On the Dynamics of Nonautonomous Parabolic Systems ... - EMIS
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 178057, 27 pages doi:10.1155/2011/178057
Research Article On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators Anh Cung The1 and Toi Vu Manh2 1
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, 10307 Hanoi, Vietnam 2 Faculty of Computer Science and Engineering, Hanoi Water Resources University, 175 Tay Son, Dong Da, 10508 Hanoi, Vietnam Correspondence should be addressed to Anh Cung The, [email protected] Received 19 December 2010; Accepted 21 February 2011 Academic Editor: Feng Qi Copyright q 2011 A. Cung The and T. Vu Manh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullback Dattractor in L2 Ωm for the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullback D-attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.
1. Introduction Nonautonomous equations are of great importance and interest as they appear in many applications in the natural sciences. One way of studying the long-time behavior of solutions of such equations is using the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of such dynamical systems see 1 and references therein. In recent years, the existence of pullback attractors for reaction-diffusion equations has been studied widely see, e.g., 2–6. However, to the best of our knowledge, little seems to be known for the asymptotic behavior of solutions of nonautonomous degenerate equations. One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of the Grushin type 7 Gs u Δx u |x|2s Δy u,
X x, y ∈ Ω ⊂ ÊN1 × ÊN2 ,
s ≥ 0.
1.1
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The global existence and long-time behavior of solutions to semilinear parabolic equations involving the Grushin operator, in both autonomous and nonautonomous cases, have been studied in some recent works 8–10. In this paper we consider the following nonautonomous semilinear parabolic system: ∂u − aGs u fu gX, t, X ∈ Ω, t > τ, ∂t uX, t 0, X ∈ ∂Ω, t > τ, uX, τ uτ X, X ∈ Ω,
1.2
where X x, y ∈ Ω ⊂ ÊN1 × ÊN2 N1 , N2 1, uτ ∈ L2 Ω is given, u u1 , . . . , um is an unknown vector-function. Here a ∈ Matm Ê, fu f 1 u1 , . . . , um , . . . , f m u1 , . . . , um , and gX, t g 1 X, t, . . . , g m X, t satisfy the following conditions: m
H1 a ∈ Matm Ê has a positive symmetric part: 1/2a a∗
Ê
H2 f :
m
→
Ê
m
βIm , β > 0;
1
is a C -vector function such that: m f j uuj , C1 |u|p − C0 ≤ fu, u
p ≥ 2,
1.3
j1
fu ≤ C2 |u|p−1 1 ,
1.4
m m ∂f i −C3 |v|2 ≤ fu uv, v uvj vi , j ∂u i1 j1
1.5
where C0 , C1 , C2 , and C3 are positive constants; 1,2 H3 g ∈ Wloc Ê; L2 Ωm such that
0 −∞
e
λ1 βs
2 gs L2 Ωm ds < ∞, 0 −∞
e
0 s −∞
−∞
2 eλ1 βr gr L2 Ωm dr ds < ∞, 1.6
2 g s
λ1 βs
L2 Ωm ds
< ∞,
where λ1 is the first eigenvalue of the operator Gs in Ω with the homogeneous Dirichlet boundary condition. In order to study problem 1.2, we will use the natural energy space defined as the complete of C0∞ Ωm in the following norm:
S10 Ωm
u Ë10Ω
Ω
2
|∇x u| |x|
2s
2 ∇y u dX
1/2 .
Ë10Ω
:
1.7
From the results in 11, we know that the embedding Ë10Ω → Äp Ω is continuous if 1 p 2∗s : 2Ns/Ns − 2, where Ns : N1 s 1N2 ; moreover, this embedding is compact if 1 p < 2∗s .
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Notations Denote Äp Ω : Lp Ωm , and Êm , we set
Ë−1Ω the dual space of Ë10Ω. For functions u, v : ÊN
∇u, ∇v :
m i1
m N ∂ui ∂vi ∇ui , ∇vi , ∂Xk ∂Xk i1 k1
→
1.8
so if a aij m i,j1 ∈ Matm Ê, then a∇u, ∇v
m N ∂ui ∂vj aij ∇ui , ∇vj aij , ∂Xk ∂Xk i,j1 i,j1 k1 m
1.9
where ·, · denotes the inner product in ÊN . Noting that by assumption H1, we have a∇u, ∇u
m m m 2 1 aij aji ∇ui , ∇uj ≥ β ∇uj , ∇uj β ∇uj . 2 i,j1 j1 j1
1.10
Hence a∇x u, ∇x u |x|2s a∇y u, ∇y u dX ≥ β u 2Ë1Ω , Ω
0
1.11
Ω
aGs u, Gs udX ≥ β Gs u 2Ä2Ω .
1.12
The aim of this paper is to study the long-time behavior of solutions to problem 1.2 by using the theory of pullback D-attractors. We first prove, under assumptions H1–H3, the existence of a pullback D-attractor in Ä2 Ω for the process Ut, τ associated to problem 1.2. Then, with an additional condition that the system has a special gradient structure, namely, a βIm and there exists a function F : Êm → Ê such that fu gradu Fu, we prove the existence of a pullback D-attractor in the space Ë10Ω∩ Ä p Ω for the process Ut, τ. Moreover, we prove the boundedness of the pulback D-attractor in Ä2p−2 Ω and in Ë20Ω, and give estimates of the fractal dimension of the pulback D-attractor. It is worth noticing that our results, in particular, extend and improve some recent results on the existence of pullback D-attractors for the reaction-diffusion equations 3–5 and for the Grushin equations 8. Let us explain the methods used in the paper. We first prove the existence of a family of pullback D-absorbing sets in Ë10Ω. Thanks to the compactness of the embedding Ë10Ω → Ä2 Ω, we immediately get the existence of a pullback D-attractor in Ä2 Ω. When the system has a special gradient structure, we are able to prove the existence of a pullback D-attractor in Ë10Ω ∩ Äp Ω. To do this, we follow the general lines of the approach used in 8, with some modifications so that we can improve conditions imposed on the external force g. In particular, we use the asymptotic a priori estimate method initiated in 12 to testify the pullback asymptotic compactness of the corresponding process. Moreover, in this case we also prove the regularity of the pullback D-attractor in the spaces Ä2p−2 Ω and Ë10Ω. Finally,
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using the recent results in 13, we give an estimate of the fractal dimension of the pullback D-attractor. It is noticed that we do not impose the restriction on the exponent p in H2 as in 13. The rest of the paper is organized as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on pullback D-attractors which we will use. In Section 3, we prove the existence of a pullback D-attractor in Ä2 Ω in the general case. In Section 4, under the additional assumption that the system has a gradient structure, we prove the regularity and fractal dimension estimates of the pullback D-attractor.
2. Preliminaries 2.1. Pullback Attractors For convenience of the reader, we recall in this section some concepts and results on the theory of pullback D-attractors, which will be used in the paper. Let X be a metric space with metric d. Denote by BX the set of all bounded subsets of X. For A, B ⊂ X, the Hausdorff semidistance between A and B is defined by distA, B sup inf d x, y . x∈A y∈B
2.1
Let {Ut, τ : t ≥ τ, τ ∈ Ê} be a process in X, that is, Ut, τ : X → X such that Uτ, τ Id and Ut, sUs, τ Ut, τ for all t ≥ s ≥ τ, τ ∈ Ê. The process {Ut, τ} is said to be normto-weak continuous if Ut, τxn Ut, τx, as xn → x in X, for all t ≥ τ, τ ∈ Ê. The following result is useful for verifying the norm-to-weak continuity of a process. Proposition 2.1 see 14. Let X, Y be two Banach spaces, X ∗ , Y ∗ be, respectively, their dual spaces. Assume that X is dense in Y , the injection i : X → Y is continuous and its adjoint i∗ : Y ∗ → X ∗ is dense, and {Ut, τ} is a continuous or weak continuous process on Y . Then {Ut, τ} is norm-toweak continuous on X if and only if for t ≥ τ, τ ∈ Ê, Ut, τ maps a compact set of X to be a bounded set of X. {Dt : t ∈ Ê} ⊂ BX. Suppose that D is a nonempty class of parameterized sets D Definition 2.2. The process {Ut, τ} is said to be pullback D-asymptotically compact if for ∈ D, and any sequence {τn }n with τn ≤ t for all n, and τn → −∞, any any t ∈ Ê, any D sequence xn ∈ Dτn , the sequence {Ut, τn xn } is relatively compact in X. Definition 2.3. A process {Ut, τ} is called pullback ω-D-limit compact if for any ε > 0, any ∈ D, there exists a τ0 τ0 D, ε, t ≤ t such that t ∈ Ê, and D α
Ut, τDτ
≤ ε,
2.2
τ≤τ0
where α is the Kuratowski measure of noncompactness of B ∈ BX, αB inf δ > 0 | B has a finite open cover of sets of diameter ≤ δ .
2.3
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Lemma 2.4 see 3. A process {Ut, τ} is pullback D-asymptotically compact if and only if it is ω-D-limit compact. ∈ D is called pullback D-absorbing for the process Definition 2.5. A family of bounded sets B ∈ D, there exists τ0 τ0 D, t ≤ t such that {Ut, τ} if for any t ∈ Ê and any D
Ut, τDτ ⊂ Bt.
τ≤τ0
Definition 2.6. A family A {At : t ∈ {Ut, τ} if
Ê}
2.4
⊂ BX is said to be a pullback D-attractor for
1 At is compact for all t ∈ Ê; 2 A is invariant, that is, Ut, τAτ At, for all t ≥ τ; 3 A is pullback D-attracting, that is, lim distUt, τDτ, At 0,
τ → −∞
2.5
∈ D and all t ∈ Ê; for all D
4 if {Ct : t ∈ Ê} is another family of closed attracting sets, then At ⊂ Ct, for all t ∈ Ê. Theorem 2.7 see 3. Let {Ut, τ} be a norm-to-weak continuous process such that {Ut, τ} is {Bt : pullback D-asymptotically compact. If there exists a family of pullback D-absorbing sets B t ∈ Ê} ∈ D, then {Ut, τ} has a unique pullback D-attractor A {At : t ∈ Ê} and At
Ut, τBτ.
s≤t τ≤s
2.6
2.2. Fractal Dimension of Pullback Attractors Given a compact K ⊂ X and ε > 0, we denote by NK, ε the minimum number of open balls in X with radius ε which are necessary to cover K. Definition 2.8. For any nonempty compact set K ⊂ X, the fractal dimension of K is the number log NK, ε . ε→0 log 1/ε
dimf K : lim
2.7
Definition 2.9. A bounded subset B0 ⊂ H is called a uniformly pullback absorbing set for process Ut, τ if for every B ⊂ H is bounded, there exists a τ0 ≥ 0 such that Ut, t − τ0 B ∈ B0 , here, τ0 does not depend on the choice of t.
∀τ ≥ τ0 ,
2.8
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Theorem 2.10 see 13. Let Ut, τ be a process in a separable Hilbert space H, B be a uniformly {At : t ∈ Ê} be a pullback attractor for Ut, τ, if there exists a pullback absorbing set in H, A finite dimensional projection P in the space H such that
P Ut, t − T0 u1 − Ut, t − T0 u2 H ≤ lT0 u1 − u2 H
2.9
for all u1 , u2 ∈ B and some T0 , lT0 > 0 and
I − P Ut, t − T0 u1 − Ut, t − T0 u2 H ≤ δ u1 − u2
2.10
for all u1 , u2 ∈ B, where δ < 1, T0 and lT0 are independent of the choice of t. Then the family of {At : t ∈ Ê} possesses a finite fractal dimension especifically pullback attractors A
8lT0 2 −1 dimf At ≤ dim P log 1
, log 1−δ 1 δ
∀t ∈ Ê.
2.11
3. Existence of Pullback D-Attractors in 2 Ω Denote V : Lp τ, T; Äp Ω ∩ L2 τ, T; Ë10Ω , V ∗ : L2 τ, T; Ë−1Ω Lp τ, T; Äp Ω ,
3.1
where p is the conjugate of p i.e., 1/p 1/p 1. Definition 3.1. Let T > 0 and uτ ∈ problem 1.2 on τ, T if
Ä2 Ω be given. A function u is called a weak solution of
u ∈ V, u|tτ uτ T τ
∂u ∈ V ∗, ∂t a.e. in Ω,
ut , ϕ a∇x u, ∇x ϕ |x|2s a∇y u, ∇y ϕ fu, ϕ dX dt Ω T gt, ϕ dX dt τ
3.2
Ω
for all test functions ϕ ∈ V . One can prove that if u ∈ V and ∂u/∂t ∈ V ∗ , then u ∈ C0, T; Ä2 Ω see 10. This makes the initial condition in 1.2 meaningful.
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Theorem 3.2. Under assumptions (H1)–(H3), for any τ ∈ Ê, T > τ, uτ ∈ Ä2 Ω given, problem 1.2 has a unique weak solution u on τ, T. Moreover, the solution u exists on the interval τ, ∞ and the following inequality holds:
u 2Ä2Ω ≤ e−λ1 βt−τ uτ 2Ä2Ω
2C0 |Ω|
e−λ1 βt λ1 β
t −∞
2 eλ1 βs gs Ä2Ω ds,
∀t ≥ τ.
3.3
Proof. The existence and uniqueness of a weak solution to problem 1.2 are proved similarly to the scalar case in 10, so it is omitted here. We now prove inequality 3.3. Multiplying 1.2 by u, integrating over Ω, and using 1.11, we have 1 d
u 2Ä2Ω β u 2Ë1Ω
0 2 dt
Ω
fu, u dX
Ω
gt, u dX.
3.4
Using condition 1.3 and the Cauchy inequality, we obtain d 2 p gt 2 2 βλ1 u 2 2 .
u 2Ä2Ω 2β u 2Ë1Ω 2C1 u Äp Ω − 2C0 |Ω| ≤ Ä Ω Ä Ω 2 0 dt βλ1
3.5
Because u 2Ë1Ω ≥ λ1 u 2Ä2Ω , so 3.5 becomes 0
d 1 gt 2 2 .
u 2Ä2Ω λ1 β u 2Ä2Ω ≤ 2C0 |Ω|
Ä Ω dt λ1 β
3.6
Applying the Gronwall inequality we get 3.3. Now, we can define the family of two-parameter mappings Ut, τ : Ä2 Ω −→ Ë10Ω ∩ Äp Ω, uτ −→ Ut, τuτ ,
3.7
where Ut, τuτ ut is the unique weak solution of 1.2 with the initial datum uτ at time τ. Then U defines a continuous process on Ä2 Ω. Let R be the set of all functions r : Ê → 0, ∞ such that limt → −∞ eλ1 βt r 2 t 0 and {Dt : t ∈ Ê} ⊂ BË1Ω such that Dt ⊂ Brt denote by D the class of all families D 0 1 for some rt ∈ R, where Brt is the closed ball in Ë Ω with radius rt. 0
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Lemma 3.3. Under assumptions (H1)–(H3), there exists a constant C > 0 such that the solution u of problem 1.2 satisfies the following inequality for all t > τ:
1 1 e−αt−τ uτ 2Ä2Ω 1
t−τ t−τ
t 2 1
1
eαs gs Ä2Ω ds e−αt t−τ −∞ t s
2 1 −αt αr
1
e gr Ä2Ω dr ds , e t−τ −∞ −∞
u 2Ë1Ω ≤ C 0
1 t − τ
3.8
where α βλ1 . This implies that there exists a family of pullback D-absorbing sets in Ë10Ω for the process {Ut, τ}. Proof. We multiply 1.2 by −Gs u and integrate over Ω. After some standard transformations we obtain 1 d
ut 2Ë1Ω β Gs ut 2Ä2Ω 0 2 dt 2s fut, Δx ut |x| fu, Δy ut dX
g, Gs ut dX. Ω
3.9
Ω
Without loss of generality, we may assume that f0 0. Otherwise we can replace fu by fu fu − f0. The function f satisfies the same conditions with modified constants Ci i 0, 1, 2, 3, because |f0| C2 see 1.4. Hence, since fut|∂Ω 0, we get Ω
N1 m fu, Δx ut dX k1 i1
Ω
f i u
∂2 ui dX ∂Xk2
N1 m m ∂f i ∂uj ∂ui − dX u j ∂Xk ∂Xk k1 i1 j1 Ω ∂u
N1
∂u ∂u − fu u dX , ∂Xk ∂Xk k1 Ω ≤ C3
3.10
N1 ∂u 2 dX C3 |∇x u|2 dX, ∂X k Ω k1 Ω
where we have used condition 1.5. Similarly, we have Ω
|x|
fu, Δy ut dX ≤ C3
2s
Ω
2 |x|2s ∇y u dX.
3.11
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Hence Ω
fu, Δx ut dX
Ω
|x|2s fu, Δy ut ≤ C3 ut 2Ë1Ω . 0
3.12
By the Cauchy inequality we have
1 g 2 2 β Gs ut 2 2 . g, Gs ut dX ≤ Ä Ω Ä Ω 2 2β Ω
3.13
From 3.9–3.13 we obtain d 1 2
ut 2Ë1Ω β Gs ut 2Ä2Ω ≤ 2C3 ut 2Ë1Ω g Ä2Ω , 0 0 dt β
3.14
thus, d
ut 2Ë1Ω α ut 2Ë1Ω 0 0 dt
2C3 ut 2Ë Ω β1 g 2Ä Ω , 1 0
3.15
2
where α βλ1 . Multiplying 3.15 by t − τeαt and integrating from τ to t, we obtain 2
t − τe u Ë1Ω 0 αt
t
t−τ ≤ 2C3 t − τ 1 e us Ë1Ω ds
0 β τ αs
2
t −∞
2 eαs gs Ä2Ω ds. 3.16
Multiplying 3.3 by αeαt and integrating from τ to t, we have t α τ
eαs us 2Ä2Ω ds ≤ αt − τeατ uτ 2Ä2Ω
2C0 |Ω| αt e
α
t s −∞
−∞
2 eαr gr Ä2Ω dr ds. 3.17
Now, from 3.5 we get 2 1 d
u 2Ä2Ω β u 2Ë1Ω ≤ gt Ä2Ω 2C1 |Ω|. 0 dt α
3.18
Multiplying this equation by eαt and integrating from τ to t, we deduce that eαt ut 2Ä2Ω β
t τ
eαs us 2Ë1Ω ds 0
2C1 |Ω| αt 1 e
≤ eατ uτ Ä2Ω
α α 2
t τ
2 eαs gs 2
Ä Ω ds α
t τ
3.19 2
eαs us Ä2Ω ds.
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Using 3.17, 3.19 becomes
eαt ut 2Ä2Ω β
t τ
eαs us 2Ë1Ω ds 0
C0 C1 |Ω| αt e αt − τeατ uτ 2Ä2Ω α t s 2 2 eατ gs Ä2Ω ds
eαr gr Ä2Ω dr ds.
≤ eατ uτ 2Ä2Ω
1 α
t −∞
−∞
3.20
−∞
Substituting 3.20 into 3.16 we obtain
1 2C3 1
uτ 2Ä2Ω
ut 2Ë1Ω ≤ e−αt−τ 2C3
t − τ
0 λ1 λ1 βt − τ t
2 C0 C1 1 1 1 −αt 2C e
2C3
α
eαs gs Ä2Ω ds
|Ω| 3 t−τ αβ αβ t−τ −∞ t s
2 1 1 2C3
e−αt eαr gr Ä2Ω dr ds.
β t−τ −∞ −∞ 3.21
Hence we get 3.8 with C Cβ, C0 , C1 , C3 , λ1 . Let r02 t be the right-hand side of 3.8, and let B0 r0 t be the closed ball in Ë10Ω ∈ D and any t ∈ Ê, by 3.8 there exists centered at 0 with radius r0 t. Obviously for any D ≤ t such that the solution u with initial datum uτ ∈ Dτ at time τ satisfies τ0 τ0 D
ut Ë10Ω ≤ r0 t for all τ ≤ τ0 ; that is, B {B0 r0 t : t ∈ Ê} is a family of bounded pullback D-absorbing sets in Ë10Ω. From the above lemma we deduce that the process {Ut, τ} maps a compact set of to be a bounded set of Ë10Ω, and thus by Proposition 2.1, the process {Ut, τ} is norm-to-weak continuous in Ë10Ω. Since {Ut, τ} has a family of pullback D-absorbing sets in Ë10Ω and the embedding Ë10Ω → Ä2 Ω is compact, we immediately get the following.
Ë10Ω
Theorem 3.4. Under assumptions (H1)–(H3), the process {Ut, τ} associated to problem 1.2 has a pullback D-attractor in Ä2 Ω.
4. Some Further Results in the Gradient Case In this section, instead of H1–H3, we assume that H1bis a βIm , where Im is the unit matrix and β > 0;
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H2bis f satisfies H2 and fu gradu Fu ∂F/∂u1 u, . . . , ∂F/∂um u, where F : Êm → Ê is a potential function satisfying C1 |u|p − C0 ≤ Fu ≤ C2 |u|p C0 ,
∀u ∈ Êm ,
4.1
with C1 , C2 , C0 being positive constants 1,2 H3bis g ∈ Wloc Ê, Ä 2 Ω satisfies
0 −∞
2 2 eαt gt Ä2Ω g t Ä2Ω dt < ∞,
4.2
where α βλ1 . The aim of this section is to prove that the pullback D-attractor obtained in Section 3 is more regular and has a finite fractal dimension.
4.1. Existence of a Pullback D-Attractor in
Ë10Ω ∩ Äp Ω
Denote by R the set of all functions r : Ê → 0, ∞ such that limt → −∞ eλ1 βt r 2 t 0 {Dt : t ∈ Ê} ⊂ BË1Ω ∩ Äp Ω such and denote by D the class of all families D 0 that Dt ⊂ Brt for some rt ∈ R, where Brt is the closed ball in Ë10Ω ∩ Äp Ω with radius rt. Thanks to the above gradient structure, one can prove the existence of a pullback D-attractor, not only in Ä2 Ω, but also in the space Ë10Ω ∩ Äp Ω for the process {Ut, τ}. We first prove the following. Lemma 4.1. Under assumptions (H1bis)–(H3bis), the solution u of problem 1.2 satisfies the following inequality for all t > τ: p
2
u Ë1Ω u Äp Ω ≤ C e 0
−αt−τ
2
uτ Ä2Ω 1 e
−αt
t −∞
e
αr
2 gr 2
Ä Ω dr ,
4.3
where C CC0 , C1 , C1 , C0 , β, λ1 . This implies that there exists a family of pullback D-absorbing sets in Ë10Ω ∩ Äp Ω for the process {Ut, τ}. Proof. Using 3.5 with α λ1 β and the fact that u 2Ë1Ω ≥ λ1 u 2Ä2Ω , we have 0
2 β d 2 p
u 2Ä2Ω α u 2Ä2Ω u 2Ë1Ω 2C1 u Äp Ω ≤ 2C0 |Ω| gt Ä2Ω , 0 dt 2 α
4.4
thus 2 2 d αt p e u 2Ä2Ω Ceαt β u 2Ë1Ω 2 u Äp Ω ≤ 2C0 |Ω|eαt eαt gt Ä2Ω . 0 dt α
4.5
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Integrating from τ to s, τ ≤ s ≤ t 1, and in particular, we have eαs us 2Ä2Ω ≤ eατ uτ 2Ä2Ω 2
C0 2 |Ω|eαs
α α
s τ
2 eαr gr Ä2Ω dr.
4.6
Furthermore, multiplying 4.5 from s to s 1 and using 4.6 we obtain s 1 s
p eαr β ur 2Ë1Ω 2 ur Äp Ω dr 0 2 C0 |Ω| αs 2 s 1 αr ≤ e us Ä2Ω 2 e
e gr Ä2Ω dr α α s s 1 s 2 2 2 ατ αs αr αr ≤ C e uτ Ä2Ω e
e ur Ä2Ω e ur Ä2Ω 2
αs
s
≤ C eατ uτ 2Ä2Ω eαt
t τ
4.7
τ
eαr ur 2Ä2Ω .
By assumption H2bis, then 4.7 becomes s 1 e s
αr
2
2
β ur Ë1Ω 2Fur dr ≤ C e uτ Ä2Ω e
0 ατ
t
αt
τ
2
e ur Ä2Ω . αr
4.8
Multiplying 1.2 by ∂u/∂t and integrating over Ω, we have
2 1 1 d 1 2 gt, ut dX ≤ gt Ä2Ω ut 2Ä2Ω , β u Ë1Ω
FudX
ut Ä2Ω
0 2 dt 2 2 Ω Ω 4.9 2
thus
2 d αt αt 1 1 e β ut Ë0Ω 2 FutdX ≤ e β ut Ë0Ω 2 FutdX eαt gt Ä2Ω . dt Ω Ω 4.10 Using 4.8, 4.10, and the uniform Gronwall inequality, we get
e
αt
2
β ut Ë1Ω 2 0
Ω
FutdX
2
≤ C e uτ Ä2Ω e
ατ
t
αt
e τ
αr
2 gr Ä2Ω dr . 4.11
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Now, using H2bis once again we have from 4.11 that
e
αt
2
p
2
β ut Ë1Ω 2 ut Äp Ω ≤ C e uτ Ä2Ω e
0 ατ
t
αt
e τ
αr
2 gr 2
Ä Ω dr .
4.12
Thus we obtain 4.3 with a suitable positive constant C C C0 , C1 , C1 , C0 , β, λ1 .
4.13
Hence, by the argument as in the end of the proof of Lemma 3.3, we obtain a family of bounded pullback D-absorbing sets in Ë10Ω ∩ Äp Ω. To prove that the process {Ut, τ} is pullback D-asymptotically compact in Äp Ω, we need the following lemma. Lemma 4.2 see 8, Lemma 3.6. Let {Ut, τ} be a norm-to-weak continuous process in and Äp Ω, and let {Ut, τ} satisfy the following two conditions:
Ä2 Ω
i {Ut, τ} is pullback D-asymptotically compact in Ä2 Ω; ∈ D, there exist constants Mε, B and τ0 ε, B ≤ t such that ii for any ε > 0, B 1/p
Ω|Ut,τuτ |≥M
|Ut, τuτ |
p
< ε,
for any uτ ∈ Bτ, τ ≤ τ0 .
4.14
Then {Ut, τ} is pullback D-asymptotically compact in Äp Ω. Theorem 4.3. Under assumptions (H1bis)–(H3bis), the process {Ut, τ} associated to problem 1.2 has a pullback D-attractor in Äp Ω. Proof. It is sufficient to show that the process {Ut, τ} satisfies the condition ii in Lemma 4.2. We will give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in 15. Let M be a positive number, we will write u ≥ M or u ≤ −M as any component of u is greater than or equal to M or as any component of u is less than or equal to −M. Using 1.3, 1.4, and for u ≥ M large enough, we have √ fu, u − M ≥ fu, u − M mfu 3 |u|p − C 2 |u|p−1 ≥C C4 α ≥ |u − M|p |u − M|2 , 2 p
4.15 C4 3 , 1. because lim|u| → ∞ C
14
International Journal of Mathematics and Mathematical Sciences Multiplying 1.2 by u − M |u − M |p−2 and integrating over Ω we obtain 1 d p dt
p
Ωu≥M
|u − M| dX
Ωu≥M
Ωu≥M
|x|2s
Ωu≥M
a∇x u − M, ∇x u − M|u − M|p−2 dX
a∇y u − M, ∇y u − M|u − M|p−2 dX
|u − M|
p−2
fu, u − M dX
Ω
4.16
gt, u − M dX,
where
u − M :
⎧ ⎨u − M
if u ≥ M,
⎩0,
in other cases.
4.17
On the other hand, by the Cauchy inequality, we have C4 1 2 g , |u − M|2p−2
gt, u − M |u − M|p−2 ≤ |u − M|p−1 gt ≤ 2 2C4
4.18
which implies that C4 1 2 g . gt, u − M |u − M|p−2 ≥ − |u − M|2p−2 − 2 2C4
4.19
Hence, from 4.15 and 4.19, we have α 1 2 g . |u − M|p−2 fu, u − M gt, u − M ≥ |u − M|p − p 2C4
4.20
From 4.16, using 4.20 and noting that
Ωu≥M
a∇x u − M, ∇x u − M|u − M|p−2 dX
Ωu≥M
|x|
2s
a∇y u − M, ∇y u − M|u − M|
p−2
4.21 dX ≥ 0,
a βIm ,
we have p d p p g 2 2 .
u − M Äp Ωu≥M α u − M Äp Ωu≥M ≤ Ä Ω dt 2C4
4.22
International Journal of Mathematics and Mathematical Sciences
15
Now, multiplying the above inequality by t − τeαt and integrating from τ to t, we get p
t − τeαt u − M Äp Ωu≥M
t 2 p ≤ e u − M Äp Ωu≥M ds
t − τ eαs gs Ä2Ω ds 2C4 τ τ t t 2 p p ≤ eαs u Äp Ω ds
t − τ eαs gs Ä2Ω ds. 2C4 τ τ t
p
αs
4.23
Then p
u − M Äp Ω ≤
1 −αt e t−τ
t τ
p
eαs u Äp Ω ds
pe−αt 2C4
t −∞
2 eαs gs Ä2Ω ds.
4.24
On the other hand, integrating 4.5 from τ to t, we have t
C0 1 e us Äp Ω ds ≤ e uτ Ä2Ω 2 |Ω|eαt
α α τ p
αs
2
ατ
t τ
2 eαs gs Ä2Ω ds.
4.25
Therefore, substituting 4.25 into 4.24, we obtain p
u − M Äp Ω ≤ C
1 −αt e−αt−τ 1
e
uτ 2Ä2Ω
t−τ t−τ t−τ
t −∞
e
αs
2 gs 2
Ä Ω ds . 4.26
Hence, for any ε > 0, there exists M1 > 0 and τ1 < t such that for any τ < τ1 and any M ≥ M1 , we have Ωut≥M
|u − M|p dx ≤ ε.
4.27
Repeating the same step above, just taking u M− instead of u−M , we deduce that there exist M2 > 0 and τ2 < t such that for any τ < τ2 and any M ≥ M2 , Ωut≤−M
|u M|p dx ≤ ε,
4.28
where
u M−
⎧ ⎨u M,
u ≤ −M,
⎩0,
in other cases.
4.29
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International Journal of Mathematics and Mathematical Sciences
Let M0 max{M1 , M2 } and τ0 min{τ1 , τ2 }, we obtain Ω|u|≥M
|u| − Mp dx ≤ ε
for τ ≤ τ0 , M ≥ M0 .
4.30
So, we have
Ω|u|≥2M
|u|p dx
Ω|u|≥2M
|u| − M Mp dx
≤2
p
p−1 Ω|u|≥2M
|u| − M dx
≤ 2p−1
p
Ω|u|≥2M
M dx
|u| − Mp dx
Ω|u|≥M
Ω|u|≥M
4.31
|u| − Mp dx
≤ 2p ε. This completes the proof. To prove the existence of a pullback D-attractor in Ë10Ω∩ Ä p Ω, we need the folowing lemma. Lemma 4.4. Under assumptions (H1bis)–(H3bis), for any t ∈ Ê and any bounded subset B ⊂ Ä2 Ω, there exists a positive constant T TB, t ≤ t such that 2
ut t Ä2Ω ≤ C 1 e
−αt
t
2 2 gs Ä2Ω g s Ä2Ω ds , e
4.32
αs
−∞
for all τ ≤ TB, t and all uτ ∈ B, where C > 0 is independent of t and B. Proof. We give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in 15. Differentiating 1.2 in time and setting v ut , we get vt − aGs v f uv g r.
4.33
Multiplying this inequality by eαr v and integrating over Ω and using 1.11, we get 1 d αr e v 2Ä2Ω βeαr v 2Ë1Ω eαr 0 2 dr
Ω
α f uv, v dX ≤ eαr v 2Ä2Ω eαr 2
Ω
g r, v dX. 4.34
By the Cauchy inequality and using 1.5, we obtain that 2 d αr e v 2Ä2Ω ≤ 2C3 α 1eαr v 2Ä2Ω eαr g r Ä2Ω . dr
4.35
International Journal of Mathematics and Mathematical Sciences
17
Let τ ≤ s ≤ t − 1. Using 3.5, we have 2 1 d αs p e u 2Ä2Ω β u 2Ë1Ω 2C1 eαs u Äp Ω ≤ eαs gs Ä2Ω C0 eαs |Ω|. 0 ds α
4.36
By H2bis we then infer from the above inequality that
2 d αs FudX ≤ C eαs gs Ä2Ω eαs . e u 2Ä2Ω C βeαs u 2Ë1Ω 2eαs 0 ds Ω
4.37
Integrating this inequality from r to r 1, we obtain r 1
βeαs u 2Ë1Ω 2eαs FudX ds Ω
0
r
2
≤ C e ur Ä2Ω
αr
r 1
e
αs
r
2 αs gs Ä2Ω e ds .
4.38
On the other hand, integrating 4.36 from τ to t, we obtain eαt u 2Ä2Ω ≤ eατ uτ 2Ä2Ω
1 α
t
2 C0 |Ω| αt e . eαs gs Ä2Ω ds
α −∞
4.39
So, substituting 4.39 into 4.38, we deduce r 1
βeαs u 2Ë1Ω 2eαs FudX ds 0
r
Ω
2
≤ C e uτ Ä2Ω
ατ
t
−∞
e
αs
2 gs Ä2Ω ds eαt
4.40
< ∞,
∀r ∈ τ, t − 1.
Now multiplying 1.2 by eαr v and integrating over Ω, we have
d βeαr u 2Ë1Ω 2eαr FudX 0 dr Ω
2 FudX eαr gr Ä2Ω . ≤ α βeαr u 2Ë1Ω 2eαr
eαr v 2Ä2Ω
4.41
Ω
0
So applying the uniform Gronwall inequality, we get
2
βe u Ë1Ω 2e 0 αr
αr Ω
2
FudX ≤ C e uτ Ä2Ω e
ατ
t
αt
−∞
e
αs
2 gs Ä2Ω ds .
4.42
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International Journal of Mathematics and Mathematical Sciences
Integrating 4.41 from r to r 1 and by 4.40–4.42, we have r 1 r
2
2
e v Ä2Ω ds ≤ C e uτ Ä2Ω e
αs
ατ
αt
t −∞
e
αs
2 gs 2
Ä Ω ds .
4.43
Therefore, by 4.35, 4.43, using the uniform Gronwall inequality once again, we get 2
2
e v Ä2Ω ≤ C e uτ Ä2Ω e
αt
ατ
t
αt
2 2 gs Ä2Ω g s Ä2Ω ds . e αs
−∞
4.44
Hence we get 4.32. Theorem 4.5. Under assumptions (H1bis)–(H3bis), the process {Ut, τ} associated to problem 1.2 has a pullback D-attractor in Ë10Ω ∩ Äp Ω. Proof. By Lemma 4.1, {Ut, τ} has a family of bounded pullback D-absorbing sets in Ë10Ω ∩ Äp Ω. It remains to show that {Ut, τ} is pullback D-asymptotically compact in Ë10Ω ∩ Äp Ω, that is, for any t ∈ Ê, any B ∈ D, and any sequence τn → −∞, any sequence uτn ∈ Bτn , the sequence {Ut, τn uτn } is precompact in Ë10Ω ∩ Äp Ω. Thanks to Theorem 4.3, we need only to show that the sequence {Ut, τn uτn } is precompact in Ë10Ω. Let un t Ut, τn uτn . By Theorem 3.4, we can assume that {un t} is a Cauchy sequence in Ä2 Ω. We have
un t − um t 2Ë1Ω 0
−Gs un t − Gs um t, un t − um t dun dum − − u − u t t, n t m t − fun t − fum t, un t − um t dt dt 2 d d 2 2 ≤ un t − um t 2 un t − um t Ä2Ω C3 un t − um t Ä2Ω , dt dt Ä Ω
4.45
where we have used condition 1.5. Because {un t} is a Cauchy sequence in Ä2 Ω and by Lemma 4.4, one gets
un t − um t Ë10Ω −→ 0,
as m, n −→ ∞.
4.46
The proof is complete.
4.2.
Ä2p−2 Ω and Ë20Ω-Boundedness of the Pullback D-Attractor
First, we prove the existence of a family of pullback D-absorbing sets for process Ut, τ in Ä2p−2 Ω.
International Journal of Mathematics and Mathematical Sciences Proposition 4.6. Under assumptions (H1bis)–(H3bis), then for any t ∈ B ⊂ Ä2 Ω, there exists a positive constant τ0 τ0 B, t ≤ t such that
2p−2
u Ä2p−2Ω
2 ≤ C 1 gt Ä2Ω e−αt
t −∞
19
Ê and any bounded subset
2 2 e gs Ä2Ω g s Ä2Ω ds
αs
4.47
for all τ ≤ τ0 and all uτ ∈ B, where C > 0 is independent of t and B. Proof. Multiplying 1.2 by |u|p−2 u and integrating over Ω we obtain Ω
a∇x u, ∇x |u|
p−2
u
p−2 fu, u |u|p−2 dX a∇y u, ∇y |u| u
|x| dX
2s
p−2 ut , |u| u dX
gt, |u|p−2 u dX. − Ω
Ω
Ω
4.48
By the Cauchy inequality, 1.3 and note that Ω
a∇x u, ∇x |u|p−2 u
|x|2s a∇y u, ∇y |u|p−2 u dX ≥ 0;
here a βIm , 4.49
then we get 2p−2
C1 u Ä2p−2Ω ≤
1 1 gt 2 2 C1 u 2p−2
ut 2Ä2Ω
Ä Ä2p−2 . C1 C1 2
4.50
Hence, by 4.32 we deduce from 4.16 that t 2 2 C1 1 1 2p−2 −αt αs gt 2 2 . gs Ä2Ω g s Ä2Ω ds
C 1 e e
u Ä2p−2Ω ≤ Ä Ω 2 C1 C 1 −∞ 4.51 Therefore, we get 4.47 and the proof is complete. And now, we denote by Ë20Ω the closure of C0∞ Ωm in the norm
u Ë20Ω
Ω
1/2 . |Δx u|2 |x|2s Δy u|2 dX
4.52
It is easy to see that Ë20Ω is a Banach space endowed with the above norm. We now prove the Ë20Ω-boundedness of the pullback D-attractor. First, we recall a lemma see 15 which is necessary for our proof.
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International Journal of Mathematics and Mathematical Sciences
Lemma 4.7. Let X, Y be Banach spaces such that X is reflexive, and the inclusion X ⊂ Y is continuous. Assume that {un } is a bounded sequence in L∞ τ, T; X such that un u weakly in Lq τ, T; X for some q ∈ 1, ∞ and u ∈ Cτ, T; Y . Then, ut ∈ X for all t ∈ τ, T and
ut X ≤ supn≥1 un L∞ τ, T; X, for all t ∈ τ, T. Let un t be the Galerkin approximations of the solution ut of 1.2 then by Lemma 4.7 with noticing that un Ut, τunτ u Ut, τuτ in L2 τ, T; Ë10Ω and the inclusion Ë20Ω ⊂ Ë10Ω is continuous, we only need the estimation on ut Ut, τuτ . Theorem 4.8. Under assumptions (H1bis)–(H3bis), the pullback D-attractor A {At : t ∈ Ê} in Ë10Ω ∩ Äp Ω of the process {Ut, τ} is bounded in Ë20Ω. More precisely, for any τ < T1 < T2 , the ! set t∈T ,T At is a bounded subset of Ë20Ω. 1
2
Proof. Let us fix a bounded set B ⊂ Ä2 Ω, τ ∈ Ê, ε > 0, t > τ ε and uτ ∈ B. Multiplying the first equation in 1.2 by Gs u and integrating over Ω, we have Ω
aGs ur, Gs urdX
Ω
u r , Gs ur dX
−
Ω
Ω
fur, Gs ur dX
gr, Gs ur dX.
4.53
By the Cauchy inequality we have −
Ω
2 β 2 gr, Gs ur dX ≤ gr Ä2Ω Gs ur 2Ä2Ω , β 8
β 2 2 ur , Gs ur dX ≤ u r Ë1Ω Gs ur 2Ä2Ω . 0 β 8 Ω
4.54
Using 3.12, 1.12, and 4.54, then from 4.53 we get β Gs ur 2Ä2Ω ≤ C3 ur 2Ë1Ω
0
β 2 u r 2 2 gr 2 2
Gs ur 2Ä2Ω Ä Ω Ä Ω β 4
C3 ≤
2C32 β
Ω
ur, −Gs urdX
ur 2Ä2Ω
β 2 u r 2 2 gr 2 2
Gs ur 2Ä2Ω Ä Ω Ä Ω β 4
2 u r 2 2 2 gr 2 2 β Gs ur 2 2 . Ä Ω Ä Ω Ä Ω 2 β β 4.55
Hence, 2 2C32 β 2 2 2
Gs ur 2Ä2Ω ≤
ur 2Ä2Ω u r Ä2Ω gr Ä2Ω . 2 β β β
4.56
International Journal of Mathematics and Mathematical Sciences
21
Differentiating the first equation in 1.2 in time t and setting vr u r, then multiplying by vr and using 1.11 we get 1 d
vr 2Ä2Ω β vr 2Ë1Ω ≤ − 0 2 dr
Ω
fu uvr, vr dX
Ω
g r, vr dX
1 ≤ C3 vr Ä2Ω vr 2Ä2Ω
2 2
1 g r 2 2 . Ä Ω 2
4.57
Hence, 2 d
vr 2Ä2Ω ≤ 2C3 1 vr 2Ä2Ω g r Ä2Ω . dr
4.58
Integrating the above inequality, we have
vr 2Ä2Ω ≤ vs 2Ä2Ω 2C3 1
t τ ε/2
vθ 2Ä2Ω
t τ ε/2
2 g θ 2 dθ, Ä Ω
4.59
for all τ ε/2 ≤ s ≤ r ≤ t. Now, integrating with respect to s between τ ε/2 and r, we get
r −τ − ≤
ε
vr 2Ä2Ω 2
r
τ ε/2
vs 2Ä2Ω ds
2 ε t ε t 2 g θ 2 dθ
2C3 1 r − τ −
vθ Ä2Ω dθ r − τ − Ä Ω 2 τ ε/2 2 τ ε/2 t 2 ε ε t g θ 2 dθ,
1 ≤ 2C3 1 t − τ −
vθ 2Ä2Ω dθ r − τ − Ä Ω 2 2 τ ε/2 τ ε/2 4.60 for all τ ε/2 ≤ r ≤ t, and in particular, for all r ∈ τ ε, t we have that from the above estimate
vr 2Ä2Ω ≤
t t 2 ε 2
1
vθ 2Ä2Ω dθ g θ Ä2Ω dθ. 2C3 1 t − τ − ε 2 τ ε/2 τ
4.61
On the other hand, multiplying the first equation in 1.2 by vr and integrating over Ω, we deduce that β d
vr Ä2Ω
ur 2Ë1Ω
0 2 dr
2
Ω
fu, v dX ≤
Ω
g, v dX,
4.62
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International Journal of Mathematics and Mathematical Sciences
where we have used 1.11. Using the Cauchy inequality and condition H2bis, then 4.62 becomes
vr 2Ä2Ω
2 d β ur 2Ë1Ω 2 FurdX ≤ gr Ä2Ω . 0 dr Ω
4.63
Integrating from τ ε/2 to t we have
t τ ε/2
vθ 2Ä2Ω dθ β ut 2Ë1Ω 2 0
ε 2 FutdX ≤ β u τ
2 Ë10Ω Ω
t ε gθ 2 2 , dX
2 F u τ
Ä Ω 2 τ ε/2 Ω
4.64
and hence because of 4.1, we get t
ε ε 2 p
vθ 2Ä2Ω dθ ≤ β u τ
1 2C2 u τ
2 Ë0Ω 2 Äp Ω τ ε/2 t 2
4C0 |Ω| gθ Ä2Ω .
4.65
τ
Now, substituting 4.65 into 4.61 we deduce
vθ 2Ä2Ω ≤
2 ε
1 2C3 1 t − τ − ε 2 t 2 ε ε 2 2 gθ Ä2Ω dθ
2C2 u τ
4C0 |Ω|
× β u τ
2 Ë10Ω 2 Äp Ω τ
t
2 g θ 2
Ä Ω dθ,
τ
4.66
for all r ∈ τ ε, t. Finally, from 4.66 and 4.56 we obtain
Gs ur 2Ä2Ω ≤
8 t−τ −
1 2C 3 β2 ε ε 2 × β u τ
2 Ë10Ω 2
β
ε
1 2 t 2 ε p
2C2 u τ
4C0 |Ω| gθ Ä2Ω dθ 2 Äp Ω τ
t
2 g θ 2 dθ 2 C2 ur 2 2 2 gr 2 2 3 Ä Ω Ä Ω Ä Ω , β2 τ
Because u 2Ë2Ω Gs u 2Ä2Ω then from 4.67, the proof is complete. 0
∀r ∈ τ ε, t. 4.67
International Journal of Mathematics and Mathematical Sciences
23
4.3. Fractal Dimensional Estimates of the Pullback D-Attractor Theorem 4.9. Under assumptions (H1bis)–(H3bis), the process Ut, τ possesses a pullback Dattractor AÄ2Ω which has a finite fractal dimension in Ä2 Ω and
8 · e2C3 dimf At ≤ k log 1
1−δ
log
2 1 δ
−1 ,
∀t ∈ Ê,
4.68
where δ < 1, k ∈ Æ , and C3 in 1.5. Proof. Let Hk span{e1 , e2 , . . . , ek } ⊂ Ä2 Ω and Pk : Ä2 Ω → Hk be the orthogonal projection, where e1 , e2 , . . . , ej , . . . are the eigenvectors of the operator −Gs corresponding to eigenvalues {λj }∞ j1 such that 0 < λ1 < λ2 ≤ λ3 ≤ · · · ≤ λj ≤ · · · and λj → ∞ as j → ∞. From 4.3, we can easily show that there exists a uniformly pullback absorbing set B of process Ut, τ in Ë10Ω. We set u1 t Ut, τu1τ and u2 t Ut, τu2τ to be solutions associated to problem 1.2 with initial datum u1τ , u2τ ∈ B. Let w u1 − u2 , because u1 , u2 being two solutions of 1.2 then we have ∂w − aGs w fu1 − fu2 0. ∂t
4.69
Multiplying 4.69 with w and integrating over Ω then we have 1 d
wt 2Ä2Ω β w 2Ë1Ω
0 2 dt
Ω
fu1 − fu2 , w dX ≤ 0,
4.70
here, we have used 1.11. Using 1.5 then we have d
wt 2Ä2Ω ≤ 2C3 wt 2Ä2Ω . dt
4.71
wt 2Ä2Ω ≤ e2C3 t−τ wτ 2Ä2Ω .
4.72
Thus,
Let wt w1 t w2 t where w1 t : Pk wt and w2 t : I − Pk wt. Therefore, by 4.72 we have
w1 t 2Ä2Ω ≤ e2C3 t−τ wτ 2Ä2Ω .
4.73
Now, taking the inner product of 4.69 with w2 in Ä2 Ω, we have 1 d 2 2
w2 t Ä2Ω β w2 Ë1Ω ≤ − fu1 − fu2 , w2 dX. 0 2 dt
4.74
24
International Journal of Mathematics and Mathematical Sciences
Using the Holder inequality and 1.4, we have ¨ −
Ω
fu1 − fu2 , w2 dX ≤
Ω
fu1 − uu2 |w2 |dX
≤
Ω
≤C
fu1 − fu2 2 dX
1/2
2
Ω
|w2 | dX
1/2
1/2 1 |u1 |2p−2 |u2 |2p−2 dX
w2 Ä2Ω
4.75
Ω
2p−2
2p−2
≤ C 1 u1 Ä2p−2Ω u2 Ä2p−2Ω
1/2
w Ä2Ω
2p−2 2p−2 ≤ C 1 u1 Ä2p−2Ω u2 Ä2p−2Ω w Ä2Ω . Therefore, by 4.74, 4.75, and Proposition 4.6 we obtain
1 d
w2 t 2Ä2Ω β w2 2Ë1Ω 0 2 dt t 2 2 2 −αt αs ≤ C 1 gt Ä2Ω e e gs Ä2Ω g s Ä2Ω ds wt Ä2Ω . −∞
4.76
Because w2 t 2Ë1Ω ≥ λk w2 t 2Ä2Ω , then 4.76 implies that 0
d
w2 t 2Ä2Ω 2βλk w2 2Ä2Ω dt t 2 2 2 −αt αs gs Ä2Ω g s Ä2Ω ds wt Ä2Ω . ≤ 2C 1 gt Ä2Ω e e −∞
4.77
Now, multiplying 4.77 by eβλk t and integrating from τ to t, we get
2
w2 t Ä2Ω ≤ e
−βλk t−τ
2
wτ Ä2Ω 2Ce
−βλk t
t
e−βλk s
τ
s 2 2 2 × 1 gs Ä2Ω e−αt eαr gr Ä2Ω g r Ä2Ω dr −∞
× ws Ä2Ω ds.
4.78
International Journal of Mathematics and Mathematical Sciences
25
Using 4.73 we have
2
w2 t Ä2Ω ≤ e
−βλk t−τ
2
wτ Ä2Ω C wτ Ä2Ω e
−βλk t
t
eβλk s eC3 s−τ
τ
s 2 2 2 −αs × 1 gs Ä2Ω e eαr gr Ä2Ω g r Ä2Ω dr ds −∞
≤ e−βλk t−τ wτ 2Ä2Ω C wτ Ä2Ω eC3 t−τ
t −∞
e−βλk t−s
s 2 2 2 −αs αr gr Ä2Ω g r Ä2Ω dr ds × 1 gs Ä2Ω e e −∞
≤ e−βλk t−τ wτ Ä2Ω C wτ Ä2Ω eC3 t−τ " t t 2 1 ×
e−βλk t−s gs Ä2Ω ds
e−βλk t−s e−αs βλk −∞ −∞
s 2 2 × eαr gr Ä2Ω g r Ä2Ω dr ds . 2
−∞
4.79
Now, because t −∞
2 eαs gs Ä2Ω ds < ∞,
4.80
we can see that, for all t ∈ Ê see, e.g., 6, Lemma 3.6, t −∞
2 e−βλk t−s gs Ä2Ω ds −→ 0 as k −→ ∞,
4.81
and we have t −∞
e
−βλk t−s −βλ1 s
≤
s
e
t −∞
e
2 2 gr Ä2Ω g r Ä2Ω dr ds e αr
−∞
−βλk t βλk −λ1 s
e−αt βλk − α
t −∞
t ds
2 2 e gr Ä2Ω g r Ä2Ω dr αr
−∞
2 2 eαs gr Ä2Ω g r Ä2Ω dr.
4.82
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International Journal of Mathematics and Mathematical Sciences
Thus, for any t ∈ Ê, from 4.82 we have t −∞
e
−βλk t−s −βλ1 s
s
e
2 2 e gr Ä2Ω g r Ä2Ω dr ds −→ 0 αr
−∞
as k −→ ∞. 4.83
Combining 4.81, 4.83 and taking T0 t − τ 1, we get k is sufficient large then from 4.79 we deduce
w2 t 2Ä2Ω ≤ δ wτ 2Ä2Ω ,
4.84
here 0 < δ < 1.
From 4.73 and 4.31, we have
w1 t 2Ä2Ω ≤ l0 wτ 2Ä2Ω ,
w2 t 2Ä2Ω ≤ δ wτ 2Ä2Ω ,
∀t ∈ Ê.
4.85
Here, l0 e2C3 ; 0 < δ < 1; T0 1. Thus, the process Ut, τ associated to 1.2 satisfies all conditions of Theorem 2.10. This completes the proof.
Acknowledgment This paper was supported by Vietnam’s National Foundation for Science and Technology Development NAFOSTED, project 101.01-2010.05.
References 1 T. Caraballo, G. Łukaszewicz, and J. Real, “Pullback attractors for asymptotically compact nonautonomous dynamical systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 3, pp. 484–498, 2006. 2 M. Anguiano, T. Caraballo, and J. Real, “H 2 -boundedness of the pullback attractor for a nonautonomous reaction-diffusion equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 876–880, 2010. 3 Y. Li and C. Zhong, “Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1020–1029, 2007. 4 Y. Li, S. Wang, and H. Wu, “Pullback attractors for non-autonomous reaction-diffusion equations in Lp ,” Applied Mathematics and Computation, vol. 207, no. 2, pp. 373–379, 2009. 5 H. Song and H. Wu, “Pullback attractors of nonautonomous reaction-diffusion equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1200–1215, 2007. 6 Y. Wang and C. Zhong, “On the existence of pullback attractors for non-autonomous reactiondiffusion equations,” Dynamical Systems, vol. 23, no. 1, pp. 1–16, 2008. 7 V. V. Gruˇsin, “A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,” Matematicheski˘ı Sbornik, vol. 84 126, pp. 163–195, 1971, English translation in Mathematics of the USSR-Sbornik, vol. 13, pp. 155–183, 1971. 8 C. T. Anh, “Pullback attractors for non-autonomous parabolic equations involving Grushin operators,” Electronic Journal of Differential Equations, vol. 2010, no. 11, pp. 1–14, 2010. 9 C. T. Anh, P. Q. Hung, T. D. Ke, and T. T. Phong, “Global attractor for a semilinear parabolic equation involving Grushin operator,” Electronic Journal of Differential Equations, vol. 2008, no. 32, pp. 1–11, 2008. 10 C. T. Anh and T. D. Ke, “Existence and continuity of global attractors for a degenerate semilinear parabolic equation,” Electronic Journal of Differential Equations, vol. 2009, no. 61, pp. 1–13, 2009.
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11 N. T. C. Thuy and N. M. Tri, “Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 365–370, 2002. 12 Q. Ma, S. Wang, and C. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1559, 2002. 13 Y. Li, S. Wang, and J. Wei, “Finite fractal dimension of pullback attractors and application to nonautonomous reaction diffusion equations,” Applied Mathematics E-Notes, vol. 10, pp. 19–26, 2010. 14 C.-K. Zhong, M.-H. Yang, and C.-Y. Sun, “The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 367–399, 2006. 15 J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001.
Research Article On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators Anh Cung The1 and Toi Vu Manh2 1
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, 10307 Hanoi, Vietnam 2 Faculty of Computer Science and Engineering, Hanoi Water Resources University, 175 Tay Son, Dong Da, 10508 Hanoi, Vietnam Correspondence should be addressed to Anh Cung The, [email protected] Received 19 December 2010; Accepted 21 February 2011 Academic Editor: Feng Qi Copyright q 2011 A. Cung The and T. Vu Manh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullback Dattractor in L2 Ωm for the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullback D-attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.
1. Introduction Nonautonomous equations are of great importance and interest as they appear in many applications in the natural sciences. One way of studying the long-time behavior of solutions of such equations is using the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of such dynamical systems see 1 and references therein. In recent years, the existence of pullback attractors for reaction-diffusion equations has been studied widely see, e.g., 2–6. However, to the best of our knowledge, little seems to be known for the asymptotic behavior of solutions of nonautonomous degenerate equations. One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of the Grushin type 7 Gs u Δx u |x|2s Δy u,
X x, y ∈ Ω ⊂ ÊN1 × ÊN2 ,
s ≥ 0.
1.1
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The global existence and long-time behavior of solutions to semilinear parabolic equations involving the Grushin operator, in both autonomous and nonautonomous cases, have been studied in some recent works 8–10. In this paper we consider the following nonautonomous semilinear parabolic system: ∂u − aGs u fu gX, t, X ∈ Ω, t > τ, ∂t uX, t 0, X ∈ ∂Ω, t > τ, uX, τ uτ X, X ∈ Ω,
1.2
where X x, y ∈ Ω ⊂ ÊN1 × ÊN2 N1 , N2 1, uτ ∈ L2 Ω is given, u u1 , . . . , um is an unknown vector-function. Here a ∈ Matm Ê, fu f 1 u1 , . . . , um , . . . , f m u1 , . . . , um , and gX, t g 1 X, t, . . . , g m X, t satisfy the following conditions: m
H1 a ∈ Matm Ê has a positive symmetric part: 1/2a a∗
Ê
H2 f :
m
→
Ê
m
βIm , β > 0;
1
is a C -vector function such that: m f j uuj , C1 |u|p − C0 ≤ fu, u
p ≥ 2,
1.3
j1
fu ≤ C2 |u|p−1 1 ,
1.4
m m ∂f i −C3 |v|2 ≤ fu uv, v uvj vi , j ∂u i1 j1
1.5
where C0 , C1 , C2 , and C3 are positive constants; 1,2 H3 g ∈ Wloc Ê; L2 Ωm such that
0 −∞
e
λ1 βs
2 gs L2 Ωm ds < ∞, 0 −∞
e
0 s −∞
−∞
2 eλ1 βr gr L2 Ωm dr ds < ∞, 1.6
2 g s
λ1 βs
L2 Ωm ds
< ∞,
where λ1 is the first eigenvalue of the operator Gs in Ω with the homogeneous Dirichlet boundary condition. In order to study problem 1.2, we will use the natural energy space defined as the complete of C0∞ Ωm in the following norm:
S10 Ωm
u Ë10Ω
Ω
2
|∇x u| |x|
2s
2 ∇y u dX
1/2 .
Ë10Ω
:
1.7
From the results in 11, we know that the embedding Ë10Ω → Äp Ω is continuous if 1 p 2∗s : 2Ns/Ns − 2, where Ns : N1 s 1N2 ; moreover, this embedding is compact if 1 p < 2∗s .
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Notations Denote Äp Ω : Lp Ωm , and Êm , we set
Ë−1Ω the dual space of Ë10Ω. For functions u, v : ÊN
∇u, ∇v :
m i1
m N ∂ui ∂vi ∇ui , ∇vi , ∂Xk ∂Xk i1 k1
→
1.8
so if a aij m i,j1 ∈ Matm Ê, then a∇u, ∇v
m N ∂ui ∂vj aij ∇ui , ∇vj aij , ∂Xk ∂Xk i,j1 i,j1 k1 m
1.9
where ·, · denotes the inner product in ÊN . Noting that by assumption H1, we have a∇u, ∇u
m m m 2 1 aij aji ∇ui , ∇uj ≥ β ∇uj , ∇uj β ∇uj . 2 i,j1 j1 j1
1.10
Hence a∇x u, ∇x u |x|2s a∇y u, ∇y u dX ≥ β u 2Ë1Ω , Ω
0
1.11
Ω
aGs u, Gs udX ≥ β Gs u 2Ä2Ω .
1.12
The aim of this paper is to study the long-time behavior of solutions to problem 1.2 by using the theory of pullback D-attractors. We first prove, under assumptions H1–H3, the existence of a pullback D-attractor in Ä2 Ω for the process Ut, τ associated to problem 1.2. Then, with an additional condition that the system has a special gradient structure, namely, a βIm and there exists a function F : Êm → Ê such that fu gradu Fu, we prove the existence of a pullback D-attractor in the space Ë10Ω∩ Ä p Ω for the process Ut, τ. Moreover, we prove the boundedness of the pulback D-attractor in Ä2p−2 Ω and in Ë20Ω, and give estimates of the fractal dimension of the pulback D-attractor. It is worth noticing that our results, in particular, extend and improve some recent results on the existence of pullback D-attractors for the reaction-diffusion equations 3–5 and for the Grushin equations 8. Let us explain the methods used in the paper. We first prove the existence of a family of pullback D-absorbing sets in Ë10Ω. Thanks to the compactness of the embedding Ë10Ω → Ä2 Ω, we immediately get the existence of a pullback D-attractor in Ä2 Ω. When the system has a special gradient structure, we are able to prove the existence of a pullback D-attractor in Ë10Ω ∩ Äp Ω. To do this, we follow the general lines of the approach used in 8, with some modifications so that we can improve conditions imposed on the external force g. In particular, we use the asymptotic a priori estimate method initiated in 12 to testify the pullback asymptotic compactness of the corresponding process. Moreover, in this case we also prove the regularity of the pullback D-attractor in the spaces Ä2p−2 Ω and Ë10Ω. Finally,
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using the recent results in 13, we give an estimate of the fractal dimension of the pullback D-attractor. It is noticed that we do not impose the restriction on the exponent p in H2 as in 13. The rest of the paper is organized as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on pullback D-attractors which we will use. In Section 3, we prove the existence of a pullback D-attractor in Ä2 Ω in the general case. In Section 4, under the additional assumption that the system has a gradient structure, we prove the regularity and fractal dimension estimates of the pullback D-attractor.
2. Preliminaries 2.1. Pullback Attractors For convenience of the reader, we recall in this section some concepts and results on the theory of pullback D-attractors, which will be used in the paper. Let X be a metric space with metric d. Denote by BX the set of all bounded subsets of X. For A, B ⊂ X, the Hausdorff semidistance between A and B is defined by distA, B sup inf d x, y . x∈A y∈B
2.1
Let {Ut, τ : t ≥ τ, τ ∈ Ê} be a process in X, that is, Ut, τ : X → X such that Uτ, τ Id and Ut, sUs, τ Ut, τ for all t ≥ s ≥ τ, τ ∈ Ê. The process {Ut, τ} is said to be normto-weak continuous if Ut, τxn Ut, τx, as xn → x in X, for all t ≥ τ, τ ∈ Ê. The following result is useful for verifying the norm-to-weak continuity of a process. Proposition 2.1 see 14. Let X, Y be two Banach spaces, X ∗ , Y ∗ be, respectively, their dual spaces. Assume that X is dense in Y , the injection i : X → Y is continuous and its adjoint i∗ : Y ∗ → X ∗ is dense, and {Ut, τ} is a continuous or weak continuous process on Y . Then {Ut, τ} is norm-toweak continuous on X if and only if for t ≥ τ, τ ∈ Ê, Ut, τ maps a compact set of X to be a bounded set of X. {Dt : t ∈ Ê} ⊂ BX. Suppose that D is a nonempty class of parameterized sets D Definition 2.2. The process {Ut, τ} is said to be pullback D-asymptotically compact if for ∈ D, and any sequence {τn }n with τn ≤ t for all n, and τn → −∞, any any t ∈ Ê, any D sequence xn ∈ Dτn , the sequence {Ut, τn xn } is relatively compact in X. Definition 2.3. A process {Ut, τ} is called pullback ω-D-limit compact if for any ε > 0, any ∈ D, there exists a τ0 τ0 D, ε, t ≤ t such that t ∈ Ê, and D α
Ut, τDτ
≤ ε,
2.2
τ≤τ0
where α is the Kuratowski measure of noncompactness of B ∈ BX, αB inf δ > 0 | B has a finite open cover of sets of diameter ≤ δ .
2.3
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Lemma 2.4 see 3. A process {Ut, τ} is pullback D-asymptotically compact if and only if it is ω-D-limit compact. ∈ D is called pullback D-absorbing for the process Definition 2.5. A family of bounded sets B ∈ D, there exists τ0 τ0 D, t ≤ t such that {Ut, τ} if for any t ∈ Ê and any D
Ut, τDτ ⊂ Bt.
τ≤τ0
Definition 2.6. A family A {At : t ∈ {Ut, τ} if
Ê}
2.4
⊂ BX is said to be a pullback D-attractor for
1 At is compact for all t ∈ Ê; 2 A is invariant, that is, Ut, τAτ At, for all t ≥ τ; 3 A is pullback D-attracting, that is, lim distUt, τDτ, At 0,
τ → −∞
2.5
∈ D and all t ∈ Ê; for all D
4 if {Ct : t ∈ Ê} is another family of closed attracting sets, then At ⊂ Ct, for all t ∈ Ê. Theorem 2.7 see 3. Let {Ut, τ} be a norm-to-weak continuous process such that {Ut, τ} is {Bt : pullback D-asymptotically compact. If there exists a family of pullback D-absorbing sets B t ∈ Ê} ∈ D, then {Ut, τ} has a unique pullback D-attractor A {At : t ∈ Ê} and At
Ut, τBτ.
s≤t τ≤s
2.6
2.2. Fractal Dimension of Pullback Attractors Given a compact K ⊂ X and ε > 0, we denote by NK, ε the minimum number of open balls in X with radius ε which are necessary to cover K. Definition 2.8. For any nonempty compact set K ⊂ X, the fractal dimension of K is the number log NK, ε . ε→0 log 1/ε
dimf K : lim
2.7
Definition 2.9. A bounded subset B0 ⊂ H is called a uniformly pullback absorbing set for process Ut, τ if for every B ⊂ H is bounded, there exists a τ0 ≥ 0 such that Ut, t − τ0 B ∈ B0 , here, τ0 does not depend on the choice of t.
∀τ ≥ τ0 ,
2.8
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Theorem 2.10 see 13. Let Ut, τ be a process in a separable Hilbert space H, B be a uniformly {At : t ∈ Ê} be a pullback attractor for Ut, τ, if there exists a pullback absorbing set in H, A finite dimensional projection P in the space H such that
P Ut, t − T0 u1 − Ut, t − T0 u2 H ≤ lT0 u1 − u2 H
2.9
for all u1 , u2 ∈ B and some T0 , lT0 > 0 and
I − P Ut, t − T0 u1 − Ut, t − T0 u2 H ≤ δ u1 − u2
2.10
for all u1 , u2 ∈ B, where δ < 1, T0 and lT0 are independent of the choice of t. Then the family of {At : t ∈ Ê} possesses a finite fractal dimension especifically pullback attractors A
8lT0 2 −1 dimf At ≤ dim P log 1
, log 1−δ 1 δ
∀t ∈ Ê.
2.11
3. Existence of Pullback D-Attractors in 2 Ω Denote V : Lp τ, T; Äp Ω ∩ L2 τ, T; Ë10Ω , V ∗ : L2 τ, T; Ë−1Ω Lp τ, T; Äp Ω ,
3.1
where p is the conjugate of p i.e., 1/p 1/p 1. Definition 3.1. Let T > 0 and uτ ∈ problem 1.2 on τ, T if
Ä2 Ω be given. A function u is called a weak solution of
u ∈ V, u|tτ uτ T τ
∂u ∈ V ∗, ∂t a.e. in Ω,
ut , ϕ a∇x u, ∇x ϕ |x|2s a∇y u, ∇y ϕ fu, ϕ dX dt Ω T gt, ϕ dX dt τ
3.2
Ω
for all test functions ϕ ∈ V . One can prove that if u ∈ V and ∂u/∂t ∈ V ∗ , then u ∈ C0, T; Ä2 Ω see 10. This makes the initial condition in 1.2 meaningful.
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Theorem 3.2. Under assumptions (H1)–(H3), for any τ ∈ Ê, T > τ, uτ ∈ Ä2 Ω given, problem 1.2 has a unique weak solution u on τ, T. Moreover, the solution u exists on the interval τ, ∞ and the following inequality holds:
u 2Ä2Ω ≤ e−λ1 βt−τ uτ 2Ä2Ω
2C0 |Ω|
e−λ1 βt λ1 β
t −∞
2 eλ1 βs gs Ä2Ω ds,
∀t ≥ τ.
3.3
Proof. The existence and uniqueness of a weak solution to problem 1.2 are proved similarly to the scalar case in 10, so it is omitted here. We now prove inequality 3.3. Multiplying 1.2 by u, integrating over Ω, and using 1.11, we have 1 d
u 2Ä2Ω β u 2Ë1Ω
0 2 dt
Ω
fu, u dX
Ω
gt, u dX.
3.4
Using condition 1.3 and the Cauchy inequality, we obtain d 2 p gt 2 2 βλ1 u 2 2 .
u 2Ä2Ω 2β u 2Ë1Ω 2C1 u Äp Ω − 2C0 |Ω| ≤ Ä Ω Ä Ω 2 0 dt βλ1
3.5
Because u 2Ë1Ω ≥ λ1 u 2Ä2Ω , so 3.5 becomes 0
d 1 gt 2 2 .
u 2Ä2Ω λ1 β u 2Ä2Ω ≤ 2C0 |Ω|
Ä Ω dt λ1 β
3.6
Applying the Gronwall inequality we get 3.3. Now, we can define the family of two-parameter mappings Ut, τ : Ä2 Ω −→ Ë10Ω ∩ Äp Ω, uτ −→ Ut, τuτ ,
3.7
where Ut, τuτ ut is the unique weak solution of 1.2 with the initial datum uτ at time τ. Then U defines a continuous process on Ä2 Ω. Let R be the set of all functions r : Ê → 0, ∞ such that limt → −∞ eλ1 βt r 2 t 0 and {Dt : t ∈ Ê} ⊂ BË1Ω such that Dt ⊂ Brt denote by D the class of all families D 0 1 for some rt ∈ R, where Brt is the closed ball in Ë Ω with radius rt. 0
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Lemma 3.3. Under assumptions (H1)–(H3), there exists a constant C > 0 such that the solution u of problem 1.2 satisfies the following inequality for all t > τ:
1 1 e−αt−τ uτ 2Ä2Ω 1
t−τ t−τ
t 2 1
1
eαs gs Ä2Ω ds e−αt t−τ −∞ t s
2 1 −αt αr
1
e gr Ä2Ω dr ds , e t−τ −∞ −∞
u 2Ë1Ω ≤ C 0
1 t − τ
3.8
where α βλ1 . This implies that there exists a family of pullback D-absorbing sets in Ë10Ω for the process {Ut, τ}. Proof. We multiply 1.2 by −Gs u and integrate over Ω. After some standard transformations we obtain 1 d
ut 2Ë1Ω β Gs ut 2Ä2Ω 0 2 dt 2s fut, Δx ut |x| fu, Δy ut dX
g, Gs ut dX. Ω
3.9
Ω
Without loss of generality, we may assume that f0 0. Otherwise we can replace fu by fu fu − f0. The function f satisfies the same conditions with modified constants Ci i 0, 1, 2, 3, because |f0| C2 see 1.4. Hence, since fut|∂Ω 0, we get Ω
N1 m fu, Δx ut dX k1 i1
Ω
f i u
∂2 ui dX ∂Xk2
N1 m m ∂f i ∂uj ∂ui − dX u j ∂Xk ∂Xk k1 i1 j1 Ω ∂u
N1
∂u ∂u − fu u dX , ∂Xk ∂Xk k1 Ω ≤ C3
3.10
N1 ∂u 2 dX C3 |∇x u|2 dX, ∂X k Ω k1 Ω
where we have used condition 1.5. Similarly, we have Ω
|x|
fu, Δy ut dX ≤ C3
2s
Ω
2 |x|2s ∇y u dX.
3.11
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Hence Ω
fu, Δx ut dX
Ω
|x|2s fu, Δy ut ≤ C3 ut 2Ë1Ω . 0
3.12
By the Cauchy inequality we have
1 g 2 2 β Gs ut 2 2 . g, Gs ut dX ≤ Ä Ω Ä Ω 2 2β Ω
3.13
From 3.9–3.13 we obtain d 1 2
ut 2Ë1Ω β Gs ut 2Ä2Ω ≤ 2C3 ut 2Ë1Ω g Ä2Ω , 0 0 dt β
3.14
thus, d
ut 2Ë1Ω α ut 2Ë1Ω 0 0 dt
2C3 ut 2Ë Ω β1 g 2Ä Ω , 1 0
3.15
2
where α βλ1 . Multiplying 3.15 by t − τeαt and integrating from τ to t, we obtain 2
t − τe u Ë1Ω 0 αt
t
t−τ ≤ 2C3 t − τ 1 e us Ë1Ω ds
0 β τ αs
2
t −∞
2 eαs gs Ä2Ω ds. 3.16
Multiplying 3.3 by αeαt and integrating from τ to t, we have t α τ
eαs us 2Ä2Ω ds ≤ αt − τeατ uτ 2Ä2Ω
2C0 |Ω| αt e
α
t s −∞
−∞
2 eαr gr Ä2Ω dr ds. 3.17
Now, from 3.5 we get 2 1 d
u 2Ä2Ω β u 2Ë1Ω ≤ gt Ä2Ω 2C1 |Ω|. 0 dt α
3.18
Multiplying this equation by eαt and integrating from τ to t, we deduce that eαt ut 2Ä2Ω β
t τ
eαs us 2Ë1Ω ds 0
2C1 |Ω| αt 1 e
≤ eατ uτ Ä2Ω
α α 2
t τ
2 eαs gs 2
Ä Ω ds α
t τ
3.19 2
eαs us Ä2Ω ds.
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Using 3.17, 3.19 becomes
eαt ut 2Ä2Ω β
t τ
eαs us 2Ë1Ω ds 0
C0 C1 |Ω| αt e αt − τeατ uτ 2Ä2Ω α t s 2 2 eατ gs Ä2Ω ds
eαr gr Ä2Ω dr ds.
≤ eατ uτ 2Ä2Ω
1 α
t −∞
−∞
3.20
−∞
Substituting 3.20 into 3.16 we obtain
1 2C3 1
uτ 2Ä2Ω
ut 2Ë1Ω ≤ e−αt−τ 2C3
t − τ
0 λ1 λ1 βt − τ t
2 C0 C1 1 1 1 −αt 2C e
2C3
α
eαs gs Ä2Ω ds
|Ω| 3 t−τ αβ αβ t−τ −∞ t s
2 1 1 2C3
e−αt eαr gr Ä2Ω dr ds.
β t−τ −∞ −∞ 3.21
Hence we get 3.8 with C Cβ, C0 , C1 , C3 , λ1 . Let r02 t be the right-hand side of 3.8, and let B0 r0 t be the closed ball in Ë10Ω ∈ D and any t ∈ Ê, by 3.8 there exists centered at 0 with radius r0 t. Obviously for any D ≤ t such that the solution u with initial datum uτ ∈ Dτ at time τ satisfies τ0 τ0 D
ut Ë10Ω ≤ r0 t for all τ ≤ τ0 ; that is, B {B0 r0 t : t ∈ Ê} is a family of bounded pullback D-absorbing sets in Ë10Ω. From the above lemma we deduce that the process {Ut, τ} maps a compact set of to be a bounded set of Ë10Ω, and thus by Proposition 2.1, the process {Ut, τ} is norm-to-weak continuous in Ë10Ω. Since {Ut, τ} has a family of pullback D-absorbing sets in Ë10Ω and the embedding Ë10Ω → Ä2 Ω is compact, we immediately get the following.
Ë10Ω
Theorem 3.4. Under assumptions (H1)–(H3), the process {Ut, τ} associated to problem 1.2 has a pullback D-attractor in Ä2 Ω.
4. Some Further Results in the Gradient Case In this section, instead of H1–H3, we assume that H1bis a βIm , where Im is the unit matrix and β > 0;
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H2bis f satisfies H2 and fu gradu Fu ∂F/∂u1 u, . . . , ∂F/∂um u, where F : Êm → Ê is a potential function satisfying C1 |u|p − C0 ≤ Fu ≤ C2 |u|p C0 ,
∀u ∈ Êm ,
4.1
with C1 , C2 , C0 being positive constants 1,2 H3bis g ∈ Wloc Ê, Ä 2 Ω satisfies
0 −∞
2 2 eαt gt Ä2Ω g t Ä2Ω dt < ∞,
4.2
where α βλ1 . The aim of this section is to prove that the pullback D-attractor obtained in Section 3 is more regular and has a finite fractal dimension.
4.1. Existence of a Pullback D-Attractor in
Ë10Ω ∩ Äp Ω
Denote by R the set of all functions r : Ê → 0, ∞ such that limt → −∞ eλ1 βt r 2 t 0 {Dt : t ∈ Ê} ⊂ BË1Ω ∩ Äp Ω such and denote by D the class of all families D 0 that Dt ⊂ Brt for some rt ∈ R, where Brt is the closed ball in Ë10Ω ∩ Äp Ω with radius rt. Thanks to the above gradient structure, one can prove the existence of a pullback D-attractor, not only in Ä2 Ω, but also in the space Ë10Ω ∩ Äp Ω for the process {Ut, τ}. We first prove the following. Lemma 4.1. Under assumptions (H1bis)–(H3bis), the solution u of problem 1.2 satisfies the following inequality for all t > τ: p
2
u Ë1Ω u Äp Ω ≤ C e 0
−αt−τ
2
uτ Ä2Ω 1 e
−αt
t −∞
e
αr
2 gr 2
Ä Ω dr ,
4.3
where C CC0 , C1 , C1 , C0 , β, λ1 . This implies that there exists a family of pullback D-absorbing sets in Ë10Ω ∩ Äp Ω for the process {Ut, τ}. Proof. Using 3.5 with α λ1 β and the fact that u 2Ë1Ω ≥ λ1 u 2Ä2Ω , we have 0
2 β d 2 p
u 2Ä2Ω α u 2Ä2Ω u 2Ë1Ω 2C1 u Äp Ω ≤ 2C0 |Ω| gt Ä2Ω , 0 dt 2 α
4.4
thus 2 2 d αt p e u 2Ä2Ω Ceαt β u 2Ë1Ω 2 u Äp Ω ≤ 2C0 |Ω|eαt eαt gt Ä2Ω . 0 dt α
4.5
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Integrating from τ to s, τ ≤ s ≤ t 1, and in particular, we have eαs us 2Ä2Ω ≤ eατ uτ 2Ä2Ω 2
C0 2 |Ω|eαs
α α
s τ
2 eαr gr Ä2Ω dr.
4.6
Furthermore, multiplying 4.5 from s to s 1 and using 4.6 we obtain s 1 s
p eαr β ur 2Ë1Ω 2 ur Äp Ω dr 0 2 C0 |Ω| αs 2 s 1 αr ≤ e us Ä2Ω 2 e
e gr Ä2Ω dr α α s s 1 s 2 2 2 ατ αs αr αr ≤ C e uτ Ä2Ω e
e ur Ä2Ω e ur Ä2Ω 2
αs
s
≤ C eατ uτ 2Ä2Ω eαt
t τ
4.7
τ
eαr ur 2Ä2Ω .
By assumption H2bis, then 4.7 becomes s 1 e s
αr
2
2
β ur Ë1Ω 2Fur dr ≤ C e uτ Ä2Ω e
0 ατ
t
αt
τ
2
e ur Ä2Ω . αr
4.8
Multiplying 1.2 by ∂u/∂t and integrating over Ω, we have
2 1 1 d 1 2 gt, ut dX ≤ gt Ä2Ω ut 2Ä2Ω , β u Ë1Ω
FudX
ut Ä2Ω
0 2 dt 2 2 Ω Ω 4.9 2
thus
2 d αt αt 1 1 e β ut Ë0Ω 2 FutdX ≤ e β ut Ë0Ω 2 FutdX eαt gt Ä2Ω . dt Ω Ω 4.10 Using 4.8, 4.10, and the uniform Gronwall inequality, we get
e
αt
2
β ut Ë1Ω 2 0
Ω
FutdX
2
≤ C e uτ Ä2Ω e
ατ
t
αt
e τ
αr
2 gr Ä2Ω dr . 4.11
International Journal of Mathematics and Mathematical Sciences
13
Now, using H2bis once again we have from 4.11 that
e
αt
2
p
2
β ut Ë1Ω 2 ut Äp Ω ≤ C e uτ Ä2Ω e
0 ατ
t
αt
e τ
αr
2 gr 2
Ä Ω dr .
4.12
Thus we obtain 4.3 with a suitable positive constant C C C0 , C1 , C1 , C0 , β, λ1 .
4.13
Hence, by the argument as in the end of the proof of Lemma 3.3, we obtain a family of bounded pullback D-absorbing sets in Ë10Ω ∩ Äp Ω. To prove that the process {Ut, τ} is pullback D-asymptotically compact in Äp Ω, we need the following lemma. Lemma 4.2 see 8, Lemma 3.6. Let {Ut, τ} be a norm-to-weak continuous process in and Äp Ω, and let {Ut, τ} satisfy the following two conditions:
Ä2 Ω
i {Ut, τ} is pullback D-asymptotically compact in Ä2 Ω; ∈ D, there exist constants Mε, B and τ0 ε, B ≤ t such that ii for any ε > 0, B 1/p
Ω|Ut,τuτ |≥M
|Ut, τuτ |
p
< ε,
for any uτ ∈ Bτ, τ ≤ τ0 .
4.14
Then {Ut, τ} is pullback D-asymptotically compact in Äp Ω. Theorem 4.3. Under assumptions (H1bis)–(H3bis), the process {Ut, τ} associated to problem 1.2 has a pullback D-attractor in Äp Ω. Proof. It is sufficient to show that the process {Ut, τ} satisfies the condition ii in Lemma 4.2. We will give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in 15. Let M be a positive number, we will write u ≥ M or u ≤ −M as any component of u is greater than or equal to M or as any component of u is less than or equal to −M. Using 1.3, 1.4, and for u ≥ M large enough, we have √ fu, u − M ≥ fu, u − M mfu 3 |u|p − C 2 |u|p−1 ≥C C4 α ≥ |u − M|p |u − M|2 , 2 p
4.15 C4 3 , 1. because lim|u| → ∞ C
14
International Journal of Mathematics and Mathematical Sciences Multiplying 1.2 by u − M |u − M |p−2 and integrating over Ω we obtain 1 d p dt
p
Ωu≥M
|u − M| dX
Ωu≥M
Ωu≥M
|x|2s
Ωu≥M
a∇x u − M, ∇x u − M|u − M|p−2 dX
a∇y u − M, ∇y u − M|u − M|p−2 dX
|u − M|
p−2
fu, u − M dX
Ω
4.16
gt, u − M dX,
where
u − M :
⎧ ⎨u − M
if u ≥ M,
⎩0,
in other cases.
4.17
On the other hand, by the Cauchy inequality, we have C4 1 2 g , |u − M|2p−2
gt, u − M |u − M|p−2 ≤ |u − M|p−1 gt ≤ 2 2C4
4.18
which implies that C4 1 2 g . gt, u − M |u − M|p−2 ≥ − |u − M|2p−2 − 2 2C4
4.19
Hence, from 4.15 and 4.19, we have α 1 2 g . |u − M|p−2 fu, u − M gt, u − M ≥ |u − M|p − p 2C4
4.20
From 4.16, using 4.20 and noting that
Ωu≥M
a∇x u − M, ∇x u − M|u − M|p−2 dX
Ωu≥M
|x|
2s
a∇y u − M, ∇y u − M|u − M|
p−2
4.21 dX ≥ 0,
a βIm ,
we have p d p p g 2 2 .
u − M Äp Ωu≥M α u − M Äp Ωu≥M ≤ Ä Ω dt 2C4
4.22
International Journal of Mathematics and Mathematical Sciences
15
Now, multiplying the above inequality by t − τeαt and integrating from τ to t, we get p
t − τeαt u − M Äp Ωu≥M
t 2 p ≤ e u − M Äp Ωu≥M ds
t − τ eαs gs Ä2Ω ds 2C4 τ τ t t 2 p p ≤ eαs u Äp Ω ds
t − τ eαs gs Ä2Ω ds. 2C4 τ τ t
p
αs
4.23
Then p
u − M Äp Ω ≤
1 −αt e t−τ
t τ
p
eαs u Äp Ω ds
pe−αt 2C4
t −∞
2 eαs gs Ä2Ω ds.
4.24
On the other hand, integrating 4.5 from τ to t, we have t
C0 1 e us Äp Ω ds ≤ e uτ Ä2Ω 2 |Ω|eαt
α α τ p
αs
2
ατ
t τ
2 eαs gs Ä2Ω ds.
4.25
Therefore, substituting 4.25 into 4.24, we obtain p
u − M Äp Ω ≤ C
1 −αt e−αt−τ 1
e
uτ 2Ä2Ω
t−τ t−τ t−τ
t −∞
e
αs
2 gs 2
Ä Ω ds . 4.26
Hence, for any ε > 0, there exists M1 > 0 and τ1 < t such that for any τ < τ1 and any M ≥ M1 , we have Ωut≥M
|u − M|p dx ≤ ε.
4.27
Repeating the same step above, just taking u M− instead of u−M , we deduce that there exist M2 > 0 and τ2 < t such that for any τ < τ2 and any M ≥ M2 , Ωut≤−M
|u M|p dx ≤ ε,
4.28
where
u M−
⎧ ⎨u M,
u ≤ −M,
⎩0,
in other cases.
4.29
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International Journal of Mathematics and Mathematical Sciences
Let M0 max{M1 , M2 } and τ0 min{τ1 , τ2 }, we obtain Ω|u|≥M
|u| − Mp dx ≤ ε
for τ ≤ τ0 , M ≥ M0 .
4.30
So, we have
Ω|u|≥2M
|u|p dx
Ω|u|≥2M
|u| − M Mp dx
≤2
p
p−1 Ω|u|≥2M
|u| − M dx
≤ 2p−1
p
Ω|u|≥2M
M dx
|u| − Mp dx
Ω|u|≥M
Ω|u|≥M
4.31
|u| − Mp dx
≤ 2p ε. This completes the proof. To prove the existence of a pullback D-attractor in Ë10Ω∩ Ä p Ω, we need the folowing lemma. Lemma 4.4. Under assumptions (H1bis)–(H3bis), for any t ∈ Ê and any bounded subset B ⊂ Ä2 Ω, there exists a positive constant T TB, t ≤ t such that 2
ut t Ä2Ω ≤ C 1 e
−αt
t
2 2 gs Ä2Ω g s Ä2Ω ds , e
4.32
αs
−∞
for all τ ≤ TB, t and all uτ ∈ B, where C > 0 is independent of t and B. Proof. We give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in 15. Differentiating 1.2 in time and setting v ut , we get vt − aGs v f uv g r.
4.33
Multiplying this inequality by eαr v and integrating over Ω and using 1.11, we get 1 d αr e v 2Ä2Ω βeαr v 2Ë1Ω eαr 0 2 dr
Ω
α f uv, v dX ≤ eαr v 2Ä2Ω eαr 2
Ω
g r, v dX. 4.34
By the Cauchy inequality and using 1.5, we obtain that 2 d αr e v 2Ä2Ω ≤ 2C3 α 1eαr v 2Ä2Ω eαr g r Ä2Ω . dr
4.35
International Journal of Mathematics and Mathematical Sciences
17
Let τ ≤ s ≤ t − 1. Using 3.5, we have 2 1 d αs p e u 2Ä2Ω β u 2Ë1Ω 2C1 eαs u Äp Ω ≤ eαs gs Ä2Ω C0 eαs |Ω|. 0 ds α
4.36
By H2bis we then infer from the above inequality that
2 d αs FudX ≤ C eαs gs Ä2Ω eαs . e u 2Ä2Ω C βeαs u 2Ë1Ω 2eαs 0 ds Ω
4.37
Integrating this inequality from r to r 1, we obtain r 1
βeαs u 2Ë1Ω 2eαs FudX ds Ω
0
r
2
≤ C e ur Ä2Ω
αr
r 1
e
αs
r
2 αs gs Ä2Ω e ds .
4.38
On the other hand, integrating 4.36 from τ to t, we obtain eαt u 2Ä2Ω ≤ eατ uτ 2Ä2Ω
1 α
t
2 C0 |Ω| αt e . eαs gs Ä2Ω ds
α −∞
4.39
So, substituting 4.39 into 4.38, we deduce r 1
βeαs u 2Ë1Ω 2eαs FudX ds 0
r
Ω
2
≤ C e uτ Ä2Ω
ατ
t
−∞
e
αs
2 gs Ä2Ω ds eαt
4.40
< ∞,
∀r ∈ τ, t − 1.
Now multiplying 1.2 by eαr v and integrating over Ω, we have
d βeαr u 2Ë1Ω 2eαr FudX 0 dr Ω
2 FudX eαr gr Ä2Ω . ≤ α βeαr u 2Ë1Ω 2eαr
eαr v 2Ä2Ω
4.41
Ω
0
So applying the uniform Gronwall inequality, we get
2
βe u Ë1Ω 2e 0 αr
αr Ω
2
FudX ≤ C e uτ Ä2Ω e
ατ
t
αt
−∞
e
αs
2 gs Ä2Ω ds .
4.42
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International Journal of Mathematics and Mathematical Sciences
Integrating 4.41 from r to r 1 and by 4.40–4.42, we have r 1 r
2
2
e v Ä2Ω ds ≤ C e uτ Ä2Ω e
αs
ατ
αt
t −∞
e
αs
2 gs 2
Ä Ω ds .
4.43
Therefore, by 4.35, 4.43, using the uniform Gronwall inequality once again, we get 2
2
e v Ä2Ω ≤ C e uτ Ä2Ω e
αt
ατ
t
αt
2 2 gs Ä2Ω g s Ä2Ω ds . e αs
−∞
4.44
Hence we get 4.32. Theorem 4.5. Under assumptions (H1bis)–(H3bis), the process {Ut, τ} associated to problem 1.2 has a pullback D-attractor in Ë10Ω ∩ Äp Ω. Proof. By Lemma 4.1, {Ut, τ} has a family of bounded pullback D-absorbing sets in Ë10Ω ∩ Äp Ω. It remains to show that {Ut, τ} is pullback D-asymptotically compact in Ë10Ω ∩ Äp Ω, that is, for any t ∈ Ê, any B ∈ D, and any sequence τn → −∞, any sequence uτn ∈ Bτn , the sequence {Ut, τn uτn } is precompact in Ë10Ω ∩ Äp Ω. Thanks to Theorem 4.3, we need only to show that the sequence {Ut, τn uτn } is precompact in Ë10Ω. Let un t Ut, τn uτn . By Theorem 3.4, we can assume that {un t} is a Cauchy sequence in Ä2 Ω. We have
un t − um t 2Ë1Ω 0
−Gs un t − Gs um t, un t − um t dun dum − − u − u t t, n t m t − fun t − fum t, un t − um t dt dt 2 d d 2 2 ≤ un t − um t 2 un t − um t Ä2Ω C3 un t − um t Ä2Ω , dt dt Ä Ω
4.45
where we have used condition 1.5. Because {un t} is a Cauchy sequence in Ä2 Ω and by Lemma 4.4, one gets
un t − um t Ë10Ω −→ 0,
as m, n −→ ∞.
4.46
The proof is complete.
4.2.
Ä2p−2 Ω and Ë20Ω-Boundedness of the Pullback D-Attractor
First, we prove the existence of a family of pullback D-absorbing sets for process Ut, τ in Ä2p−2 Ω.
International Journal of Mathematics and Mathematical Sciences Proposition 4.6. Under assumptions (H1bis)–(H3bis), then for any t ∈ B ⊂ Ä2 Ω, there exists a positive constant τ0 τ0 B, t ≤ t such that
2p−2
u Ä2p−2Ω
2 ≤ C 1 gt Ä2Ω e−αt
t −∞
19
Ê and any bounded subset
2 2 e gs Ä2Ω g s Ä2Ω ds
αs
4.47
for all τ ≤ τ0 and all uτ ∈ B, where C > 0 is independent of t and B. Proof. Multiplying 1.2 by |u|p−2 u and integrating over Ω we obtain Ω
a∇x u, ∇x |u|
p−2
u
p−2 fu, u |u|p−2 dX a∇y u, ∇y |u| u
|x| dX
2s
p−2 ut , |u| u dX
gt, |u|p−2 u dX. − Ω
Ω
Ω
4.48
By the Cauchy inequality, 1.3 and note that Ω
a∇x u, ∇x |u|p−2 u
|x|2s a∇y u, ∇y |u|p−2 u dX ≥ 0;
here a βIm , 4.49
then we get 2p−2
C1 u Ä2p−2Ω ≤
1 1 gt 2 2 C1 u 2p−2
ut 2Ä2Ω
Ä Ä2p−2 . C1 C1 2
4.50
Hence, by 4.32 we deduce from 4.16 that t 2 2 C1 1 1 2p−2 −αt αs gt 2 2 . gs Ä2Ω g s Ä2Ω ds
C 1 e e
u Ä2p−2Ω ≤ Ä Ω 2 C1 C 1 −∞ 4.51 Therefore, we get 4.47 and the proof is complete. And now, we denote by Ë20Ω the closure of C0∞ Ωm in the norm
u Ë20Ω
Ω
1/2 . |Δx u|2 |x|2s Δy u|2 dX
4.52
It is easy to see that Ë20Ω is a Banach space endowed with the above norm. We now prove the Ë20Ω-boundedness of the pullback D-attractor. First, we recall a lemma see 15 which is necessary for our proof.
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International Journal of Mathematics and Mathematical Sciences
Lemma 4.7. Let X, Y be Banach spaces such that X is reflexive, and the inclusion X ⊂ Y is continuous. Assume that {un } is a bounded sequence in L∞ τ, T; X such that un u weakly in Lq τ, T; X for some q ∈ 1, ∞ and u ∈ Cτ, T; Y . Then, ut ∈ X for all t ∈ τ, T and
ut X ≤ supn≥1 un L∞ τ, T; X, for all t ∈ τ, T. Let un t be the Galerkin approximations of the solution ut of 1.2 then by Lemma 4.7 with noticing that un Ut, τunτ u Ut, τuτ in L2 τ, T; Ë10Ω and the inclusion Ë20Ω ⊂ Ë10Ω is continuous, we only need the estimation on ut Ut, τuτ . Theorem 4.8. Under assumptions (H1bis)–(H3bis), the pullback D-attractor A {At : t ∈ Ê} in Ë10Ω ∩ Äp Ω of the process {Ut, τ} is bounded in Ë20Ω. More precisely, for any τ < T1 < T2 , the ! set t∈T ,T At is a bounded subset of Ë20Ω. 1
2
Proof. Let us fix a bounded set B ⊂ Ä2 Ω, τ ∈ Ê, ε > 0, t > τ ε and uτ ∈ B. Multiplying the first equation in 1.2 by Gs u and integrating over Ω, we have Ω
aGs ur, Gs urdX
Ω
u r , Gs ur dX
−
Ω
Ω
fur, Gs ur dX
gr, Gs ur dX.
4.53
By the Cauchy inequality we have −
Ω
2 β 2 gr, Gs ur dX ≤ gr Ä2Ω Gs ur 2Ä2Ω , β 8
β 2 2 ur , Gs ur dX ≤ u r Ë1Ω Gs ur 2Ä2Ω . 0 β 8 Ω
4.54
Using 3.12, 1.12, and 4.54, then from 4.53 we get β Gs ur 2Ä2Ω ≤ C3 ur 2Ë1Ω
0
β 2 u r 2 2 gr 2 2
Gs ur 2Ä2Ω Ä Ω Ä Ω β 4
C3 ≤
2C32 β
Ω
ur, −Gs urdX
ur 2Ä2Ω
β 2 u r 2 2 gr 2 2
Gs ur 2Ä2Ω Ä Ω Ä Ω β 4
2 u r 2 2 2 gr 2 2 β Gs ur 2 2 . Ä Ω Ä Ω Ä Ω 2 β β 4.55
Hence, 2 2C32 β 2 2 2
Gs ur 2Ä2Ω ≤
ur 2Ä2Ω u r Ä2Ω gr Ä2Ω . 2 β β β
4.56
International Journal of Mathematics and Mathematical Sciences
21
Differentiating the first equation in 1.2 in time t and setting vr u r, then multiplying by vr and using 1.11 we get 1 d
vr 2Ä2Ω β vr 2Ë1Ω ≤ − 0 2 dr
Ω
fu uvr, vr dX
Ω
g r, vr dX
1 ≤ C3 vr Ä2Ω vr 2Ä2Ω
2 2
1 g r 2 2 . Ä Ω 2
4.57
Hence, 2 d
vr 2Ä2Ω ≤ 2C3 1 vr 2Ä2Ω g r Ä2Ω . dr
4.58
Integrating the above inequality, we have
vr 2Ä2Ω ≤ vs 2Ä2Ω 2C3 1
t τ ε/2
vθ 2Ä2Ω
t τ ε/2
2 g θ 2 dθ, Ä Ω
4.59
for all τ ε/2 ≤ s ≤ r ≤ t. Now, integrating with respect to s between τ ε/2 and r, we get
r −τ − ≤
ε
vr 2Ä2Ω 2
r
τ ε/2
vs 2Ä2Ω ds
2 ε t ε t 2 g θ 2 dθ
2C3 1 r − τ −
vθ Ä2Ω dθ r − τ − Ä Ω 2 τ ε/2 2 τ ε/2 t 2 ε ε t g θ 2 dθ,
1 ≤ 2C3 1 t − τ −
vθ 2Ä2Ω dθ r − τ − Ä Ω 2 2 τ ε/2 τ ε/2 4.60 for all τ ε/2 ≤ r ≤ t, and in particular, for all r ∈ τ ε, t we have that from the above estimate
vr 2Ä2Ω ≤
t t 2 ε 2
1
vθ 2Ä2Ω dθ g θ Ä2Ω dθ. 2C3 1 t − τ − ε 2 τ ε/2 τ
4.61
On the other hand, multiplying the first equation in 1.2 by vr and integrating over Ω, we deduce that β d
vr Ä2Ω
ur 2Ë1Ω
0 2 dr
2
Ω
fu, v dX ≤
Ω
g, v dX,
4.62
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International Journal of Mathematics and Mathematical Sciences
where we have used 1.11. Using the Cauchy inequality and condition H2bis, then 4.62 becomes
vr 2Ä2Ω
2 d β ur 2Ë1Ω 2 FurdX ≤ gr Ä2Ω . 0 dr Ω
4.63
Integrating from τ ε/2 to t we have
t τ ε/2
vθ 2Ä2Ω dθ β ut 2Ë1Ω 2 0
ε 2 FutdX ≤ β u τ
2 Ë10Ω Ω
t ε gθ 2 2 , dX
2 F u τ
Ä Ω 2 τ ε/2 Ω
4.64
and hence because of 4.1, we get t
ε ε 2 p
vθ 2Ä2Ω dθ ≤ β u τ
1 2C2 u τ
2 Ë0Ω 2 Äp Ω τ ε/2 t 2
4C0 |Ω| gθ Ä2Ω .
4.65
τ
Now, substituting 4.65 into 4.61 we deduce
vθ 2Ä2Ω ≤
2 ε
1 2C3 1 t − τ − ε 2 t 2 ε ε 2 2 gθ Ä2Ω dθ
2C2 u τ
4C0 |Ω|
× β u τ
2 Ë10Ω 2 Äp Ω τ
t
2 g θ 2
Ä Ω dθ,
τ
4.66
for all r ∈ τ ε, t. Finally, from 4.66 and 4.56 we obtain
Gs ur 2Ä2Ω ≤
8 t−τ −
1 2C 3 β2 ε ε 2 × β u τ
2 Ë10Ω 2
β
ε
1 2 t 2 ε p
2C2 u τ
4C0 |Ω| gθ Ä2Ω dθ 2 Äp Ω τ
t
2 g θ 2 dθ 2 C2 ur 2 2 2 gr 2 2 3 Ä Ω Ä Ω Ä Ω , β2 τ
Because u 2Ë2Ω Gs u 2Ä2Ω then from 4.67, the proof is complete. 0
∀r ∈ τ ε, t. 4.67
International Journal of Mathematics and Mathematical Sciences
23
4.3. Fractal Dimensional Estimates of the Pullback D-Attractor Theorem 4.9. Under assumptions (H1bis)–(H3bis), the process Ut, τ possesses a pullback Dattractor AÄ2Ω which has a finite fractal dimension in Ä2 Ω and
8 · e2C3 dimf At ≤ k log 1
1−δ
log
2 1 δ
−1 ,
∀t ∈ Ê,
4.68
where δ < 1, k ∈ Æ , and C3 in 1.5. Proof. Let Hk span{e1 , e2 , . . . , ek } ⊂ Ä2 Ω and Pk : Ä2 Ω → Hk be the orthogonal projection, where e1 , e2 , . . . , ej , . . . are the eigenvectors of the operator −Gs corresponding to eigenvalues {λj }∞ j1 such that 0 < λ1 < λ2 ≤ λ3 ≤ · · · ≤ λj ≤ · · · and λj → ∞ as j → ∞. From 4.3, we can easily show that there exists a uniformly pullback absorbing set B of process Ut, τ in Ë10Ω. We set u1 t Ut, τu1τ and u2 t Ut, τu2τ to be solutions associated to problem 1.2 with initial datum u1τ , u2τ ∈ B. Let w u1 − u2 , because u1 , u2 being two solutions of 1.2 then we have ∂w − aGs w fu1 − fu2 0. ∂t
4.69
Multiplying 4.69 with w and integrating over Ω then we have 1 d
wt 2Ä2Ω β w 2Ë1Ω
0 2 dt
Ω
fu1 − fu2 , w dX ≤ 0,
4.70
here, we have used 1.11. Using 1.5 then we have d
wt 2Ä2Ω ≤ 2C3 wt 2Ä2Ω . dt
4.71
wt 2Ä2Ω ≤ e2C3 t−τ wτ 2Ä2Ω .
4.72
Thus,
Let wt w1 t w2 t where w1 t : Pk wt and w2 t : I − Pk wt. Therefore, by 4.72 we have
w1 t 2Ä2Ω ≤ e2C3 t−τ wτ 2Ä2Ω .
4.73
Now, taking the inner product of 4.69 with w2 in Ä2 Ω, we have 1 d 2 2
w2 t Ä2Ω β w2 Ë1Ω ≤ − fu1 − fu2 , w2 dX. 0 2 dt
4.74
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International Journal of Mathematics and Mathematical Sciences
Using the Holder inequality and 1.4, we have ¨ −
Ω
fu1 − fu2 , w2 dX ≤
Ω
fu1 − uu2 |w2 |dX
≤
Ω
≤C
fu1 − fu2 2 dX
1/2
2
Ω
|w2 | dX
1/2
1/2 1 |u1 |2p−2 |u2 |2p−2 dX
w2 Ä2Ω
4.75
Ω
2p−2
2p−2
≤ C 1 u1 Ä2p−2Ω u2 Ä2p−2Ω
1/2
w Ä2Ω
2p−2 2p−2 ≤ C 1 u1 Ä2p−2Ω u2 Ä2p−2Ω w Ä2Ω . Therefore, by 4.74, 4.75, and Proposition 4.6 we obtain
1 d
w2 t 2Ä2Ω β w2 2Ë1Ω 0 2 dt t 2 2 2 −αt αs ≤ C 1 gt Ä2Ω e e gs Ä2Ω g s Ä2Ω ds wt Ä2Ω . −∞
4.76
Because w2 t 2Ë1Ω ≥ λk w2 t 2Ä2Ω , then 4.76 implies that 0
d
w2 t 2Ä2Ω 2βλk w2 2Ä2Ω dt t 2 2 2 −αt αs gs Ä2Ω g s Ä2Ω ds wt Ä2Ω . ≤ 2C 1 gt Ä2Ω e e −∞
4.77
Now, multiplying 4.77 by eβλk t and integrating from τ to t, we get
2
w2 t Ä2Ω ≤ e
−βλk t−τ
2
wτ Ä2Ω 2Ce
−βλk t
t
e−βλk s
τ
s 2 2 2 × 1 gs Ä2Ω e−αt eαr gr Ä2Ω g r Ä2Ω dr −∞
× ws Ä2Ω ds.
4.78
International Journal of Mathematics and Mathematical Sciences
25
Using 4.73 we have
2
w2 t Ä2Ω ≤ e
−βλk t−τ
2
wτ Ä2Ω C wτ Ä2Ω e
−βλk t
t
eβλk s eC3 s−τ
τ
s 2 2 2 −αs × 1 gs Ä2Ω e eαr gr Ä2Ω g r Ä2Ω dr ds −∞
≤ e−βλk t−τ wτ 2Ä2Ω C wτ Ä2Ω eC3 t−τ
t −∞
e−βλk t−s
s 2 2 2 −αs αr gr Ä2Ω g r Ä2Ω dr ds × 1 gs Ä2Ω e e −∞
≤ e−βλk t−τ wτ Ä2Ω C wτ Ä2Ω eC3 t−τ " t t 2 1 ×
e−βλk t−s gs Ä2Ω ds
e−βλk t−s e−αs βλk −∞ −∞
s 2 2 × eαr gr Ä2Ω g r Ä2Ω dr ds . 2
−∞
4.79
Now, because t −∞
2 eαs gs Ä2Ω ds < ∞,
4.80
we can see that, for all t ∈ Ê see, e.g., 6, Lemma 3.6, t −∞
2 e−βλk t−s gs Ä2Ω ds −→ 0 as k −→ ∞,
4.81
and we have t −∞
e
−βλk t−s −βλ1 s
≤
s
e
t −∞
e
2 2 gr Ä2Ω g r Ä2Ω dr ds e αr
−∞
−βλk t βλk −λ1 s
e−αt βλk − α
t −∞
t ds
2 2 e gr Ä2Ω g r Ä2Ω dr αr
−∞
2 2 eαs gr Ä2Ω g r Ä2Ω dr.
4.82
26
International Journal of Mathematics and Mathematical Sciences
Thus, for any t ∈ Ê, from 4.82 we have t −∞
e
−βλk t−s −βλ1 s
s
e
2 2 e gr Ä2Ω g r Ä2Ω dr ds −→ 0 αr
−∞
as k −→ ∞. 4.83
Combining 4.81, 4.83 and taking T0 t − τ 1, we get k is sufficient large then from 4.79 we deduce
w2 t 2Ä2Ω ≤ δ wτ 2Ä2Ω ,
4.84
here 0 < δ < 1.
From 4.73 and 4.31, we have
w1 t 2Ä2Ω ≤ l0 wτ 2Ä2Ω ,
w2 t 2Ä2Ω ≤ δ wτ 2Ä2Ω ,
∀t ∈ Ê.
4.85
Here, l0 e2C3 ; 0 < δ < 1; T0 1. Thus, the process Ut, τ associated to 1.2 satisfies all conditions of Theorem 2.10. This completes the proof.
Acknowledgment This paper was supported by Vietnam’s National Foundation for Science and Technology Development NAFOSTED, project 101.01-2010.05.
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