Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity

J. Math. Anal. Appl. 338 (2008) 1243–1254 www.elsevier.com/locate/jmaa

Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity ✩ Lu Yang School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China Received 5 December 2006 Available online 19 June 2007 Submitted by D. Russell

Abstract In this paper, we consider dynamical behavior of non-autonomous plate-type evolutionary equations with critical nonlinearity. We prove the existence of a uniform attractor in the space H02 (Ω) × L2 (Ω). © 2007 Elsevier Inc. All rights reserved. Keywords: Uniform attractor; Plate equation; Critical exponent

1. Introduction In this paper, we consider the following non-autonomous plate equation with critical nonlinearity: ⎧ utt + a(x)ut + Δ2 u + λu + f (u) = g(x, t), x ∈ Ω, ⎪ ⎪ ⎪ ⎨ ∂  u = 0, u|∂Ω = ⎪ ∂ν ∂Ω ⎪ ⎪ ⎩ u(x, τ ) = u0τ (x), ut (x, τ ) = u1τ (x).

(1.1)

Here Ω ⊂ Rn is a bounded domain with a sufficiently smooth boundary, λ > 0, and the functions a(x) and f satisfy the following conditions: a(x) ∈ L∞ (Ω), a(x)  α0 > 0 in Ω,     f ∈ C 1 (R), f  (s)  C 1 + |s|p , 0 < p 

(1.2) 4 , n−4

f (s) > −λ21 . |s|→∞ s The external force g satisfies   g(x, t) ∈ L∞ R; L2 (Ω) lim inf

✩

Partially supported by the National Science Foundation of China Grant 10471056. E-mail address: [email protected].

0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.06.011

(1.3) (1.4)

(1.5)

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L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

and   ∂t g ∈ Lrb R; Lr (Ω)

with r >

2n . n+4

(1.6)

The parameter α0 in (1.2) is a constant and λ1 in (1.4) is the first eigenvalue of −Δ with zero Dirichlet data. The 4 is called the critical exponent, since the nonlinearity f is not compact in this case, which is an essential number n−4 difficulty in studying the asymptotic behavior even for the autonomous case (e.g., [6,7,10,17,23] etc.). It is clear that the external force g satisfying (1.5) and (1.6) is translation bounded but not translation compact (the definition can be found in the beginning of the next section). In present paper, we consider the non-autonomous system (1.1) via the uniform attractors of the corresponding family of processes {Uσ (t, τ )}, σ ∈ Σ . The feature of the model (1.1) is that: (i) the nonlinearity f (u) has critical exponent, and (ii) the external forcing g(x, t) is not translation compact in L2loc (R; L2 (Ω)). Let us recall some relevant research in this area. For the autonomous case, when a(x) ≡ α0 > 0, the existence of a global attractor for plate equation with subcritical exponent was studied in [1] and the critical exponent case was studied in [15]. Khanmamedov [14] showed the global attractor for Eq. (1.1) with critical nonlinearity. In the case of semilinear wave equations, if a(x) ≡ constant, the long-time behavior of weak solutions was investigated by many authors, see, for example, [6,10] for the linear damping and [7,17,23] for the nonlinear damping. In the locally distributed damping form (i.e. a(x) ≡ constant), the exponential decay property of solutions was successively treated in [4,5,22] and references therein. Using finite speed of propagation and a unique continuation result for the wave equations, the existence of a global attractor was established in [3]. Other results of global attractor can be found in [2,12,18] and references therein. For the special case of wave equations with localized damping, i.e., f (u) = g(x, t) ≡ 0, the problem was also considered extensively, for instance, [20] for the localized nonlinear damping and [19,21] for the localized linear dissipation. The study of wave equation with localized damping in autonomous case has attracted much attention in recent years. However, non-autonomous hyperbolic equation with localized damping is less discussed, especially for the plate equation. When a(x) ≡ constant, in Chepyzhov and Vishik [8], the authors obtained the existence of a uniformly attractor when g is translation compact and nonlinearity f is subcritical. In Zelik [9], under some assumptions (i.e., g, ∂t g ∈ L∞ (R; L2 (Ω)), f ∈ C 2 (R), f (0) = 0, and f is critical exponent) and by using a bootstrap argument, the author obtained some regularity estimates for the solutions, which naturally implies the existence of a uniform attractor. Belleri and Pata [24] considered the strongly damped semilinear wave equation with localized dissipation and proved the existence of uniform absorbing set. Our main goal in this paper is to prove the existence of an uniform attractor for the problem (1.1)–(1.6) in H02 (Ω) × L2 (Ω), by using a “contractive function” method, which was proposed by Sun et al. in [16]. They used the method to testify the uniform asymptotic compactness. The initial idea of the method occurred in the reference [13] for autonomous case. The key idea of “contractive function” is to verify asymptotic compactness of the corresponding semigroup (or process). Comparing to the autonomous case, the main complexity of non-autonomous system is placed on the external force. Under appropriate assumptions, we need to prove that the external force belongs to the contractive function. To our best knowledge, the present paper is the first one of dealing with the non-autonomous plate equation with critical nonlinearity. This paper is organized as follows: In Section 2, we give some preparations for our consideration; in Section 3, we prove the existence of an uniformly absorbing set in H02 (Ω) × L2 (Ω); in the last section, we derive uniform asymptotic compactness of the corresponding family of processes {Uσ (t, τ )}, σ ∈ Σ , generated by problem (1.1). 2. Preliminaries In this section, we first recall some basic concepts about non-autonomous systems, we refer to [8] for more details. Space of translation bounded functions in Lrloc (R; Lk (Ω)), with r, k  1: Lrb



r

t+1

k k      k r k   g(x, s) dx ds < ∞ . R; L (Ω) = g ∈ Lloc R; L (Ω) : sup t∈R

t

Ω

L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

1245

Space of translation compact functions in L2loc (R; L2 (Ω)):      L2c R; L2 (Ω) = g ∈ L2loc R; L2 (Ω) : for any interval [t1 , t2 ] ⊂ R,     is precomputed in L2 t1 , t2 ; L2 (Ω) . g(x, h + s): h ∈ R  [t1 ,t2 ]

Let X be a Banach space, and Σ a parameter set. The operators {Uσ (t, τ )}, σ ∈ Σ , are said to be a family of processes in X with symbol space Σ if for any σ ∈ Σ Uσ (t, s) ◦ Uσ (s, τ ) = Uσ (t, τ ), Uσ (τ, τ ) = Id(identity),

∀t  s  τ, τ ∈ R,

∀τ ∈ R.

(2.1) (2.2)

Let {T (s)}s0 be the translation semigroup on Σ. We say that a family of processes {Uσ (t, τ )}, σ ∈ Σ , satisfies the translation identity if Uσ (t + s, τ + s) = UT (s)σ (t, τ ), T (s)Σ = Σ,

∀σ ∈ Σ, t  τ, τ ∈ R, s  0,

∀s  0.

(2.3) (2.4)

We denote by B(X) the collection of the bounded sets of X, and Rτ = {t ∈ R, t  τ }. Definition 2.1. (See [8].) A bounded set B0 ∈ B(X) is said to be a bounded uniformly (w.r.t.  σ ∈ Σ ) absorbing set for {Uσ (t, τ )}, σ ∈ Σ , if for any τ ∈ R and B ∈ B(X) there exists T0 = T0 (B, τ ) such that σ ∈Σ Uσ (t; τ )B ⊂ B0 for all t  T0 . Definition 2.2. (See [8].) A set A ⊂ X is said to be uniformly (w.r.t σ ∈ Σ) attracting for the family of processes {Uσ (t, τ )}, σ ∈ Σ, if for any fixed τ ∈ R and any B ∈ B(X)    lim sup dist Uσ (t; τ )B; A = 0, t→+∞

σ ∈Σ

where dist(·,·) is the usual Hausdorff semidistance in X between two sets. In particular, a closed uniformly attracting set AΣ is said to be the uniform (w.r.t. σ ∈ Σ ) attractor of the family of processes {Uσ (t, τ )}, σ ∈ Σ , if it is contained in any closed uniformly attracting set (minimality property). Similar to the autonomous cases (e.g., see [14]), we can easily obtain the following existence and uniqueness results. The proof is based on the Galerkin approximation method (e.g., see [25]) and the time-dependent terms make no essential complications. Lemma 2.3. Let Ω be a bounded domain of Rn with smooth boundary, a(x) and f satisfy (1.2)–(1.4), g ∈ L∞ (R; L2 (Ω)). Then for any initial data (u0τ , u1τ ) ∈ H02 (Ω) × L2 (Ω), the problem (1.1) has an unique solution u(t) which satisfies (u(t), ut (t)) ∈ C(Rτ ; H02 (Ω) × L2 (Ω)) and ∂tt u(t) ∈ L2loc (Rτ ; H −2 (Ω)). For convenience, hereafter, the norm and scalar product in L2 (Ω) are denoted by  ·  and (·,·), respectively. We use the notations as in Chepyzhov and Vishik [8]. Let y(t) = (u(t), ut (t)), yτ = (u0τ , u1τ ), E0 = H02 (Ω) × L2 (Ω) with finite energy norm yE0 = Δu2 + ut 2 . Then system (1.1) is equivalent to ∂t ut = −Δ2 u − a(x)ut − λu − f (u) + g(x, t), for any t  τ, ∂  u|∂Ω = u = 0, u(x, τ ) = u0τ (x), ut (x, τ ) = u1τ (x). ∂ν ∂Ω We can also write (2.5) in the operator form ∂t y = Aσ (t) (y),

y|t=τ = yτ ,

where σ (t) = g(x, t) is symbol of Eq. (2.6).

(2.5)

(2.6)

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L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

We now define the symbol space for (2.6). Take a fixed symbol σ0 (s) = g0 (x, s), g0 ∈ L∞ (R; L2 (Ω)) ∩ 2n for some r > n+4 . Set   Σ0 = (x, t) → g0 (x, t + h): h ∈ R (2.7)

Wb1,r (R; Lr (Ω))

and

    Σ the ∗ -weakly closure of Σ0 in L∞ R; L2 (Ω) ∩ Wb1,r R; Lr (Ω) .

(2.8)

Then we have the following simple properties. Proposition 2.4. Σ is bounded in L∞ (R; L2 (Ω)) ∩ Wb1,r (R; Lr (Ω)), and for any σ ∈ Σ , the following estimate holds: σ L∞ (R;L2 (Ω))∩W 1,r (R;Lr (Ω))  g0 L∞ (R;L2 (Ω))∩W 1,r (R;Lr (Ω)) . b

b

Thus, from Lemma 2.3, we know that (1.1) is well posed for all σ (s) ∈ Σ and generates a family of processes {Uσ (t, τ )}, σ ∈ Σ, given by the formula Uσ (t, τ )yτ = y(t). The y(t) is the solution of (1.1)–(1.6) and {Uσ (t, τ )}, σ ∈ Σ , satisfies (2.1)–(2.2). At the same time, due to the unique solvability, we know {Uσ (t, τ )}, σ ∈ Σ , satisfies the translation identity (2.3)–(2.4). In what follows, we denote by {Uσ (t, τ )}, σ ∈ Σ , the family of processes which is generated by (2.6)–(2.8). 3. Uniformly (w.r.t. σ ∈ Σ) absorbing set in E0 Theorem 3.1. Under assumptions (1.2)–(1.6), the family of processes {Uσ (t, τ )}, σ ∈ Σ , corresponding to (1.1) has a bounded uniformly (w.r.t. σ ∈ Σ ) absorbing set B0 in E0 . Proof. From the definition of Σ we know that for all σ ∈ Σ, σ 2L2  g0 2L2 . b

(3.1)

b

s Let F (s) = 0 f (τ ) dτ , taking the scalar product with v = ut + δu in L2 , where 0 < δ  δ0 (the parameter δ0 will be determined later), we get 

  1 d v2 + Δu2 + λu2 + 2 F (u) dx + a(x)v, v − δv2 + δΔu2 2 dt Ω

  2 2 + δλu − δ a(x)u, v + δ (u, v) + δ f (u)u dx = (g, v). (3.2) Ω

From (1.3) and (1.4) we know that there are λ21 > λ > 0 and C0 > 0 such that

  λ f (u), u  λ F (u) dx − C0 , F (u)  − u2 − C0 . 2 Ω

Due to (1.2), there exists L > 0, such that     −δ a(x)u, v  −δL(u, v), a(x)v, v  α0 v2 . From now on, δ0 is chosen as   α0 λ21 α0 δ0 = min . , 4 2L2 Consequently, we obtain the estimate by using Hölder and Young inequalities

L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

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    a(x)v, v − δv2 + δΔu2 − δ a(x)u, v + δ 2 (u, v)  (α0 − δ)v2 + δΔu2 − δ(L − δ)(u, v)  3 δ δL2 2 2 2 2 Δu + 2 v  δΔu + α0 v − 4 2 2λ1 δ α0  Δu2 + v2 . 2 2

(3.3)

Inserting the inequality (3.3) in (3.2), we have 

δ α0 1 d 2 2 2 v + Δu + λu + 2 F (u) dx + Δu2 + v2 2 dt 2 2 Ω

α0 1 + δλu2 + δλ F (u) dx  δC0 + v2 + g2 . 4 α0

(3.4)

Ω

 Choosing γ = min{δ, δλ } and setting z(t) = v2 + Δu2 + λu2 + 2 Ω F (u) dx, we get      2 y(t)  z(t)  z(τ ) exp −γ (t − τ ) + 1 + γ −1 2δC0 + g2L2 , α0 b

(3.5)

where we have also used the Gronwall’s inequality. From (3.1), we know g2L2  g0 2L2 b

b

for all g ∈ Σ.

Therefore, we obtain the uniformly (w.r.t. σ ∈ Σ) absorbing set B0 in E0 ,    B0 = y = (u, ut )  y2  ρ02 , where ρ0 = 2(1 + γ −1 )(2δC0 + t0 (τ, B)  τ such that  Ug (t, τ )B ⊂ B0 ,

2 2 α0 g0 L2 ).

∀t  t0 .

b

That is to say, for any bounded subset B in E0 , there exists a t0 =

2

g∈Σ

4. Uniform (w.r.t. σ ∈ Σ) asymptotic compactness in E0 In this section, we first give some useful preliminaries and then obtain some a priori estimates about the energy inequalities based on the idea presented in [7,13,16,23]. Finally, we use Theorem 4.5 to establish the uniform (w.r.t. σ ∈ Σ ) asymptotic compactness in E0 . Hereafter, we always denote by B0 the bounded uniformly (w.r.t. σ ∈ Σ) absorbing set obtained in Theorem 3.1. 4.1. Preliminaries We now recall the simple criterion developed in [16]. Definition 4.1. (See [16].) Let X be a Banach space, B a bounded subset of X and Σ a symbol (or parameter) space. We call a function φ(·,· ; ·,·), defined on (X × X) × (Σ × Σ), to be a contractive function on B × B if for any sequence ∞ ∞ ∞ ∞ {xn }∞ n=1 ⊂ B and any {σn } ⊂ Σ, there is a subsequence {xnk }k=1 ⊂ {xn }n=1 and {σnk }k=1 ⊂ {σn }n=1 such that lim lim φ(xnk , xnl ; σnk , σnl ) = 0.

k→∞ l→∞

We denote the set of all contractive functions on B × B by Contr(B, Σ).

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L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

Theorem 4.2. (See [16].) Let {Uσ (t, τ )}, σ ∈ Σ , be a family of processes which satisfies the translation identity (2.3)– (2.4) on Banach space X and has a bounded uniformly (w.r.t. σ ∈ Σ) absorbing set B0 ⊂ X. Moreover, assume that for any ε > 0 there exist T = T (B0 , ε) and φT ∈ Contr(B0 , Σ) such that   Uσ (T , 0)x − Uσ (T , 0)y   ε + φT (x, y; σ1 , σ2 ), ∀x, y ∈ B0 , ∀σ1 , σ2 ∈ Σ. 1

2

Then {Uσ (t, τ )}, σ ∈ Σ , is uniformly (w.r.t. σ ∈ Σ ) asymptotically compact in X. Applying Proposition 7.1 of Robinson [11], we immediately get the following results. The proof is based on the technique in [16], for reader’s convenience, we replicate it here and only make a few minor changes for our problem. Proposition 4.3. Let g ∈ L∞ (R; L2 (Ω)) ∩ Wb1,r (R; Lr (Ω))(r >   supg(x, t + s)L2 (Ω)  M for all s ∈ R.

2n n+4 ).

Then there is an M > 0 such that

t∈R

2n ), {un (t) | t  0, n = Proposition 4.4. Let si ∈ R (i = 1, 2, . . .), g ∈ L∞ (R; L2 (Ω)) ∩ Wb1,r (R; Lr (Ω)) (r > n+4 2 ∞ 1, 2, . . .} be bounded in H0 (Ω), and for any T1 > 0, {unt (t) | n = 1, 2, . . .} bounded in L (0, T1 ; L2 (Ω)). Then for ∞ ∞ ∞ any T > 0, there exist subsequences {unk }∞ k=1 of {un }n=1 and {snk }k=1 of {sn }n=1 such that

T T

lim lim

k→∞ l→∞ 0

  g(x, τ + snk ) − g(x, τ + snl ) (unk − unl )t (τ ) dx dτ ds = 0.

s Ω

Proof. Since {un (t) | t  0, n = 1, 2, . . .} is bounded in H02 (Ω) and for any T1 > 0, {unt (t) | n = 1, 2, . . .} is bounded in L∞ (0, T1 ; L2 (Ω)), we assume for any T > 0, without loss of generality (at most by passing subsequence), that un (T ) → u0

in L2 (Ω)

and   in Lk 0, T ; Lk (Ω)

un → v where k <

 this require r >

2n , n+4

2n n−4 .

1,2 Note that for any w ∈ Wloc (R; L2 (Ω)),   g(x, t + si ) − g(x, t + sj ) wt (t)     d  g(x, t + si ) − g(x, t + sj ) w(t) − gt (x, t + si ) − gt (x, t + sj ) wt (t). = dt Then by using Hölder inequality, we obtain

T T

lim lim

n→∞ m→∞ 0



 g(x, τ + sn ) − g(x, τ + sm ) (un − um )t (τ ) dx dτ ds

s Ω



 lim lim 2MT n→∞ m→∞

Ω

T

+ lim lim T n→∞ m→∞

  un (T ) − um (T )2 dx

1 2

+ lim lim 2MT

1 2

 T

n→∞ m→∞

0 Ω

    gt (x, s + sn ) − gt (x, s + sm ) un (s) − um (s)  dx ds

0 Ω

T



= lim lim T n→∞ m→∞

0 Ω

  unx (s) − um (s)2 dx ds

    gt (x, s + sn ) − gt (x, s + sm ) un (s) − um (s)  dx ds

1 2

L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

 T

 lim lim T n→∞ m→∞

  gt (x, s + sn ) − gt (x, s + sm )r

0 Ω

= 0.

 1  T

r

1249

  un (s) − um (s)k

1 k

0 Ω

2

4.2. A priori estimates The main purpose of this subsection is to establish (4.10)–(4.12), which will be used to obtain the uniform (w.r.t. σ ∈ Σ) asymptotic compactness. For any (ui0 , v0i ) ∈ B0 , let (ui (t), uit (t)) be the corresponding solution to σi with respect to initial data (ui0 , v0i ), i = 1, 2, that is, (ui (t), uit (t)) is the solution of the following equation: ⎧   utt + a(x)ut + Δ2 u + λu + f u(t) = σi (x, t), ⎪ ⎪ ⎪ ⎨ ∂  u = 0, u|∂Ω = (4.1) ⎪ ∂ν ∂Ω ⎪ ⎪     ⎩ u(0), ut (0) = ui0 , v0i . Lemma 4.5. Assume the condition (1.2) is satisfied. Then for any fixed T > 0, there exist a constant CT and a function φT = φT ((u10 , υ01 ), (u20 , υ02 ); σ1 , σ2 ) such that        u1 (T ) − u2 (T )  CT + φT u1 , υ 1 , u2 , υ 2 ; σ1 , σ2 , 0 0 0 0 E 0

where CT and φT depend on T . Proof. For convenience, we denote gi (t) = σi (x, t),

t  0, i = 1, 2,

and w(t) = u1 (t) − u2 (t). Then w(t) satisfies     ⎧ wtt + a(x)wt + Δ2 w + λw + f u1 (t) − f u2 (t) = g1 (t) − g2 (t), ⎪ ⎪ ⎨ ∂  w  = 0, w|∂Ω = ⎪ ∂ν  ∂Ω ⎪    ⎩ w(0), wt (0) = u10 , v01 − u20 , v02 . Set Ew (t) =

1 2



  wt (t)2 + 1 2

Ω



(4.2)

  Δw(t)2 .

Ω

At first, multiplying (4.2) by w and integrating over [0, T ] × Ω, we get

T

0 Ω

  Δw(s)2 dx ds =



wt (0)w(0) dx −

Ω

1 wt (T )w(T ) dx + 2

Ω



1 − 2

Ω

T



− 0 Ω

2  a(x)w(T ) dx − λ



2  a(x)w(0) dx

Ω

T

  w(s)2 dx ds +

0 Ω

     f u1 (s) − f u2 (s) w(s) dx ds +

T

|wt (s)|2 dx ds 0 Ω

T

(g1 − g2 )w dx ds.

0 Ω

(4.3)

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L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

Then we have

T Ew (s) ds =

1 2

wt (0)w(0) dx −

1 2

Ω

0

1 − 4

wt (T )w(T ) dx + Ω



1 2



2  a(x)w(0) dx

Ω

2  1 a(x)w(T ) dx − λ 2

Ω

−

1 4

T

  w(s)2 dx ds +

0 Ω

T



T

  wt (s)2 dx ds

0 Ω

     1 f u1 (s) − f u2 (s) w(s) dx ds + 2

0 Ω

T

(g1 − g2 )w dx ds.

(4.4)

0 Ω

From condition (1.2), we get

T

  wt (s)2 dx ds  c˜

T

s Ω

 2 a(x)wt (s) dx ds,

s Ω

where c˜ is a constant. So we have

T Ew (s) ds 

1 2

wt (0)w(0) dx − Ω

0

1 − 4

1 2

wt (T )w(T ) dx + Ω



−

1 2



2  a(x)w(0) dx

Ω

2  1 a(x)w(T ) dx − λ 2

Ω

1 4

T



  w(s)2 dx ds + c˜

T

0 Ω

T

     1 f u1 (s) − f u2 (s) w(s) dx ds + 2

0 Ω

 2 a(x)wt (s) dx ds

0 Ω

T

(g1 − g2 )w dx ds.

(4.5)

0 Ω

Secondly, multiplying (4.2) by wt and integrating over [s, T ] × Ω, we get

T

Ew (T ) +

2  2 1  a(x)wt (τ ) dx dτ + λw(T ) 2

s Ω

T

= Ew (s) −

     f u1 (τ ) − f u2 (τ ) wt (τ ) dx dτ

s Ω

2 1  + λw(s) + 2

T

(g1 − g2 )wt dx dτ,

(4.6)

s Ω

where 0  s  T . Integrating (4.6) over [0, T ] with respect to s, we obtain that

T T Ew (T ) 

T T

Ew (s) ds −

0

1 + λ 2

0

T



0 Ω

From (4.6) we also have

     f u1 (τ ) − f u2 (τ ) wt (τ ) dx dτ ds

s Ω

  w(s)2 dx ds +

T T

(g1 − g2 )wt dx dτ ds. 0

s Ω

(4.7)

L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

T

 2 a(x)wt (τ ) dx dτ  Ew (0) −

0 Ω

T

1251

     f u1 (τ ) − f u2 (τ ) wt (τ ) dx dτ

0 Ω

2 1  + λw(0) + 2

T

(g1 − g2 )wt dx dτ.

(4.8)

0 Ω

Therefore, from (4.5), (4.7) and (4.8), we get

2 1 2  1  T Ew (T )  cE ˜ w(0) + ˜ w (0) + cλ a(x)w(0) dx 2 4 Ω



2  1 1 1 − a(x)w(T ) dx + wt (0)w(0) dx − wt (T )w(T ) dx 4 2 2 Ω

Ω

Ω

T

+ c˜

T

(g1 − g2 )wt dx ds − c˜

0 Ω

1 2

+

0 Ω

T

T T

(g1 − g2 )w dx ds −

0 Ω

T T

+

0

T

     f u1 (s) − f u2 (s) w(s) dx ds.

2 1 1  ˜ w(0) + = cE ˜ w (0) + cλ 2 2



1 wt (0)w(0) dx − 2

Ω

φT

(4.9)

0 Ω

Set CM

     f u1 (τ ) − f u2 (τ ) wt (τ ) dx dτ ds

s Ω

1 (g1 − g2 )wt dx dτ ds − 2

s Ω

0

     f u1 (s) − f u2 (s) wt (s) dx ds

wt (T )w(T ) dx,

(4.10)

Ω

 1 1   2 2   u0 , υ0 , u0 , υ0 ; σ1 , σ2 1 = 4



2  1 a(x)w(0) dx − 4

Ω



2  a(x)w(T ) dx +

Ω

T

+ c˜

(g1 − g2 )wt dx ds − c˜

1 + 2

T

− 0

s Ω

     f u1 (s) − f u2 (s) wt (s) dx ds

0 Ω

1 (g1 − g2 )w dx ds − 2

0 Ω

T T

(g1 − g2 )wt dx dτ ds 0

T

0 Ω

T T

T

     f u1 (s) − f u2 (s) w(s) dx ds

0 Ω

     f u1 (τ ) − f u2 (τ ) wt (τ ) dx dτ ds.

(4.11)

s Ω

Then we have Ew (T ) 

    1  CM + φT u10 , υ01 , u20 , υ02 ; σ1 , σ2 . T T

The proof is complete.

2

(4.12)

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L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

4.3. Uniform asymptotic compactness In this subsection, we shall prove the uniform (w.r.t. σ ∈ Σ ) asymptotic compactness in H02 (Ω) × L2 (Ω), which is given in the following theorem. 2n Theorem 4.6. Assume that a(x) and f satisfy (1.2)–(1.4). If g0 ∈ L∞ (R; L2 (Ω))∩Wb1,r (R; Lr (Ω)) for some r > n+4 and Σ is defined by (2.8), then the family of processes {Uσ (t, τ )}, σ ∈ Σ , corresponding to (1.1), is uniformly (w.r.t. σ ∈ Σ) asymptotically compact in H02 (Ω) × L2 (Ω).

Proof. Since the family of processes {Uσ (t, τ )}, σ ∈ Σ , has a bounded uniformly absorbing set and from Lemma 4.5, for any fixed ε > 0, we can choose T large enough, such that CM  ε. T Hence, thanks to Theorem 4.2, it is sufficient to prove φT (·,· ; ·,·) ∈ Contr(B0 , Σ) for each fixed T . From the proof procedure of Theorem 3.1, we can deduce that for any fixed T ,   Uσ (t, 0)B0 is bounded in E0 ,

(4.13)

σ ∈Σ t∈[0,T ]

and the bound depends on T . Let (un , unt ) be the solutions corresponding to initial data (un0 , v0n ) ∈ B0 with respect to symbol σn ∈ Σ, n = 1, 2, . . . . Then, from (4.13), without loss of generality (at most by passing subsequence), we assume that   un → u  -weakly in L∞ 0, T ; H02 (Ω) , (4.14)   ∞ 2 unt → ut  -weakly in L 0, T ; L (Ω) , (4.15)   2 2 un → u in L 0, T ; L (Ω) , (4.16)   k k (4.17) un → u in L 0, T ; L (Ω) , un (0) → u(0)

and un (T ) → u(T )

in Lk (Ω),

2n n−4 ,

(4.18) H02

→ Lk .

where we have used the compact embedding for k < We now deal with each term on the right-hand side of (4.11). At first, from Proposition 4.3 and (4.17), by the similar method used in the proof of Proposition 4.4, we can obtain

T

lim lim

n→∞ m→∞

   gn (x, s) − gm (x, s) un (s) − um (s) dx ds = 0,

(4.19)

0 Ω

and from Proposition 4.4 we can get

T

lim lim

n→∞ m→∞

   gn (x, s) − gm (x, s) unt (s) − umt (s) dx ds = 0,

(4.20)

0 Ω

T T

lim lim

n→∞ m→∞ 0



  gn (x, τ ) − gm (x, τ ) unt (τ ) − umt (τ ) dx dτ ds = 0.

(4.21)

s Ω

Secondly, from (4.16), we have

T

lim lim

n→∞ m→∞

      f un (s) − f um (s) un (s) − um (s) dx ds = 0.

0 Ω

From (4.18), noticing the condition (1.2), we obtain

(4.22)

L. Yang / J. Math. Anal. Appl. 338 (2008) 1243–1254

lim lim

n→∞ m→∞

1253

 2 a(x)un (T ) − um (T ) dx = 0,

(4.23)

 2 a(x)un (0) − um (0) dx = 0.

(4.24)

Ω

and

lim lim

n→∞ m→∞ Ω

Finally, using the similar method in the proof of Lemma 2.2 in [23], we get

T

lim lim

n→∞ m→∞

      f un (s) − f um (s) unt (s) − umt (s) dx ds = 0,

(4.25)

0 Ω

T T

lim lim

n→∞ m→∞ 0

      f un (τ ) − f um (τ ) unt (τ ) − umt (τ ) dx dτ ds = 0.

(4.26)

s Ω

Hence, combining (4.19)–(4.26), we get φT (·,· ; ·,·) ∈ Contr(B0 , Σ) immediately.

2

4.4. Existence of uniform attractor 2n Theorem 4.7. Assume that a(x) and f satisfy (1.2)–(1.4). If g0 ∈ L∞ (R; L2 (Ω))∩Wb1,r (R; Lr (Ω)) for some r > n+4 and Σ is defined by (2.8), then the family of processes {Uσ (t, τ )}, σ ∈ Σ, corresponding to (1.1) has a compact uniform (w.r.t. σ ∈ Σ ) attractor AΣ .

Proof. Theorems 3.1 and 4.6 imply the existence of an uniform attractor immediately.

2

Acknowledgments The author expresses her gratitude to Professor Chengkui Zhong for his guidance and encouragement, also to the anonymous referee for his/her valuable comments and suggestions, which lead to the present improved version of our paper.

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