≥0 , which is a weakly continuous semigroup. If {Uσ (t, τ )}σ ∈Σ acting on X is (X, Y )-uniformly (w.r.t. σ ∈ Σ ) asymptotically compact, then it possesses a (X, Y )-uniform (w.r.t. σ ∈ Σ ) attractor AΣ , AΣ is compact in Y , and attracts the bounded subset of X in the topology of Y ; moreover, [ Kσ (s), ∀s ∈ R. AΣ = ωτ,Σ (B0 ) = σ ∈Σ

Here B0 is a bounded neighborhood of the compact (X, Y )-uniformly attracting set in Y ; i.e., B0 is a bounded (X, Y )uniformly (w.r.t. σ ∈ Σ ) absorbing set of {Uσ (t, τ )}σ ∈Σ . Kσ (s) is the section at t = s of kernel Kσ of the process {Uσ (t, τ )} with symbol σ ∈ Σ . Furthermore, Kσ is nonempty for all σ ∈ Σ . Proof. Existence is given in Theorem 3.6. We now prove the last equality. Let S(t)(u, σ ) = (Uσ (t, 0)u, T (t)σ ),

t ≥ 0,

(u, σ ) ∈ X × Σ .

(3.3)

It is easy to verify that {S(t)}t≥0 is a weakly continuous semigroup on X × Σ , and B0 × Σ is a weakly compact set which is weakly attracting B × Σ for every B ∈ B(X ). Applying Theorem X.I.3.1 in [6], we have \[ w in Y ×Σ A = ω(B0 × Σ ) = S(t)(B0 × Σ ) s≥0 t≥s

= {γ (0)|γ (·) is a complete bounded trajectory of {S(t)}t≥0 }

(3.4)

is a weakly compact attractor which is invariant and weakly attracts B × Σ for every B ∈ B(X ) in the weak topology of Y × Σ . Denote by PA the projection of A from X × Σ to Y . For any y ∈ PA, from (3.3) and (3.4), we know that there are sequences {xn } ⊂ B0 , {σn } ⊂ Σ , {tn } ⊂ R+ , tn → +∞ as n → ∞, such that Uσn (tn , 0)xn + y in Y . Thanks to (3.1) and Lemma 3.5, {Uσn (tn , 0)xn } is precompact in Y ; hence Uσn (tn , 0)xn → y. So we have PA ⊂ AΣ . Conversely, we find AΣ ⊂ PA. Thus AΣ = PA. As in the proof of Theorem IV.5.1 in [6], we obtain [ PA = Kσ (s), s ∈ R σ ∈Σ

and Kσ (s) is nonempty for all σ ∈ Σ .



3.2. Abstract results From the ideas of [8,9], we give the following results, which are very useful for the existence of a uniform attractor in L p (Ω ). Proposition 3.10 (See [8,9]). For any ε > 0, the bounded subset B of L q (Ω ) (q > 0) has a finite ε-net in L q (Ω ) if there exists a positive constant M = M(ε) which depends on ε, such that: (1) B has a finite (3M)( p−q)/ p ( 2ε )q/ p -net in L p (Ω ) for some p, q > 0; and R 1/q (2) Ω (|u|≥M) |u|q < 2−(2q+2)/q ε for all u ∈ B. Proposition 3.11. Let {Uσ (t, τ )}σ ∈Σ be a family of processes on L p (Ω )( p ≥ 1), and suppose {Uσ (t, τ )}σ ∈Σ has a bounded (L p (Ω ), L p (Ω ))-uniformly (w.r.t. σ ∈ Σ ) absorbing set in L p (Ω ). Then, for any ε > 0, τ ∈ R and any bounded subset B ⊂ L p (Ω ), there exist positive constants T = T (B, τ ) and M = M(ε) such that: m(Ω (|Uσ (t, τ )u τ | ≥ M)) ≤ ε,

for any u τ ∈ B, t ≥ T, σ ∈ Σ .

Proof. From the assumption that {Uσ (t, τ )}σ ∈Σ has a bounded (L p (Ω ), L p (Ω ))-uniformly (w.r.t. σ ∈ Σ ) absorbing set in L p (Ω ), we know that there exists a bounded set B0 and M0 , where M0 is the upper bound of B0 in L p (Ω ), such

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that for any τ ∈ R and any bounded subset B of L p (Ω ), we can find a constant T = T (B, τ ) such that when t ≥ T [ Uσ (t, τ )B ⊂ B0 . σ ∈Σ

Then for ∀u τ ∈ B, ∀σ ∈ Σ and t ≥ T , we have p

kUσ (t, τ )u τ k p ≤ M0 . So we have Z M0 ≥ Ω

|Uσ (t, τ )u τ | p ≥

Z ≥ Ω (|Uσ (t,τ )u τ |≥M1 )

Z Ω (|Uσ (t,τ )u τ |≥M1 )

p M1

|Uσ (t, τ )u τ | p

p

= M1 · m(Ω (|Uσ (t, τ )u τ | ≥ M1 )).

This inequality implies that m(Ω (|Uσ (t, τ )u τ | ≥ M1 )) ≤ ε if we choose M1 large enough such that M1 ≥ ( Mε0 )1/ p .  Notice the definition of (X, Y )-uniformly (w.r.t. σ ∈ Σ ) asymptotically compact for the family of processes; combining this with the Proposition 3.10, we have the following result Corollary 3.12. Let {Uσ (t, τ )}σ ∈Σ be a family of processes on L p (Ω ) and be (L p (Ω ), L p (Ω ))-uniformly (w.r.t. σ ∈ Σ ) asymptotically compact; then {Uσ (t, τ )}σ ∈Σ is (L p (Ω ), L q (Ω ))-uniformly asymptotically compact, p ≤ q < ∞, if (1) {Uσ (t, τ )}σ ∈Σ has a bounded (L p (Ω ), L q (Ω ))-uniformly (w.r.t. σ ∈ Σ ) absorbing set B0 , (2) for any ε > 0, τ ∈ R and any bounded subset B ⊂ L p (Ω ), there exist positive constants M = M(ε, B) and T = T (ε, B, τ ), such that Z |Uσ (t, τ )u τ |q < ε for any u τ ∈ B, t ≥ T, σ ∈ Σ . Ω (|Uσ (t,τ )u τ |≥M)

Proposition 3.13. Let {Uσ (t, τ )}σ ∈Σ be a family of processes on L p (Ω ) and have a (L p (Ω ), L p (Ω ))-uniform (w.r.t. σ ∈ Σ ) attractor; Then, for any ε > 0, τ ∈ R, and any bounded subset B of L p (Ω ), we can find a constant M = M(ε) and T = T (B, τ, ε) such that Z |Uσ (t, τ )u τ | p < 2 p+1 ε for any u τ ∈ B, t ≥ T, σ ∈ Σ . Ω (|Uσ (t,τ )u τ |≥M)

Proof. Let A be the compact (L p (Ω ), L p (Ω ))-uniform (w.r.t. σ ∈ Σ ) attractor; then there is a T0 = T0 (ε, B, τ ), such that   ε 1/ p , for all t ≥ t0 , σ ∈ Σ Uσ (t, τ )B ⊂ N p A, 2 where N p (A, ε 2 ) means the ε 2 neighborhood of A with respect to the L p -norm. Since A is compact in L p (Ω ), there exist u 1 , u 2 , . . . , u k ∈ A such that for any u ∈ A, we can find some u i , (1 ≤ i ≤ k) satisfying Z 1/ p ε1/ p . (3.5) |u − u i | p dx < 2 Ω 1/ p

1/ p

Hence, for every u ∈ N p (A, ε 2 ), we can find u 0 ∈ A and some u i , (1 ≤ i ≤ k) such that Z 1/ p Z 1/ p Z 1/ p p p p |u − u i | dx ≤ |u − u 0 | dx + |u 0 − u i | dx 1/ p

Ω

Ω

≤

ε 1/ p 2

Ω

+

ε 1/ p 2

= ε1/ p .

(3.6)

G.-x. Chen, C.-K. Zhong / Nonlinear Analysis 68 (2008) 3349–3363

3357

At the same time, for the fixed ε > 0, there is a δ > 0, such that for each u i ∈ A, (1 ≥ i ≥ k), we have Z |u i | p < ε, e

provided that m(e) < δ (e ⊂ Ω ). 1/ p On the other hand, since N p (A, ε 2 ) is bounded in L p (Ω ). For the δ > 0 given above, there exists an M > 0, such that m(Ω (|u| ≥ M)) < δ hold for each u ∈ N p (A, ε 2 ). Therefore, we have, Z |Uσ (t, τ )u τ | p Ω (|Uσ (t,τ )u τ |≥M) Z Z p p p ≤2 |u i | p |Uσ (t, τ )u τ − u i | + 2 1/ p

Ω (|Uσ (t,τ )u τ |≥M)

Ω (|Uσ (t,τ )u τ |≥M)

≤ 2 p+1 ε

for all u τ ∈ B, t ≥ T0 , σ ∈ Σ .

Thus the result holds.



4. (L 2 (Ω), L 2 (Ω))-uniform attractor 4.1. Construction of the associated family of processes From Theorem 2.3, we know that the problem (1.1)–(1.3) generates a process {Uσ (t, τ )}, acting in L 2 (Ω ), and the 2 time symbol is σ (s) = g(s, x). We denote by L 2,w loc (R; L s (Ω )) the space L loc (R; L s (Ω )) endowed with a local weak 2,w convergence topology. Let Hw (g) be the hull of g in L loc (R; L s (Ω )), i.e., the closure of the set {g(s + h)|h ∈ R} in 2 L 2,w loc (R; L s (Ω )), and g(s, x) ∈ L b (R, L s (Ω )). Proposition 4.1 (See [6,7]). If E is reflective separable, ϕ ∈ L 2b (R; E) then (1) for all ϕ1 ∈ Hw (ϕ), kϕ1 k2 2 ≤ kϕk2 2 ; Lb

Lb

(2) the translation group {T (h)} is weakly continuous on Hw (ϕ); (3) T (h)Hw (ϕ) = Hw (ϕ) for h ≥ 0; (4) Hw (ϕ) is weakly compact. Due to the Proposition 4.1, Hw (g) is weakly compact and the translation semigroup {T (h)|h ∈ R+ } satisfies that T (h)Hw (g) = Hw (g) and is weakly continuous on Hw (g). Because of the uniqueness of solution, the following translation identity holds Uσ (t + h, τ + h) = UT (h)σ (t, τ ) ∀σ ∈ Hw (g), t ≥ τ, τ ∈ R, h ≥ 0.

(4.1)

Theorem 4.2. The family of processes {Uσ (t, τ )}σ ∈Hw (g) corresponding to problem (1.1)–(1.3) is (L 2 (Ω ) × 1, p Hw (g), L 2 (Ω ))-weakly continuous, and (L 2 (Ω ) × Hw (g), L q (Ω ) ∩ W0 (Ω ))-weakly continuous. Proof. For any fixed t1 and τ , t1 ≥ τ, τ ∈ R, let u τn * u τ (n → ∞) weakly in L 2 (Ω ), and σn + σ0 weakly in Hw (g) as n → ∞; denote by u n (t) = Uσn (t, τ )u τn . The same estimates for u n ∈ E n given in the Galerkin approximations (in Section 2) are valid for the u n (t) here. Therefore, for some subsequence {m} ⊂ {n} and w(t), such that for any 1, p t1 , τ ≤ t1 ≤ t, u m (t1 ) + w(t1 ) weakly in L 2 (Ω ) and L q (Ω ) ∩ W0 (Ω ). And the sequence {u m (s)}, τ ≤ s ≤ t, is 1, p bounded in the class L ∞ (τ, t; L 2 (Ω )) ∩ L p (τ, t; W0 (Ω )) ∩ L q (τ, t; L q (Ω )). Denote by η1 (s), and η0 (s) the weak 1, p limits of Au m (s) and f (u m (s)) in L p0 (τ, t; (W0 (Ω ))∗ ) and L q 0 (τ, t; L q 0 (Ω )) respectively. So we get the equation for w(s) ∂t w + η1 + η0 = σ0 . By the same method as in Theorem 2.3.2 in [1] and the proof of the Theorem 2.3, we know that η1 = Aw and 1, p η0 = f (w), which means that w(s) in L ∞ (τ, t; L 2 (Ω )) ∩ L p (τ, t; W0 (Ω )) ∩ L q (τ, t; L q (Ω )) is the weak solution

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of (1.1) with initial condition u τ . Due to the uniqueness of the solution, we state that Uσm (t1 , τ )u τm + Uσ0 (t1 , τ )u τ 1, p weakly in L 2 (Ω ) and L q (Ω ) ∩ W0 (Ω ). For any other subsequence {u τm 0 } and {σm 0 } satisfies u τm 0 + u τ weakly in L 2 (Ω ) and σm 0 + σ0 ; by the same process we obtain the analogous relation Uσm 0 (t1 , τ )u τm 0 + Uσ0 (t1 , τ )u τ weakly 1, p

in L 2 (Ω ) and L q (Ω ) ∩ W0 (Ω ) holds. Then it can be easily seen that for any weakly convergent initial sequence {u τn } ∈ L 2 (Ω ) and weakly convergent sequence {σn } ∈ Hw (g), we have Uσn (t1 , τ )u τn + Uσ0 (t1 , τ )u τ weakly in 1, p L 2 (Ω ) and L q (Ω ) ∩ W0 (Ω )).  1, p

4.2. (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniformly absorbing set and (L 2 (Ω ), L 2 (Ω ))-uniform attractor Theorem 4.3. The family of process {Uσ (t, τ )}σ ∈Hw (g) corresponding to problem (1.1)–(1.3) has a bounded 1, p (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniformly (w.r.t. σ ∈ Hw (g)) absorbing set. Proof. Multiplying (1.1) with an external force σ ∈ Hw (g) by u, and integrating on Ω , we have Z Z 1 d p kuk22 + k∇uk p + f (u)u = σ (t)u. 2 dt Ω Ω Combining this with assumption (1.5), the H¨older inequality and the embedding theorem when s ≥ 2, implies that Z 1 1 d p kuk22 + k∇uk p + c1 |u|q ≤ Ckσ (t)k2L s + kuk22 + k|Ω | 2 dt 2 Ω by inequality (2.5); setting λ = 1 and µ = 21 c1 , we deduce that Z 1 c1 1 d p 2 |u|q + kuk22 ≤ Ckσ (t)k2L s + k|Ω | + C1 kuk2 + k∇uk p + 2 dt 2 Ω 2

(4.2)

that is d kuk22 + kuk22 ≤ Ckσ (t)k2L s + C2 (|Ω |, C1 ). dt

(4.3)

Applying the Gronwall lemma and Proposition 4.1, we get Z t 2 2 −(t−τ ) −(t−τ ) ku(t)k2 ≤ ku(τ )k2 e + C2 (1 − e )+C e−(t−s) kσ (s)k2L s (Ω ) ds τ

≤

ku(τ )k22 e−(t−τ )

+ C2 (1 − e

−(t−τ )

) + Ckg(t)k2L 2 .

(4.4)

b

From this inequality, we know that the family of processes {Uσ (t, τ )}σ ∈Hw (g) has a (L 2 (Ω ), L 2 (Ω ))-uniformly absorbing set, i.e., for an arbitrary bounded subset B in L 2 (Ω ), there exists T1 = T1 (B, τ ) such that ku(t)k22 ≤ ρ0 (kg(t)k2L 2 , C2 ) b

for all t ≥ T1 , u τ ∈ B, σ ∈ Hw (g).

Taking t ≥ T1 , integrating (4.2) on [t, t + 1] and combining with (4.5), we have Z t+1 p q (k∇uk p + kukq ) ≤ C(|Ω |, ρ0 , kg(t)k2L 2 ) for any t ≥ T1 .

(4.5)

(4.6)

b

t

Meanwhile, let F(s) =

Rs 0

f (τ )dτ ; then by (1.5), we deduce that

c1 |u|q − k1 ≤ F(u) ≤ c2 |u|q + k1

(4.7)

and then q

Z

q

c1 kukq − k1 |Ω | ≤

F(u) ≤ c2 kukq + k1 |Ω |. Ω

(4.8)

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Hence from (4.6), we get  Z t+1  Z p k∇uk p + F(u) ds ≤ C(|Ω |, ρ0 , k1 , c2 , kg(t)k2L 2 ) b

Ω

t

for any t ≥ T1 .

(4.9)

On the other hand, multiplying (1.1) by u t and using the H¨older inequality and embedding theorem, we obtain   Z Z d 1 1 2 p F(u) ≤ Ckσ (t)k2L s + ku t k22 . ku t k2 + |∇u| + (4.10) dt p Ω 2 Ω Therefore,  Z  Z d 1 F(u) ≤ Ckσ (t)k2L s . |∇u| p + dt p Ω Ω

(4.11)

From (4.9) and (4.11), by virtue of the uniform Gronwall lemma, we get Z Z 1 |∇u| p + F(u) ≤ C(|Ω |, ρ0 , k1 , c2 , kg(t)k2L 2 ). p Ω b Ω

(4.12)

By (4.8) and (4.12), we deduce that for all t ≥ T1 + 1, Z Z p |∇u| + |u|q ≤ C(|Ω |, ρ0 , c2 , c1 , k1 , kg(t)k2L 2 ). Ω

(4.13)

b

Ω

1, p

From (4.13), we get the (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniformly absorbing set and thus complete the proof.



1, p

From the Theorem 4.3 and the compactness of the Sobolev embedding W0 (Ω ) ,→ L 2 (Ω ), and Theorem 3.9 we have the following result: Corollary 4.4. The family of processes {Uσ (t, τ )}σ ∈Hw (g) generated by (1.1)–(1.3) with initial data u τ in L 2 (Ω ) has a (L 2 (Ω ), L 2 (Ω ))-uniform (w.r.t. σ ∈ Hw (g)) attractor A2 , A2 which is compact in L 2 (Ω ) and attracts every bounded subset of L 2 (Ω ) in the topology of L 2 (Ω ). Moreover, [ A2 = ωτ,Hw (g) (B0 ) = Kσ (s), ∀s ∈ R, σ ∈Hw (g)

where B0 is the (L 2 (Ω ), L 2 (Ω ))-uniformly (w.r.t. σ ∈ Hw (g)) absorbing set in L 2 (Ω ), and Kσ (s) is the section at t = s of kernel of Kσ of the process {Uσ (t, τ )} with symbol σ ∈ Hw (g). 5. (L 2 (Ω), L q (Ω))-uniform attractor The main purpose of this section is to give an asymptotic a priori estimate for the unbounded part of the modular |u| for the solution u of problem (1.1)–(1.3) in the L q -norm. Lemma 5.1. Assume f (u) and g(t) satisfy the conditions in Section 1; for any ε > 0, τ ∈ R and any bounded subset B ⊂ L 2 (Ω ), there exist two positive constants T = T (B, ε, τ ) and M = M(ε), such that Z |Uσ (t, τ )u τ |q ≤ Cε, ∀t ≥ T, u τ ∈ B, σ ∈ Hw (g). (5.1) Ω (|Uσ (t,τ )u τ |≥M)

Proof of Lemma 5.1. From the assumption imposed on g(t), we can easily deduce that, for any σ ∈ Hw (g), sup kσ (t)k L s (Ω ) ≤ sup kg(t)k L s (Ω ) ≤ K . t∈R

(5.2)

t∈R

Moreover, from Proposition 3.11, we know that there exists T1 = T1 (B, ε, τ ) and M1 = M1 (ε) such that for any u τ ∈ B and t ≥ T1 and σ ∈ Hw (g), m(Ω (|Uσ (t, τ )u τ | ≥ M1 )) ≤ ε.

(5.3)

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In addition, by (1.5) we can take M0 large enough such that c1 |u|q−1 ≤ f (u) in Ω (u > M0 ), let M2 = max{M0 , M1 } and t ≥ T1 . Let (u − M2 )+ denotes the positive part of u − M2 , that is  u − M2 u ≥ M2 (u − M2 )+ = 0 u ≤ M2 . q−1

Multiplying (1.1) with external force σ ∈ Hw (g), by (u − M2 )+ and integrating over Ω , we have Z 1 d q |∇(u − M2 )+ | p |(u − M2 )+ |q−2 k(u − M2 )+ kq + (q − 1) q dt Ω (u≥M2 ) Z Z σ (t)|(u − M2 )+ |q−1 c1 |u|q−1 |(u − M2 )+ |q−1 ≤ +

(5.4)

using Young’s inequality, we get Z Z σ (t)|(u − M2 )+ |q−1 ≤ c1

(5.5)

Ω (u≥M2 )

Ω (u≥M2 )

Ω (u≥M2 )

for q ≥ 2, Z Ω (u≥M2 )

q−1

c1 |u|

|(u − M2 )+ |

q−1

|(u − M2 )+ |

2q−2

Ω (u≥M2 )

Z ≥ Ω (u≥M2 )

Z +C

s

2/s

|σ | Ω (u≥M2 )

c1 |(u − M2 )+ |2q−2

combining (5.2)–(5.6), we have 2/s Z d q s ≤ Cε. |σ | k(u − M2 )+ kq ≤ C dt Ω (u≥M2 ) R t+1 R t+1 q q From t k(u − M2 )+ kq ≤ t kukq ≤ C, by uniform Gronwall’s inequality, we obtain q

k(u − M2 )+ kq ≤ Cε.

(5.6)

(5.7)

(5.8)

Repeating the same steps above, just taking |(u + M2 )− |q−2 (u + M2 )− instead of |(u − M2 )+ |q−1 , we can deduce that q

k(u + M2 )− kq ≤ Cε

(5.9)

from (5.8) and (5.9), we have Z Z ||u| − M2 |q ≤ Ω (|u|≥M2 )

Let M = 2M2 Z Ω (|u|≥M)

q

Ω (u≥M2 )

Z

|u| dx =

Ω (|u|≥M) q−1

≤2

≤ 2q−1

Z Ω (u≤−M2 )

|(u + M2 )− |q ≤ 2Cε.

(5.10)

|(|u| − M2 ) + M2 |q dx

Z

q

Ω (|u|≥M)

Z Ω (|u|≥M)

Z

||u| − M2 | dx + ||u| − M2 |q dx +

q

Ω (|u|≥M)

Z Ω (|u|≥M)

M2 dx ||u| − M2 |q dx (5.11)

≤ Cε where C is independent of σ .

|(u − M2 )+ |q +



Remark. In this lemma, we assume that the external force g(t) ∈ L 2b (R, L s (Ω )) ∩ L ∞ (R, L s (Ω )), in fact, this is not necessary. In [7] the authors presented the normal external force (see [7] for details), which belongs to translation bounded space, but contains translation compact space in general topology. Here in this lemma, if g(t) is normal, the conclusion also holds.

G.-x. Chen, C.-K. Zhong / Nonlinear Analysis 68 (2008) 3349–3363

3361

From the lemma above, combining with Theorem 3.9, Corollary 3.12 and Theorem 4.3, we have the existence and structure of the (L 2 (Ω ), L q (Ω ))-uniform (w.r.t. σ ∈ Σ ) attractor. Theorem 5.2. The family of processes {Uσ (t, τ )}σ ∈Hw (g) corresponding to problem (1.1)–(1.3) with initial data u τ ∈ L 2 (Ω ) has a (L 2 (Ω ), L q (Ω ))-uniform (w.r.t. σ ∈ Hw (g)) attractor Aq , Aq which is compact in L q (Ω ) and attracts every bounded subset B of L 2 (Ω ) in the topology of L q (Ω ); moreover, [ Kσ (s), ∀s ∈ R, Aq = ωτ,Hw (g) (B0 ) = σ ∈Hw (g)

where B0 is the (L 2 (Ω ), L q (Ω ))-uniformly (w.r.t. σ ∈ Hw (g)) absorbing set, and Kσ (s) is the section at t = s of the kernel of Kσ of the process {Uσ (t, τ )} with symbol σ ∈ Hw (g). 1, p

6. (L 2 (Ω), W0 (Ω) ∩ L q (Ω))-uniform attractor 1, p

In this section, we prove the existence of The (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniform attractor. For this purpose, first we will give a priori estimates about u t endowed with an L 2 -norm. Lemma 6.1. Assume f (u) and g(t) satisfy the conditions in Section 1; then for any bounded subset B ⊂ L 2 (Ω ), and any τ ∈ R and σ ∈ Hw (g), there exists a positive constant T = T (B, τ ) ≥ τ , such that ku t (s)k22 ≤ ρ, where u t (s) =

for any u τ ∈ B, s ≥ T, σ ∈ Hw (g).

d dt (Uσ (t, τ )u τ )|t=s

and ρ is a positive constant which is independent of B and σ .

Proof. We only give some formal calculations. In fact, by use of the Galerkin approximation, we can get a rigorous proof. By differentiating (1.1) with the external force σ in time, and denoting v = u t , we get, vt − div(|∇u| p−2 ∇v) − ( p − 2) div(|∇u| p−4 ∇u(∇u · ∇v)) + f 0 (u)v = σ 0 (t)

(6.1)

where the “·” denote the dot product in Multiplying the above equality by v, integrating over Ω and using (1.4), we obtain that Z Z 1 d 2 p−2 2 kvk2 + |∇u| |∇v| + ( p − 2) |∇u| p−4 (∇u · ∇v)2 2 dt Ω Ω Z 2 0 ≤ lkvk2 + σ (t)v Rn .

Ω

1 ≤ lkvk22 + Ckσ 0 (t)k2L s + kvk22 . 2

(6.2)

Hence,   1 1 d 2 kvk2 ≤ l + kvk22 + Ckσ 0 (t)k2L s . 2 dt 2

(6.3)

On the other hand, integrating (4.10) from t to t + 1, t large enough and using (4.12), we have Z t+1 ku t k22 ≤ C(|Ω |, ρ0 , k1 , c2 , kgk2L 2 )

(6.4)

as t large enough. Combining (6.3) and (6.4), and using the uniform Gronwall lemma, we get Z |u t |2 ≤ C(|Ω |, ρ0 , k1 , c2 , kgk2L 2 , khk2L 2 , l)

(6.5)

b

t

b

Ω

as t large enough, independent of σ .



b

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Theorem 6.2. The family of processes {Uσ (t, τ )}σ ∈Hw (g) corresponding to problem (1.1)–(1.3) with initial data 1, p u τ ∈ L 2 (Ω ) is (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniformly (w.r.t. σ ∈ Hw (g)) asymptotically compact, i.e., there exists a 1, p compact uniformly attracting set in W0 (Ω ) ∩ L q (Ω ), which attracts any bounded subset B ⊂ Ł2 (Ω ) in the topology 1, p of W0 (Ω ) ∩ L q (Ω ). 1, p

Proof. Let B0 be a (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniformly (w.r.t. σ ∈ Hw (g)) absorbing set obtained in Theorem 4.3; then we need only to show that for any {u τn } ⊂ B0 , {σn } ⊂ Hw (g) and tn → ∞, 1, p

{Uσn (tn , τ )u τn }∞ n=1 is precompact in W0 (Ω ) ∩ L q (Ω ). Thanks to Theorem 5.2, it is sufficient to verify that for any {u τn } ⊂ B0 , {σn } ⊂ Hw (g) and tn → ∞, 1, p

{Uσn (tn , τ )u τn }∞ n=1 is precompact in W0 (Ω ).

(6.6)

is precompact in L 2 (Ω ) and L q (Ω ). In fact, from Corollary 4.4 and Theorem 5.2, we know that is a Cauchy sequence in L q (Ω ) and L 2 (Ω ). Without loss of generality, we assume that {Uσn (tn , τ )u τn }∞ n=1 1, p is a Cauchy sequence in W (Ω ). Now, we prove that {Uσn (tn , τ )u τn }∞ n=1 0 Denote by u σnn (tn ) := Uσn (tn , τ )u τn , and from (2.13), which is the property of a p-Laplacian operator when p ≥ 2, we have {Uσn (tn , τ )u τn }∞ n=1

δku σnn (tn ) − u σmm (tm )k ≤

hAu σnn (tn ) −

p 1, p

W0 Au σmm (tm ), u σnn (tn ) − u σmm (tn )i



d d ≤ − u σnn (tn ) − f (u σnn (tn )) + σn + u σmm (tm ) + f (u σmm (tm )) − σm , u σnn (tn ) − u σmm (tn ) dt dt Z d σ d u n (tn ) − u σm (tm ) |u σn (tn ) − u σm (tm )| ≤ m n n dt m Ω dt Z + | f (u σnn (tn )) − f (u σmm (tm ))| |u σnn (tn ) − u σmm (tm )| Ω Z + |σn − σm | |u σnn (tn ) − u σmm (tm )| Ω



d σ d σm σm σn n

≤ u n (tn ) − u m (tm )

ku n (tn ) − u m (tm )k2 dt dt 2 + Ckσn − σm ks ku σnn (tn ) − u σmm (tm )k2 + C(1 + ku σnn (tn )kq + ku σmm (tm )kq )ku σnn (tn ) − u σmm (tm )kq q

q

which, combined with Lemma 6.1, yields (6.6) immediately.



(6.7)



Theorem 6.3. The family of processes {Uσ (t, τ )}σ ∈Hw (g) corresponding to problem (1.1)–(1.3) with initial data 1, p u τ ∈ L 2 (Ω ) has a (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniform (w.r.t. σ ∈ Hw (g)) attractor A, A is compact in 1, p 1, p W0 (Ω ) ∩ L q (Ω ) and attracts every bounded subset B of L 2 (Ω ) in the topology of W0 (Ω ) ∩ L q (Ω ); moreover, [ A = ωτ,Hw (g) (B0 ) = Kσ (s), ∀s ∈ R, σ ∈Hw (g)

1, p where B0 is the (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-uniformly (w.r.t. σ ∈ Hw (g)) t = s of kernel of Kσ of the process {Uσ (t, τ )} with symbol σ ∈ Hw (g).

absorbing set, and Kσ (s) is the section at 1, p

Remark. Although the family of processes {Uσ (t, τ )}σ ∈Hw (g) is only (L 2 (Ω ), W0 (Ω ) ∩ L q (Ω ))-weakly continuous, we can also obtain that A2 , Aq and A coincide with each other; so the bounded complete trajectories 1, p of the process {Uσ (t, τ )}, ∀σ ∈ Hw (g) are in W0 (Ω ) ∩ L q (Ω ).

G.-x. Chen, C.-K. Zhong / Nonlinear Analysis 68 (2008) 3349–3363

3363

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