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Uniform attractors for a non-autonomous viscoelastic equation with a past history
Nonlinear Analysis 101 (2014) 1–15
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Uniform attractors for a non-autonomous viscoelastic equation with a past history Yuming Qin a,∗ , Baowei Feng b , Ming Zhang c a
Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China
b
College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, PR China
c
College of Information Science and Technology, Donghua University, Shanghai 201620, PR China
article
abstract
info
Article history: Received 27 May 2013 Accepted 9 January 2014 Communicated by Enzo Mitidieri
This paper investigates the existence of uniform attractors for a non-autonomous viscoelastic equation with a past history in a bounded domain Ω ⊆ Rn (n ≥ 1) by establishing the uniformly asymptotic compactness of the semi-process generated by the global solutions. Our results improve those results by Araújo et al. (2013). © 2014 Elsevier Ltd. All rights reserved.
Keywords: Viscoelastic equation Uniform attractors Past history
1. Introduction In this paper we investigate the existence of uniform attractors for a nonlinear non-autonomous viscoelastic equation with a past history
|ut |ρ utt − 1u − 1utt +
+∞
g (s)1u(t − s)ds + ut = σ (x, t ),
x ∈ Ω, t > τ ,
(1.1)
0
u(x, t ) = 0,
on ∂ Ω × [τ , +∞),
τ
τ
u(x, τ ) = u0 (x),
ut (x, τ ) = u1 (x),
(1.2) u(x, t ) = uτ (x, t ),
x ∈ Ω, τ ∈ R , +
(1.3)
where Ω ⊆ R (n = 1, 2) is a bounded domain with smooth boundary ∂ Ω , uτ (x, t ) (the past history of u) is a given datum which has to be known for all t ≤ τ , the function g represents the kernel of a memory, σ = σ (x, t ) is a non-autonomous term, called a symbol, and ρ is a real number such that n
1<ρ≤
2 n−2
if n ≥ 3;
ρ > 1 if n = 1, 2.
(1.4)
Our problem is of the form f (ut )utt − 1u − 1utt = 0,
(1.5)
which has several modeling features. In the case f (ut ) is a constant, Eq. (1.5) has been used to model extensional vibrations of thin rods (see Love [1, Chapter 20]). In the case f (ut ) is not a constant, Eq. (1.5) can model materials whose density depends
∗
Corresponding author. Tel.: +86 21 67792089 545; fax: +86 21 67823227. E-mail addresses: [email protected] (Y. Qin), [email protected] (B. Feng), [email protected] (M. Zhang).
0362-546X/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2014.01.006
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Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
on the velocity ut . For instance, a thin rod which possesses a rigid surface and with an interior which can deform slightly. We refer the reader to Fabrizio and Morro [2] for several other related models. Let us recall some results concerning viscoelastic wave equations. In [3], Cavalcanti et al. studied the following equation with Dirichlet boundary conditions
|ut |ρ utt − 1u − 1utt + g ∗ 1u − γ 1ut = 0 (1.6) t where g ∗ 1u = 0 g (t − s)1u(s)ds. They established a global existence result for γ ≥ 0 and an exponential decay of energy for γ > 0, and studied the interaction within the |ut |ρ utt and the memory term g ∗ 1u. Messaoudi and Tatar [4] established, for small initial data, the global existence and uniform stability of solutions to the equation
|ut |ρ utt − 1u − 1utt + g ∗ 1u = b|u|p−2 u,
(1.7)
with Dirichlet boundary condition, where γ ≥ 0, ρ, b > 0, p > 2 are constants. In the case b = 0 in (1.7), Messaoudi and Tatar [5] proved the exponential decay of global solutions to (1.7) without smallness of initial data, considering only the dissipation effect given by the memory. Considering nonlinear dissipation, Han and Wang [6] proved the energy decay for the viscoelastic equation with nonlinear damping
|ut |ρ utt − 1u − 1utt + g ∗ 1u + |ut |m ut = 0,
(1.8)
with Dirichlet boundary condition, where ρ > 0, m > 0 are constants. Then Park and Park [7] established the general decay for the viscoelastic problem with nonlinear weak damping
|ut |ρ utt − 1utt − 1u + g ∗ 1u + h(ut ) = 0,
(1.9)
with the Dirichlet boundary condition, where ρ > 0 is a constant. Recently, Araújo et al. [8] studied the following equation
|ut |ρ utt − 1u − 1utt +
+∞
µ(s)1u(t − s)ds + f (u) = h(x), 0
and proved the global existence, uniqueness and exponential stability, and the global attractor was also established, but they did not establish the uniform attractors for the non-autonomous equation. Moreover, we would like to mention some results in [9–13]. +∞ For problem (1.1)–(1.3) with σ (x, t ) = 0, when 0 g (s)1u(t − s)ds was replaced by g ∗ 1u, Han and Wang [14] established the global existence of weak solutions and the uniform decay estimates for the energy by using the Faedo–Galerkin method and the perturbed energy method, respectively. To the best of our knowledge, there is no result on the existence of uniform attractors for the non-autonomous viscoelastic problem (1.1)–(1.3). Therefore in this paper, we shall establish the existence of uniform attractors for problem (1.1)–(1.3) by establishing uniformly asymptotic compactness of the semiprocess generated by their global solutions. Noting that the symbol σ (x, t ), which is dependent on t, so our estimates are more complicated than [8] and we must use new methods to deal with the symbol σ (x, t ) as the change of time. Therefore we improved the results in [8]. For more results concerning attractors, we can refer to [15–21]. Motivated by [22–24], we shall add a new variable η = ηt (x, s) to the system which corresponds to the relative displacement history. Let us define
η = ηt (x, s) = u(x, t ) − u(x, t − s),
t ≥ τ , (x, s) ∈ Ω × R+ .
(1.10)
A direct computation yields
ηtt (x, s) = −ηst (x, s) + ut (x, t ),
t ≥ τ , ( x, s) ∈ Ω × R + ,
(1.11)
and we can take as initial condition (t = τ )
ητ (x, s) = uτ0 (x) − uτ0 (x, τ − s),
( x, s) ∈ Ω × R + .
(1.12)
Thus, the original memory term can be written as +∞
g (s)1u(t − s)ds = 0
+∞
g (s)ds · 1u −
+∞
0
g (s)1ηt (s)ds,
(1.13)
0
and we get a new system ρ
+∞
g (s)ds 1u − 1utt −
|ut | utt − 1 − 0
+∞
g (s)1ηt (s)ds + ut = σ (x, t ),
(1.14)
0
ηtt + ηst = ut ,
(1.15)
with the boundary conditions u=0
on ∂ Ω × R+ ,
ηt = 0 on ∂ Ω × R+ × R+ ,
(1.16)
and initial conditions u(x, τ ) = uτ0 (x),
ut (x, τ ) = uτ1 (x),
ηt (x, 0) = 0,
ητ (x, s) = uτ0 (x) − u(x, τ − s).
(1.17)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
3
The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statements and the proofs of our main results will be given in Sections 3 and 4, respectively. For convenience, we denote the norm and scalar product in L2 (Ω ) by ∥ · ∥ and (·, ·), respectively. C1 denotes a general positive constant, which may be different in different estimates. 2. Preliminaries and the main result We assume the memory kernel g : R+ → R+ is a bounded C 1 function such that +∞
g (s) < +∞,
g (s)ds > 0
l=1−
(2.1)
0
and suppose that there exist a positive constant ξ2 verifying g ′ (t ) ≤ −ξ2 g (t ),
∀ t ≥ 0,
(2.2)
and the symbol
σ ∈ E1 ≡ L2 (R+ , L2 (Ω )).
(2.3)
In order to consider the relative displacement η as a new variable, one introduces the weighted L -space 2
M=
L2g
(R ; +
H01
(Ω )) = u : R → +
H01
(Ω )
+∞
g (s)∥∇ u(s)∥ ds < +∞ 2
,
0
which is a Hilbert space equipped with the inner product and the norm
(u, v)M =
+∞
g (s)
Ω
0
+∞
∇ u(s)∇v(s)dx ds and ∥u∥2M =
g (s)∥∇ u(s)∥2 ds, 0
respectively. Let H = H01 (Ω ) × H01 (Ω ) × M .
(2.4)
The energy of problem (1.14)–(1.17) is given by F (t ) =
1
ρ+2
l
ρ+2
1
1
∥ut (t )∥ρ+2 + ∥∇ u(t )∥2 + ∥∇ ut (t )∥2 + ∥ηt ∥2M .
(2.5)
2 2 2 First, we give the following existence and uniqueness results.
Theorem 2.1. Let (uτ0 , uτ1 , ητ ) ∈ H (∀ τ ∈ R+ ), Rτ = [τ , +∞), and any fixed σ ∈ E1 . Assume (2.1) and (2.2) hold. Then problem (1.14)–(1.17) admits a unique global solution (u, ut , ηt ) ∈ C ([0, T ], H ) such that u ∈ L∞ (Rτ , H01 (Ω )),
ut ∈ L∞ (Rτ , H01 (Ω )),
utt ∈ L2 (Rτ , H01 (Ω )),
ηt ∈ L∞ (Rτ , M).
(2.6)
We now define the symbol space for (1.14)–(1.17). Observe the following important fact: the properly defined (uniform) attractor A of problem (1.14)–(1.17) with the symbol σ0 must be simultaneously the attractor of each problem (1.14)–(1.17) with the symbol σ (t ) ∈ H+ (σ0 ), which is called the hull of σ0 and defined as
Σ = H+ (σ0 ) = [σ0 (t + h)|h ∈ R+ ]E1 where [ · ]E1 denotes the closure in Banach space E1 .
(2.7)
We note that
σ0 ∈ E1 ⊆ E1 = L2loc (R+ , L2 (Ω )) where σ0 is a translation compact function in E1 in the weak topology, which means that σ0 is compact in E1 . We consider p the Banach space Lloc (R+ , E1 ) of functions σ (s), s ∈ R+ with values in a Banach space E1 that are locally p-power integrable in the Bochner sense. In particular, for any time interval [t1 , t2 ] ⊆ R+ ,
t2 t1
∥σ (s)∥pE1 ds < +∞. p
Let σ (s) ∈ Lloc (R+ , E1 ), consider the quantity
ησ (h) = sup
t ∈R+
t +h
t
∥σ (s)∥pE1 ds.
Lemma 2.1. Let Σ be defined as before and σ0 ∈ E1 , then (1) σ0 is a translation compact in E1 and any σ ∈ Σ = H+ (σ0 ) is also a translation compact in E1 , moreover, H+ (σ ) ⊆ H+ (σ0 ); (2) the set H+ (σ0 ) is bounded in L2 (R+ , L2 (Ω )) such that
ησ (h) ≤ ησ0 (h) < +∞,
for all σ ∈ Σ .
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Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
Proof. See, e.g., Chepyzhov and Vishik [25].
Lemma 2.2. For every τ ∈ R, every non-negative locally summable function φ0 on Rτ ≡ [τ , +∞) and every ν > 0, we have t
sup t ≥τ
τ
φ0 (s)e−ν(t −s) ds ≤
t +1
1
sup 1 − e−ν t ≥τ
φ0 (s)ds
t
for a.a. t ≥ τ . Proof. See, e.g., Chepyzhov, Pata and Vishik [26].
Similar to Theorem 2.1, we have the following existence and uniqueness result. Theorem 2.2. Let Σ = H+ (σ0 ) = [σ0 (t + h)|h ∈ R+ ]E1 , where σ0 ∈ E1 is an arbitrary but fixed symbol function. Assume (2.1) and (2.2) hold. Then for any σ ∈ Σ and for any (uτ0 , uτ1 , ητ ) ∈ H (∀ τ ∈ R+ ), problem (1.14)–(1.17) admits a unique global solution (u, ut , ηt ) ∈ H , which generates a unique semi-process {Uσ (t , τ )}, (t ≥ τ ∈ R+ , σ ∈ Σ ) on H of a two-parameter family of operators such that for any t ≥ τ , τ ∈ R+ , Rτ = [τ , +∞), Uσ (t , τ )(uτ0 , uτ1 , ητ ) = (u, ut , ηt ) ∈ H , u ∈ L∞ (Rτ , H01 (Ω )),
(2.8)
ut ∈ L∞ (Rτ , H01 (Ω )),
utt ∈ L2 (Rτ , H01 (Ω )),
ηt ∈ L∞ (Rτ , M).
(2.9)
Now we give the following result concerning the uniform attractors. Theorem 2.3. Assume that σ ∈ E1 and Σ is defined by (2.7), then the family of processes {Uσ (t , τ )} (σ ∈ Σ , t ≥ τ , τ ∈ R+ ) corresponding to (1.14)–(1.17) has a uniformly (w.r.t. σ ∈ Σ ) compact attractor AΣ . 3. The well-posedness The global existence of solutions is the same as in [8,14,24], so we omit the details here. Next we prove the uniqueness of solutions. We consider two symbols σ1 and σ2 and the corresponding solutions (u, ηt ) and (v, ξ t ) of problem (1.14)–(1.17) with initial data (uτ0 , uτ1 , ητ ) and (v0τ , v1τ , ξ τ ) respectively. Let ω(t ) = u(t ) − v(t ) = Uσ1 (t , τ )uτ0 − Uσ2 (t , τ )v0τ , ζ t (x, s) = ηt (x, s) − ξ t (x, s). Then (ω, ζ t ) verifies
|ut |ρ ωtt + vtt (|ut |ρ − |vt |ρ ) − l1ω − 1ωtt −
+∞
g (s)1ζ (s)ds + ωt = σ1 − σ2 ,
x ∈ Ω, t > τ ,
(3.1)
0
ζtt + ζst = ωt ,
(3.2)
with Dirichlet boundary conditions and initial conditions
ω(x, τ ) = ω0τ ,
ωt (x, τ ) = ω1τ ,
ζ τ = ητ − ξ τ .
(3.3)
The corresponding energy for (3.1)–(3.3) is defined Eω (t ) =
1
2
Ω
l
1
1
2
2
2
|ut |ρ ωt2 dx + ∥∇ω∥2 + ∥∇ωt ∥2 + ∥ζ t ∥2M .
(3.4)
It is easy to see that +∞
|∇ζ t (s)|2 ds dx 2 Ω ds 0 +∞ 1 =− g ′ (s)|∇ζ t (s)|2 ds dx.
(ζst , ζ t )M =
1
2
g (s)
Ω
d
0
ρ
Note that x → |x| is differentiable since ρ > 1. Then 1 d 2 dt
ρ
Ω
|ut | ω
2 t dx
ρ
= Ω
|ut | ωtt ωt dx +
ρ 2
Ω
|ut |ρ−2 ut utt ωt2 dx,
and clearly d dt
Eω (t ) = −∥ωt ∥2 +
+
ρ 2
Ω
1 2
+∞
g ′ (s)∥∇ζ t (s)∥2 ds + 0
|ut |ρ−1 utt ωt2 dx −
Ω
Ω
(σ1 − σ2 )ωt dx
vtt ωt (|ut |ρ − |vt |ρ )dx.
(3.5)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
5
To simplify notations, let us say that the norm of the initial data is bounded by some R > 0. Then given T > τ we use CRT to denote several positive constants which depend on R and T . By Young’s inequality and the interpolation inequalities, we derive
(σ1 − σ2 )ωt dx ≤ ∥σ1 − σ2 ∥ ∥ωt ∥ Ω
≤ ∥σ1 − σ2 ∥2 + CRT Eω (t ), ρ ρ ρ−1 ρ−1 2 ∥ut ∥2(ρ+1) ∥utt ∥ ∥ωt ∥22(ρ+1) | ut | utt ωt dx ≤ 2 2
(3.6)
≤ CRT ∥∇ utt ∥ ∥∇ωt ∥2 , ρ ρ − vtt (|ut | −|vt | )ωt dx ≤ C1 |vtt |(|ut |ρ−1 + |vt |ρ−1 )ωt2 dx
(3.7)
Ω
Ω
Ω
ρ−1
ρ−1
≤ C1 ∥vtt ∥(∥ut ∥2(ρ+1) + ∥vt ∥2(ρ+1) )∥ωt ∥22(ρ+1) ≤ C1 ∥∇vtt ∥ ∥∇ωt ∥2 ,
which, together with (3.5)–(3.7), yields for some C1 > 0 large d Eω (t ) ≤ ∥σ1 − σ2 ∥2 + C1 (1 + ∥∇ utt ∥ + ∥∇vtt ∥)Eω (t ). dt Integrating (3.8) from τ to t and using Hölder’s inequality, we have Eω (t ) ≤ Eω (τ ) +
≤ Eω (τ ) +
t
∥σ1 (s) − σ2 (s)∥2 ds + C1
τ
∥σ1 (s) − σ2 (s)∥ ds + C1 2
τ
t
T
(3.8)
τ
(1 + ∥∇ utt ∥ + ∥∇vtt ∥)Eω (s)ds t
τ
21
(1 + ∥∇ utt ∥ + ∥∇vtt ∥) ds 2
t
Eω (s)ds 2
τ
12
.
(3.9)
Note that T
(1 + ∥∇ utt ∥ + ∥∇vtt ∥)2 ds ≤ CRT ,
τ
then we get for any t ∈ [τ , T ]
Eω (t ) ≤ 2 Eω (τ ) + 2
T
∥σ1 (s) − σ2 (s)∥ ds 2
τ
2
t
+ CRT
τ
Eω2 (s)ds.
(3.10)
Applying Gronwall’s inequality, we see that Eω (t ) ≤ Using
Ω
√
2 Eω (τ ) +
T
τ
CRT ∥σ1 (s) − σ2 (s)∥2 ds exp T , 2
∀ t ∈ [τ , T ].
(3.11)
ρ
|ut |ρ |ωt |2 dx ≤ ∥ut ∥ρ+2 ∥ωt ∥2ρ+2 ≤ CRT ∥∇ωt ∥2 , we know that Eω (t ) is equivalent to the norm of u in H and we get
Eω (τ ) ≤ CRT ∥(ω0τ , ω1τ , ζ τ )∥2H , which, together with (3.11), gives for all τ ≤ t ≤ T
∥u(t ) − v(t )∥2H 1 + ∥ut (t ) − vt (t )∥2H 1 + ∥ηt − ξ t ∥2M 0
0
≤ CRT (∥uτ0 − v0τ ∥2H 1 + ∥uτ1 − v1τ ∥2H 1 + ∥ητ − ξ τ ∥2M + ∥σ1 − σ2 ∥2L2 (τ ,T ;L2 (Ω )) ). 0
0
This shows that solutions of (1.14)–(1.17) depend continuously on the initial data. We complete the proof of Theorem 2.1. 4. Uniform attractors In this section, we shall prove Theorem 2.3. To this end, we shall introduce some basic conceptions and basic lemmas. For more results concerning uniform attractors, we can refer to [15,20,21,27,28]. be a parameter set. The operators {Uσ (t , τ )}(t ≥ τ , τ ∈ R+ , σ ∈ Σ ) are said to be a Let X be a Banach space, and Σ if for any σ ∈ Σ , family of processes in X with symbol space Σ Uσ (t , s)Uσ (s, τ ) = Uσ (t , τ ), Uσ (τ , τ ) = Id (identity),
∀ t ≥ s ≥ τ , τ ∈ R+ ,
∀τ ∈R . +
(4.1) (4.2)
6
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
, we say that a family of processes {Uσ (t , τ )} (t ≥ τ , τ ∈ R+ , σ ∈ Σ ) satisfies Let {T (s)} be the translation semigroup on Σ the translation identity if Uσ (t + s, τ + s) = UT (s)σ (t , τ ),
=Σ , T (s)Σ
, t ≥ τ , τ , s ∈ R+ , ∀σ ∈Σ
(4.3)
∀s∈R . +
(4.4)
By B(X ) we denote the collection of the bounded sets of X , and Rτ = [τ , +∞), τ ∈ R . +
) absorbing set for {Uσ (t , τ )} (σ ∈ Definition 4.1. A bounded set B0 ∈ B(X ) is said to be a bounded uniformly (w.r.t. σ ∈ Σ , t ≥ τ , τ ∈ R+ ) if for any τ ∈ R+ and B ∈ B(X ), there exists a time T0 = T0 (B, τ ) ≥ τ such that Σ
Uσ (t , τ )B ⊆ B0 ,
(4.5)
σ ∈Σ
for all t ≥ T0 . In the following, as usual, (w.r.t.) will represent ‘‘with respect to’’.
) attracting for the family of semi-processes {Uσ (t , τ )} (t ≥ Definition 4.2. A set A ⊆ X is said to be uniformly (w.r.t. σ ∈ Σ ) if for any fixed τ ∈ R+ and any B ∈ B(X ), τ , τ ∈ R+ , σ ∈ Σ lim (sup dist(Uσ (t , τ )B, A)) = 0,
(4.6)
t →+∞
here dist(·, ·) stands for the usual Hausdorff semidistance between two sets in X . In particular, a closed uniformly attracting ) attractor of the family of the semi-process {Uσ (t , τ )} (t ≥ τ , τ ∈ R+ , σ ∈ Σ ) set AΣ is said to be the uniform (w.r.t. σ ∈ Σ if it is contained in any closed uniformly attracting set (minimality property).
be a symbol (or parameter) space. We call a Definition 4.3. Let X be a Banach space and B be a bounded subset of X , Σ ×Σ ) to be a contractive function on B × B if for any sequence {xn }∞ function φ(·, ·; ·, ·), defined on (X × X ) × (Σ n =1 ⊆ B ∞ ∞ ∞ , there is a subsequence {xnk }∞ and any {σn } ⊆ Σ k=1 ⊂ {xn }n=1 and {σnk }k=1 ⊂ {σn }n=1 such that lim lim φ(xnk , xnl ; σnk , σnl ) = 0.
(4.7)
k→∞ l→∞
). We denote the set of all contractive functions on B × B by Contr(B, Σ ) be a family of semi-processes satisfying the translation identities (4.3) and Lemma 4.1. Let {Uσ (t , τ )} (t ≥ τ , τ ∈ R+ , σ ∈ Σ ) absorbing set B0 ⊆ X . Moreover, assuming that for any (4.4) on Banach space X and has a bounded uniformly (w.r.t. σ ∈ Σ ) such that ε > 0, there exist T = T (B0 , ε) > 0 and φT ∈ Contr(B0 , Σ ∥Uσ1 (T , 0)x − Uσ2 (T , 0)y∥ ≤ ε + φT (x, y; σ1 , σ2 ),
, t ≥ τ , τ ∈ R+ . ∀σ ∈Σ
(4.8)
) is uniformly (w.r.t. σ ∈ Σ ) asymptotically compact in X . Then {Uσ (t , τ )} (t ≥ τ , τ ∈ R , σ ∈ Σ +
Proof. This lemma is a version for semi-processes of a result by Khanmamedov [29]. A proof can be found in Sun et al. [27, Theorem 4.2]. Next, we will divide into two subsections to prove Theorem 2.3. 4.1. Uniformly (w.r.t. σ ∈ Σ ) absorbing set in H In this subsection we shall establish the family of processes {Uσ (t , τ )} has a bounded uniformly absorbing set given in the following theorem. Theorem 4.1. Assume that σ ∈ E1 and Σ is defined by (2.7), then the family of processes {Uσ (t , τ )} (σ ∈ Σ , t ≥ τ , τ ∈ R+ ) corresponding to (1.14)–(1.17) has a bounded uniformly (w.r.t. σ ∈ Σ ) absorbing set B in H . Proof. We define F (t ) =
1
ρ+2
l
1
1
2
2
2
ρ+2
∥ut ∥ρ+2 + ∥∇ u∥2 + ∥∇ ut ∥2 + ∥ηt ∥2M .
(4.9)
Multiplying (1.14) by ut , integrating the resulting equation over Ω and Young’s inequality, we derive F ′ (t ) = −∥ut ∥2 − (ηst , ηt )M + (σ , ut ) 1
1
2
2
≤ − ∥ ut ∥ 2 +
0
+∞
g ′ (s)∥∇ηt (s)∥2 ds +
1 2
∥σ ∥2 .
(4.10)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
7
Let 1
F1 (t ) = F (t ) +
+∞
2
∥σ (s)∥2 ds,
for all t ≥ τ .
(4.11)
t
Then (4.11) gives F1′ (t ) ≤ 0, whence from (4.9), for t ≥ τ > 0 F (t ) ≤ F1 (t ) ≤ F1 (τ ) = F (τ ) + l 2
1
1
2
2
1
+∞
1
∥σ (s)∥2 ds = F (τ ) + ∥σ ∥2L2 (Rτ ,L2 (Ω )) ,
2 τ
(4.12)
2
∥∇ u∥2 + ∥∇ ut ∥2 + ∥ηt ∥2M ≤ F (t ) ≤ F1 (t ) ≤ F1 (τ ).
(4.13)
Now we define
Φ (t ) =
1
ρ+1
Ω
|ut |ρ ut udx +
Ω
∇ ut · ∇ udx.
(4.14)
From (1.14), integration by parts and Young’s inequality, we derive for any ε ∈ (0, 1),
Φ ′ (t ) =
l1u +
+∞
g (s)1ηt (s)ds − ut + σ , u
1
+
ρ+1
0
= −l∥∇ u∥ − ∇ u, 2
+∞
g (s)∇η (s)ds t
(|ut |ρ ut , ut ) − (1ut , ut )
− (ut , u) +
0
1
ρ+2
ρ+1
∥ut ∥ρ+2 + ∥∇ ut ∥2 + (σ , u).
(4.15)
Using Young’s inequality, Hölder’s inequality and Poincaré’s inequality, we deduce
| − (ut , u)| ≤ ε∥u∥2 + − ∇ u,
1 4ε
+∞
g (s)∇ηt (s)ds ≤ ε∥∇ u∥2 +
0
≤ ε∥∇ u∥2 + |(σ , u)| ≤ ε∥u∥2 +
1 4ε
1
∥ut ∥2 ≤ ελ2 ∥∇ u∥2 +
∥σ ∥2 ≤ ελ2 ∥∇ u∥2 +
4ε
1
∥ ut ∥ 2 ,
1 4ε
2
+∞
4ε Ω 0 1 − l +∞ 4ε
(4.16) g (s)∇ηt (s)ds
dx
g (s)∥∇ηt (s)∥2 ds,
(4.17)
0
∥σ ∥2 ,
(4.18)
hereinafter we use λ to represent the Poincaré constant. From the expression of F (t ), we get
∥∇ u∥2 =
2 l
F (t ) −
2
ρ+2
l(ρ + 2)
1
1
l
l
∥ut ∥ρ+2 − ∥∇ ut ∥2 − ∥ηt ∥2M ,
which, together with (4.15)–(4.18), yields 1
Φ ′ (t ) ≤ − (l − 2ελ2 − ε)∥∇ u∥2 − 2
1
2
4ε
4ε
∥ηt ∥2M +
2 (l − 2ελ2 − ε) F (t ) − l
1
2 l(ρ + 2)
ρ+2 ∥ut ∥ρ+2
ρ+2
1
1
l
l
∥ut ∥ρ+2 − ∥∇ ut ∥2 − ∥ηt ∥2M
ρ+1
1
+ ∥∇ ut ∥2 + ∥σ ∥2 4η 1 l − 2ελ2 − ε 1 ρ+2 = − (l − 2ελ2 − ε)∥∇ u∥2 − (l − 2ελ2 − ε)F (t ) + + ∥ut ∥ρ+2 2 l(ρ + 2) ρ+1 1 1 1 1−l 1 2 2 2 + 1 + (l − 2ελ − ε) ∥∇ ut ∥ + ∥ηt ∥2M + ∥σ ∥2 + ∥ut ∥2 . (l − 2ελ − ε) + 2 2l 4ε 4ε 4ε +
∥ ut ∥ 2 +
1−l
1
(4.19)
Noting that ∥∇ ut ∥2 ≤ 2F (t ) ≤ 2F1 (τ ) and the embedding theorem H 1 (Ω ) ↩→ L2(ρ+1) (Ω ), we have for any ε ∈ (0, 1),
1 − 2ελ2 − ε l(ρ + 2)
+
1
ρ+1
ρ+2 ρ+1 ∥ut ∥ρ+2 ≤ C1 ∥ut ∥ ∥ut ∥2(ρ+1) ≤ C1 ∥∇ ut ∥ρ+1 ∥ut ∥ ≤ ε∥∇ ut ∥2(ρ+1) + Cε ∥ut ∥2 ρ ≤ 2ρ+1 ε F1 (τ )F (t ) + Cε ∥ut ∥2 ,
8
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
which, together with (4.19) and Poincaré’s inequality, gives 1
Φ ′ (t ) ≤ − (l − 2ελ2 − ε)∥∇ u∥2 −
l − 2ελ2 − ε
2
l
λ + ∥∇ ut ∥2 + 4ε
l − 2ελ − ε
2
2
+ 1 + Cε λ + 2
ρ − 2ρ+1 εF1 (τ ) F (t )
2l
l − 2ελ2 − ε 2l
+
1−l 4ε
∥ηt ∥2M +
1 4η
∥σ ∥2 .
(4.20)
Now we take ε ∈ (0, 1) so small that l − 2ελ2 − ε ≥
l 2
,
l − 2ελ2 − ε l
ρ
− 2ρ+1 ε F1 (τ ) ≥
1 4
.
(4.21)
Hence from (4.20)–(4.21), it follows l
1
4
4
Φ ′ (t ) ≤ − ∥∇ u∥2 −
1
F (t ) + C1 ∥ηt ∥2M + C1 ∥∇ ut ∥2 +
4ε
∥σ ∥2 .
(4.22)
We define the functional
Ψ (t ) =
Ω
1ut −
1
ρ+1
|ut |ρ ut
+∞
g (s)ηt (s)dsdx.
(4.23)
0
It follows from (1.14) that
Ψ (t ) = ′
+∞
−l1u − Ω + 1ut − Ω
g (s)1η (s)ds + ut − σ t
+∞
0
g (s)ηt (s)ds
0
1
ρ+1
+∞
ρ
g (s)ηtt (s)ds
|ut | ut 0
:= I1 + I2 .
(4.24)
From Young’s inequality, Hölder’s inequality and Poincaré’s inequality, we derive for any δ ∈ (0, 1),
+∞ 2 t (−l1u) ≤ δ∥∇ u∥2 + l (1 − l) ∥ηt ∥2 , g ( s )η ( s ) ds M 4δ Ω 0 +∞ 2 g (s)1ηt (s)ds ds ≤ (1 − l)∥ηt ∥2M , − Ω 0 +∞ 2 t ut · ≤ δ∥ut ∥2 + λ (1 − l) ∥ηt ∥2 , g ( s )η ( s ) ds M 4δ Ω 0 +∞ 1 1−l t 2 (−σ ) g (s)ηt (s)ds ≤ ∥σ ∥2 + ∥η ∥M , Ω
2
0
(4.25)
(4.26)
(4.27)
2
which, together with (4.25)–(4.27), gives 1
I1 ≤ δ(∥ut ∥2 + ∥∇ u∥2 ) +
2
∥σ ∥2 +
3(1 − l) 2
+
(λ2 + l2 )(1 − l) ∥ηt ∥2M . 4δ
(4.28)
Note that +∞
g (s)ηtt (s)ds = −
+∞
0
g (s)ηst (s)ds +
+∞
0
g (s)ut (t )ds 0
+∞
g ′ (s)ηt (s)ds + (1 − l)ut ,
= 0
then we have I2 = −(1 − l)∥∇ ut ∥2 −
+
1
ρ+1
Ω
|ut |ρ ut
1−l
ρ+1 +∞ 0
ρ+2
∥ut ∥ρ+2 +
+∞
g ′ (s) 0
g ′ (s)ηt (s)dsdx.
Ω
1ut (t )ηt (s)dxds (4.29)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
9
By Young’s inequality, we derive +∞
g (s) ′
Ω
0
1ut (t )η (s)dxds ≤ − t
+∞
g ′ (s)∥∇ ut (t )∥ ∥∇ηt (s)∥ds
0
1−l
≤
4
∥∇ ut (t )∥2 −
+∞
1 1−l
g ′ (s)∥∇ηt (s)∥2 ds,
(4.30)
0
and for any ε > 0 1
ρ+1
Ω
+∞
|ut |ρ ut
g ′ (s)ηt (s)dsdx ≤ ε∥∇ ut ∥2 − Cε
+∞
g ′ (s)∥∇ηt (s)∥2 ds, 0
0
which, together with (4.29)–(4.30) and taking ε > 0 small enough, yields I2 ≤ −
1−l 2
∥∇ ut ∥2 −
1−l
ρ+1
+∞
ρ+2
∥ut ∥ρ+2 − C1
g ′ (s)∥∇ηt (s)∥2 ds.
(4.31)
0
Inserting (4.28) and (4.31) into (4.24), we arrive at 1−l
Ψ ′ (t ) ≤ −
1
∥∇ ut ∥2 + δ(∥∇ u∥2 + ∥ut ∥2 ) + C1 ∥ηt ∥2M + ∥σ ∥2
4
2
+∞
g ′ (s)∥∇ηt (s)∥2 ds −
− C1 0
1−l
ρ+2
ρ+1
∥ut ∥ρ+2 .
(4.32)
Set G(t ) = MF (t ) + ϵ Φ (t ) + Ψ (t ),
(4.33)
where M and ϵ are positive constants. Then it follows from (4.10), (4.22), (4.32) and (2.2) that G′ (t ) ≤ −
lϵ ϵ 1−l − δ ∥ ut ∥ 2 − − δ ∥∇ u∥2 − F (t ) − − C1 ϵ ∥∇ ut ∥2
M 2
4
M
+
2
− C1 − C1 ξ2
4
4
+∞
g (s)∥∇η (s)∥ ds + ′
t
2
ϵ 1 1−l ρ+2 + + ∥σ ∥2 − ∥ut ∥ρ+2 . 2 4ε 2 ρ+1
M
0
(4.34)
Now we claim that there exist two constants β1 , β2 > 0 such that
β1 F (t ) ≤ G(t ) ≤ β2 F (t ),
t ≥ 0.
(4.35)
In fact,
|Φ (t )| ≤ ≤
1
ρ+1
ρ+1
1
l
2
2
∥ut ∥2(ρ+1) ∥u∥ + ∥∇ ut ∥2 + ∥∇ u∥2
1
1
2(ρ+1)
2(ρ + 1)
∥ut ∥2(ρ+1) +
ρ
1
2(ρ + 1)
≤ C1 F (τ ) ∥∇ ut ∥ + C1 ∥∇ u∥ + 2
2
l
∥u∥2 + ∥∇ ut ∥2 + ∥∇ u∥2 2
1 2(ρ + 1)
2
2
1
l
2
2
∥u∥ + ∥∇ ut ∥2 + ∥∇ u∥2
≤ C0 F (t ).
(4.36)
Using Hölder’s inequality, Young’s inequality and Poincaré’s inequality, we get
+∞
g ( s) 0
1 1 1ut η (s)dxds ≤ ∥∇ ut ∥2 + ∥ηt ∥M , 2 2 t
Ω
and
ρ+1
+∞
1
g (s) 0
|ut (t )| ut (t )η (s)dxds ≤ C1 F (τ )ρ ∥∇ ut ∥2 + C1 ∥∇ηt (s)∥2 , ρ
Ω
which, along with (4.37), gives
|Ψ (t )| ≤ C1 F (t ). It follows that for some constant C2 > 0
|ϵ Φ (t ) + Ψ (t )| ≤ C2 F (t ).
t
(4.37)
10
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
Then taking M > C2 we get (4.35) with β1 = M − C2 and β2 = M + C2 . For any t ≥ τ , we take ε so small that 1−l 4
− C1 ϵ > 0.
(4.38)
For fixed ε , we choose δ small enough and M so large that M 2
lϵ
− δ > 0,
4
M
− δ > 0,
− C1 − C1 ξ 2 > 0 .
2
Then there exist a constant γ > 0 such that G′ (t ) ≤ −γ F (t ) + C1 ∥σ ∥2 ,
(4.39)
which, together with (4.35), gives G′ (t ) ≤ −
γ G(t ) + C1 ∥σ ∥2 . β2
(4.40)
Integrating (4.40) over [τ , t ] with respect to t and using Lemmas 2.1–2.2, we obtain − βγ (t −τ )
G(t ) ≤ G(τ )e
≤ CB0 e ≤ CB0
2
− βγ (t −τ )
t
+ C1
− γ (t −τ ) e β2
2
τ
1
+ C1
2
− βγ (t −s)
e
1−e 1
+ C1
1−e
− βγ
2
− βγ
∥σ (s)∥2 ds t +1 sup ∥σ (s)∥2 ds t ≥τ
t
ησ (1).
(4.41)
2
Now for any bounded set B0 ⊆ H , for any (uτ0 , uτ1 , ητ ) ∈ B0 , there exists a constant CB0 > 0 such that F (τ ) ≤ CB0 ≤ C1 . Take
R20
= 2 2C1
1−e t0 = τ −
ησ0 (1)
γ β2
− βγ
+1 ,
2
−1
C1 ησ0 (1) + 1
log
C B0 1 − e
− βγ
2
,
then for any t ≥ t0 ≥ 1, we have − βγ (t −τ )
G(t ) ≤ CB0 e
2
1
+ C1 1−
−γ e β2
ησ0 (1) ≤
R20 2
,
which gives
∥(u, ut , ηt )∥H ≤ 2G(t ) = R20 , i.e., B = B(0, R0 ) = {(u, ut , ηt ) ∈ H : ∥(u, ut , ηt )∥2H ≤ R20 } ⊆ H is a uniform absorbing ball for any σ ∈ E1 . The proof is now complete. 4.2. Uniformly (w.r.t. σ ∈ Σ ) asymptotic compactness in H In this subsection, we will prove the uniformly (w.r.t. σ ∈ Σ ) asymptotic compactness in H , which is given in the following theorem. Theorem 4.2. Assume that σ ∈ E1 and Σ is defined by (2.7), then the family of processes {Uσ (t , τ )} (σ ∈ Σ , t ≥ τ , τ ∈ R+ ) corresponding to (1.14)–(1.17) is uniformly (w.r.t. σ ∈ Σ ) asymptotically compact in H . Proof. For any (uτ0i , uτ1i , ηiτ ) ∈ B, i = 1, 2, we consider two symbols σ1 and σ2 and the corresponding solutions u1 and u2 of problem (1.14)–(1.17) with initial data (uτ0i , uτ1i , ηiτ ), i = 1, 2, respectively. Let ω(t ) = u1 (t ) − u2 (t ) = Uσ1 (t , τ )uτ01 − Uσ2 (t , τ )uτ02 , ζ t (x, s) = η1t (x, s) − η2t (x, s). Then (ω, ζ t ) verifies ρ
ρ
ρ
|u1t | ωtt + u2tt (|u1t | − |u2t | ) − l1ω − 1ωtt −
+∞
g (s)1ζ (s)ds + ωt = σ1 − σ2 ,
x ∈ Ω, t > τ ,
(4.42)
0
ζtt + ζst = ωt ,
(4.43)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
11
with Dirichlet boundary conditions and initial conditions
ω(x, τ ) = ω0τ ,
ωt (x, τ ) = ω1τ ,
ζ τ = η1τ − η2τ .
(4.44)
The corresponding energy for (4.42)–(4.44) is defined 1
Eω (t ) =
2
l
1
1
2
2
2
|u1t |ρ ωt2 dx + ∥∇ω∥2 + ∥∇ωt ∥2 + ∥ζ t ∥2M .
Ω
(4.45)
Clearly, d dt
Eω (t ) = −∥ωt ∥ + 2
+
ρ 2
Ω
1
+∞
2
g (s)∥∇ζ (s)∥ ds + ′
t
2
Ω
0
|u1t |ρ−1 u1tt ωt2 dx −
Ω
(σ1 − σ2 )ωt dx
u2tt ωt (|u1t |ρ − |u2t |ρ )dx.
(4.46)
Using Hölder’s inequality, Poincaré’s inequality and Theorem 4.1, we derive
(σ1 − σ2 )ωt dx ≤ ∥σ1 − σ2 ∥ ∥ωt ∥, Ω ρ ρ ρ−1 ρ−1 2 u1tt ωt dx ≤ ∥u1t ∥2(ρ+1) ∥u1tt ∥ ∥ωt ∥2(ρ+1) ∥ωt ∥ |u1t | 2
(4.47)
2
Ω
≤ CB ∥∇ u1t ∥ρ−1 ∥∇ωt ∥ ∥∇ u1tt ∥ ∥ωt ∥ ≤ CB ∥ωt ∥ ∥∇ u1tt ∥, ρ ρ ρ − ≤ C1 ∥u2tt ∥2(ρ+1) (∥u1t ∥ρ | | u (| u −| u )ω dx 2tt 1t 2t t 2(ρ+1) + ∥u2t ∥2(ρ+1) )∥ωt ∥ Ω ≤ C1 ∥∇ u2tt ∥(∥∇ u1t ∥ρ + ∥∇ u2t ∥ρ )∥ωt ∥ ≤ CB ∥∇ u2tt ∥ ∥ωt ∥,
(4.48)
which, combined with (4.46)–(4.48), yields d dt
Eω (t ) ≤ −∥ωt ∥2 +
1 2
+∞
g ′ (s)∥∇ζ t (s)∥2 ds + ∥ωt ∥ ∥σ1 − σ2 ∥ + CB0 (∥∇ u1tt ∥ + ∥∇ u2tt ∥)∥ωt ∥.
(4.49)
0
We define
Φω (t ) =
|u1t |ρ ωt ωdx +
Ω
Ω
∇ωt · ∇ωdx +
1 2
Ω
ω2 dx.
(4.50)
It is very easy to verify
|Φω (t )| ≤
1 2
1
ρ
(∥∇ω∥2 + ∥∇ωt ∥2 ) + ∥u1t ∥2(ρ+1) ∥ωt ∥2(ρ+1) ∥ω∥ + ∥ω∥2 2
≤ CB (∥∇ω∥2 + ∥∇ωt ∥2 ) ≤ CB Eω (t ).
(4.51)
Taking the derivative of Φω (t ), it follows from (4.42) that
Φω′ (t ) = −
Ω
+ Ω
=
4
u2tt (|u1t |ρ − |u2t |ρ )ωdx − l∥∇ω∥2 −
(σ1 − σ2 )ωdx +
Ω
+∞
∇ω Ω
g (s)∇ζ t (s)dsdx
0
(ρ|u1t |ρ−1 u1tt ωt ω + |u1t |ρ ωt2 )dx + ∥∇ωt ∥2
Ai − l∥∇ω∥2 + ∥∇ωt ∥2 .
(4.52)
i=1
Applying Hölder’s inequality, Young’s inequality, Poincaré’s inequality and Theorem 4.1, we get ρ−1
ρ−1
|A1 | ≤ C1 ∥u2tt ∥2(ρ+1) (∥u1t ∥2(ρ+1) + ∥u2t ∥2(ρ+1) )∥ωt ∥2(ρ+1) ∥ω∥ ≤ C1 ∥∇ u2tt ∥(∥∇ u1t ∥ρ−1 + ∥∇ u2t ∥ρ−1 )∥∇ωt ∥ ∥ω∥ ≤ CB ∥∇ u2tt ∥ ∥ω∥,
(4.53)
12
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
+∞ t g (s)∇ζ (s)ds 0 1−l t 2 ≤ ε∥∇ω∥2 + ∥ζ ∥ , ∀ ε ∈ (0, 1), 4ε |A3 | ≤ ∥σ1 − σ2 ∥ ∥ω∥,
|A2 | ≤ ∥∇ω∥
(4.54) (4.55)
ρ−1 C1 ∥u1tt ∥2(ρ+1) ∥u1t ∥2(ρ+1) ∥ωt ∥2(ρ+1) ∥ω∥
|A4 | ≤ ≤ CB ∥∇ u1tt ∥ ∥ω∥ + CB ∥∇ωt ∥2 .
ρ C1 ∥u1t ∥ρ+2 ∥ωt ∥2ρ+2
+
(4.56)
By virtue of (4.45), we have 2
∥∇ω∥2 =
Eω (t ) −
l
1
l
Ω
1
1
l
l
|u1t |ρ ωt2 dx − ∥∇ωt ∥2 − ∥ζ t ∥2M .
(4.57)
Then from (4.52)–(4.57), we can conclude
Φω′ (t ) ≤ −
l−ε 2
∥∇ω∥2 −
l−ε
2
2
l
1
Eω (t ) −
l
1
Ω
1
|u1t |ρ ωt2 dx − ∥∇ωt2 ∥ − ∥ζ t ∥2M l
l
+ CB ∥∇ωt ∥2 + CB ∥ω∥(∥∇ u1tt ∥ + ∥∇ u2tt ∥) + C1 ∥σ1 − σ2 ∥ ∥ω∥ + ≤−
l−ε 2
1−l 4ε
∥ζ t ∥2M
∥∇ω∥2 − (l − ε)Eω (t ) + Cε ∥∇ωt ∥2 + Cε ∥ζ t ∥2M
+ C1 ∥σ1 − σ2 ∥ ∥ω∥ + CB ∥ω∥(∥∇ u1tt ∥ + ∥∇ u2tt ∥).
(4.58)
Now we define
Ψω (t ) =
ρ
Ω
(1ωt − |u1t | ωt )
+∞
g (s)ζ (s)ds dx. t
(4.59)
0
From (4.42) and integration by parts, we derive
Ψω′ (t ) =
u2tt (|u1t |ρ − |u2t |ρ )
Ω
+∞
g (s)ζ t (s)dsdx + l
Ω
0
+∞
g (s)ζ t (s)ds
+ Ω
2
dx + Ω
0
ωt
+∞
g (s)ζ t (s)dsdx
∇ω 0
+∞
g (s)ζ t (s)dsdx 0
+∞ +∞ (σ1 − σ2 ) g (s)ζ t (s)dsdx − ρ |u1t |ρ−1 u1tt ωt g (s)ζ t (s)dsdx Ω 0 Ω 0 +∞ +∞ + 1ωt g (s)ζtt (s)dsdx − |u1t |ρ ωt g (s)ζtt (s)dsdx
−
Ω
=
8
Ω
0
0
Bi .
(4.60)
i=1
Using Hölder’s inequality, Poincaré’s inequality and Theorem 4.1, we derive for any δ ∈ (0, 1), ρ−1
ρ−1
B1 ≤ ∥u2tt ∥2(ρ+1) (∥u1t ∥2(ρ+1) + ∥u2t ∥2(ρ+1) )∥ωt ∥
≤ CB ∥∇ u2tt ∥ ∥ωt ∥
+∞ 0
0
+∞
g (s)ζ t (s)ds
2(ρ+1)
t g (s)∇ζ (s)ds
1
≤ CB (1 − l) 2 ∥∇ u2tt ∥ ∥ωt ∥ ∥ζ t ∥M 1
≤ CB (1 − l) 2 ∥∇ u2tt ∥ ∥ωt ∥(∥η1t ∥M + ∥η2t ∥M ) ≤ CB′ ∥∇ u2tt ∥ ∥ωt ∥, B2 ≤ δ∥∇ω∥2 +
(1 − l)l ∥ζ t ∥2M 4δ
B3 ≤ (1 − l)∥ζ t ∥2M , B4 ≤ δ∥ωt ∥2 +
λ
(4.61)
2
2
4δ
(1 − l)∥ζ t ∥2M ,
(4.62) (4.63) (4.64)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
13
1
B5 ≤ λ(1 − l) 2 ∥σ1 − σ2 ∥ ∥ζ t ∥M ≤ C1 ∥σ1 − σ2 ∥2 + C1 ∥ζ t ∥2M
(4.65)
+∞ ρ−1 t B6 ≤ ∥u1t ∥2(ρ+1) ∥u1tt ∥2(ρ+1) ∥ωt ∥ g ( s )ζ ( s ) ds 0 2(ρ+1) +∞ t ≤ CB ∥∇ u1tt ∥ ∥ωt ∥ g (s)∇ζ (s)ds 0
1
≤ CB (1 − l) 2 ∥∇ u1tt ∥ ∥ωt ∥ ∥ζ t ∥M ≤ CB′ ∥∇ u1tt ∥ ∥ωt ∥.
(4.66)
Note that +∞
g (s)ζtt (s)ds =
+∞
g (s)ds · ωt −
g (s)ζst (s)ds 0
0
0
+∞
= (1 − l)ωt +
+∞
g ′ (s)ζ t (s)ds, 0
then we see that B7 ≤ −(1 − l)∥∇ωt ∥2 −
+∞
g ′ (s)∥∇ωt (t )∥ ∥∇ζ t (s)∥ds 0
≤ −
1−l 2
∥∇ωt ∥2 −
1
2(1 − l)
+∞
g ′ (s)∥∇ζ t (s)∥2 ds,
(4.67)
0
+∞ ρ ′ t |ut |ρ ωt2 dx + ∥u1t ∥2(ρ+1) ∥ωt ∥2(ρ+1) g ( s )ζ ( s ) ds 0 Ω +∞ 2 C1 λ g (0) ≤ −(1 − l) |ut |ρ ωt2 dx + δ∥∇ωt ∥2 − g ′ (s)∥∇ζ t (s)∥2 ds. 4δ Ω 0
B8 ≤ −(1 − l)
(4.68)
Plugging (4.61)–(4.68) into (4.60), we get
Ψω′ (t ) ≤ CB′ (∥∇ u1tt ∥ + ∥∇ u2tt ∥)∥ωt ∥ + δ∥∇ω∥2 + C1 ∥ζ t ∥2M + δ∥ωt ∥2 + C1 ∥σ1 − σ2 ∥2 − +∞
g ′ (s)∥∇ζ t (s)∥2 ds − (1 − l)
− C1
0
Ω
1−l 2
|u1t |ρ ωt2 dx.
− δ ∥∇ωt ∥2 (4.69)
On the other hand, we can get
1 ρ |Ψω (t )| ≤ (1 − l) 2 ∥∇ωt ∥ ∥ζ t ∥M + ∥u1t ∥ρ+2 ∥ωt ∥ρ+2 1 2
1 2
+∞ 0
g (s)ζ t (s)ds
ρ
≤ (1 − l) ∥∇ωt ∥ ∥ζ ∥M + C1 (1 − l) λ∥∇ u1t ∥ ∥∇ωt ∥ ∥ζ t ∥M t
≤ CB (∥∇ωt ∥2 + ∥ζ t ∥2M ) ≤ CB Eω (t ).
(4.70)
Define Gω (t ) = MEω (t ) + ϵ Φω (t ) + Ψω (t ),
(4.71)
which, together with (4.51) and (4.70), yields
(M − CB ϵ − CB )Eω (t ) ≤ Gω (t ) ≤ (M + CB ϵ + CB )Eω (t ).
(4.72)
Now we take ε > 0 so small and M so large that M 2
Eω (t ) ≤ Gω (t ) ≤ 2MEω (t ).
(4.73)
Then for any t ≥ τ , we have G′ω (t ) ≤ −(l − ε)ϵ Eω (t ) − (M − δ)∥ωt ∥2 −
−
1−l 2
M 2
l−ε ξ2 − C1 ξ2 − C1 ∥ζ t ∥2M − − δ ∥∇ω∥2 2
− δ − Cε ϵ ∥∇ωt ∥2 + C (B, M , ε)(∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥)
+ C (B, M , ε)(∥ωt ∥ + ∥ω∥)∥σ1 − σ2 ∥ + C1 ∥σ1 − σ2 ∥2 .
(4.74)
14
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
Now we take δ > 0 and ϵ > 0 so small that 1−l
− δ − Cε ϵ > 0,
2
l−ε 2
− δ > 0.
For fixed ϵ and δ , we choose M so large that M
ξ2 − C1 ξ2 − C1 > 0,
2
M − δ > 0.
Then there exist some constant β > 0 such that G′ω (t ) ≤ −β Eω (t ) + C1 (∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥) + C1 ∥σ1 − σ2 ∥2 + C1 ∥σ1 − σ2 ∥(∥ωt ∥ + ∥ω∥)
≤−
β 2M
Gω (t ) + C1 (∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥) + C1 ∥σ1 − σ2 ∥2 + C1 ∥σ1 − σ2 ∥(∥ωt ∥ + ∥ω∥). (4.75)
Integrating (4.75) over (τ , t ) with respect to t, we derive β
Gω (t ) ≤ Gω (τ )e− 2M (t −τ ) + C1 t
+ C1
β
− 2M (t −s)
e τ
≤ Gω (τ )e− 2M (t −τ ) + C1 + C1
τ
β
e− 2M (t −s) ∥σ1 − σ2 ∥2 ds
(∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥)ds + C1
β
t
t
t
τ
(∥ωt ∥2 + ∥ω∥2 )ds
21
∥σ1 − σ2 ∥ ds 2
τ
21
t
τ
12
(∥ωt ∥ + ∥ω∥ )ds 2
2
τ
β
e− 2M (t −s) ∥σ1 − σ2 ∥(∥ωt ∥ + ∥ω∥)ds
t
+ C1
t
τ
∥σ1 − σ2 ∥2 ds
.
(4.76)
For any fixed ε ∈ (0, 1), we choose T > τ so large that β
Gω (τ )e− 2M (T −τ ) ≤ ε, which, together with (4.73) and (4.76), gives Eω (t ) ≤ ε + C1 t
C1
τ
t
τ
21
(∥ωt ∥ + ∥ω∥ )ds 2
2
+ C1
τ
t
∥σ1 − σ2 ∥2 ds
21 12 t (∥ωt ∥2 + ∥ω∥2 )ds := ε + φT ((uτ01 , uτ11 , η1τ ), (uτ02 , uτ12 , η2τ ); σ1 , σ2 ). ∥σ1 − σ2 ∥2 ds
(4.77)
τ
It suffices to show φT (·, ·, ·, ·) ∈ Contr(B, Σ ) for each fixed T > τ . From the proof of existence theorem, we can deduce that for any fixed T > τ , and the bound B depends on T ,
Uσ (t , τ )B
(4.78)
σ ∈Σ t ∈[τ ,T ]
is bounded in H . Let (un , unt , ηnt ) be the solutions corresponding to initial data (uτ0n , uτ1n , ηnτ ) ∈ B with respect to symbol σn ∈ Σ , n = 1, 2, . . . . Then from (4.78), we get un → u
⋆ -weakly in L∞ (0, T ; H01 (Ω )),
unt → ut
⋆ -weakly in L (0, T ; ∞
H01
(Ω )).
(4.79) (4.80)
Taking u1 = un , u2 = um , σ1 = σn , σ2 = σm , noting that compact embedding H01 (Ω ) ↩→↩→ L2 (Ω ), passing to a subsequence if necessary, we have un and unt converge strongly in C ([τ , T ]; L2 (Ω )). Therefore we get T
∥unt − umt ∥2 ds → 0,
τ
as m, n → +∞,
(4.81)
T
τ
∥un − um ∥2 ds → 0,
as m, n → +∞.
(4.82)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
15
On the other hand, by σm , σn ∈ Σ , we see that T
τ
∥σn − σm ∥2 ds → 0,
as m, n → +∞.
(4.83)
Hence it follows from (4.81)–(4.83) that t φT ((un , unt , ηnt ), (um , umt , ηm ); σn , σm ) → 0 as m, n → +∞,
(4.84)
that is, φT ∈ Contr(B, Σ ). Therefore by Lemma 3.1, the semigroup {Uσ (t , τ )}(t ≥ τ > 0, σ ∈ Σ ) is uniformly asymptotically compact and the proof is now complete. Proof of Theorem 2.3. Combining Theorems 4.1–4.2, we can complete the proof of Theorem 2.3.
Acknowledgments This work was in part supported by the NNSF of China with contract numbers 11031003, 11271066 and the grant from Shanghai Education Commission with contract number 13ZZ048. References [1] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1944. [2] M. Fabrizio, A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, No. 12, Philadelphia, 1992. [3] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043–1053. [4] S.A. Messaoudi, Nasser-eddine Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci. 30 (2007) 665–680. [5] S.A. Messaoudi, Nasser-eddine Tatar, Exponential decay for a quasilinear viscoelastic equation, Math. Nachr. 282 (2009) 1443–1450. [6] X. Han, M. Wang, General decay of energy for a viscoelastic equation with nonlinear damping, Math. Methods Appl. Sci. 32 (3) (2009) 346–358. [7] J.Y. Park, S.H. Park, General decay for quasiliear viscoelastic equations with nonlinear weak damping, J. Math. Phys. 50 (2009) 1–10. Article ID 083505. [8] R.O. Araújo, T. Ma, Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations 254 (2013) 4066–4087. [9] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Prates Filho, J.A. Soriano, Existence and uniform decay rates for viscoelastic problems with non-linear boundary damping, Differential Integral Equations 14 (1) (2001) 85–116. [10] I. Chueshov, I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, in: Mem. Amer. Math. Soc., Vol. 195, No. 12, Providence, 2008. [11] J.E. Munoz Rivera, E.C. Lapa, R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity 44 (1996) 61–87. [12] S.T. Wu, General decay of solutions for a viscoelastic equation with nonlinear damping and source terms, Acta Math. Sci. 31 (4) (2011) 1436–1448. [13] L. Yang, C. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal. TMA 69 (2008) 3802–3810. [14] X. Han, M. Wang, Global existence and uniform decay for a nonlinar viscoelastic equation with damping, Nonlinear Anal. TMA 70 (9) (2009) 3090–3098. [15] A.V. Babin, M.I. Vishik, Attractors of Evolution Equatioins, North-Holland, Amesterdam, London, New York, Tokyo, 1992. [16] T.F. Ma, V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal. TMA 73 (2010) 3402–3412. [17] Yuming Qin, Universal attractor in H 4 for the nonlinear one-dimensional compressible Navier–Stokes equations, J. Differential Equations 207 (1) (2004) 21–72. [18] Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and their Attractors, in: Operator Theory, Advances in PDEs, vol. 184, Birkhäuser, Basel, Boston, Berlin, 2008. [19] Y. Qin, S. Deng, W. Schulze, Uniform compact attractors for a nonlinear non-autonomous equation of viscoelasticity, J. Partial Differential Equations 22 (2) (2009) 153–198. [20] L. Yang, Uniform attractor for non-autonomous hyperbolic equation with critical exponent, Appl. Math. Comput. 203 (2008) 895–902. [21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, in: Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1988. [22] S. Borini, V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal. 20 (1999) 263–277. [23] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970) 297–308. [24] V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11 (2001) 505–529. [25] V.V. Chepyzhov, M.I. Vishik, Attractors of Equations of Mathematical Physics, in: Colloquium Publications, vol. 49, American Mathematical Society, Providence, Rhode Island, 2001. [26] V.V. Chepyzhov, V. Pata, M.I. Vishik, Averaging of 2D Navier–Stokes equations with singularly oscillating external forces, Nonlinearity 22 (2009) 351–370. [27] C. Sun, D. Cao, J. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst. 6 (2007) 293–318. [28] C. Sun, D. Cao, J. Duan, Non-autonomous wave dynamics with memory—asympototic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B 9 (2008) 743–761. [29] A.Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl. 318 (2006) 92–101.
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Uniform attractors for a non-autonomous viscoelastic equation with a past history Yuming Qin a,∗ , Baowei Feng b , Ming Zhang c a
Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China
b
College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, PR China
c
College of Information Science and Technology, Donghua University, Shanghai 201620, PR China
article
abstract
info
Article history: Received 27 May 2013 Accepted 9 January 2014 Communicated by Enzo Mitidieri
This paper investigates the existence of uniform attractors for a non-autonomous viscoelastic equation with a past history in a bounded domain Ω ⊆ Rn (n ≥ 1) by establishing the uniformly asymptotic compactness of the semi-process generated by the global solutions. Our results improve those results by Araújo et al. (2013). © 2014 Elsevier Ltd. All rights reserved.
Keywords: Viscoelastic equation Uniform attractors Past history
1. Introduction In this paper we investigate the existence of uniform attractors for a nonlinear non-autonomous viscoelastic equation with a past history
|ut |ρ utt − 1u − 1utt +
+∞
g (s)1u(t − s)ds + ut = σ (x, t ),
x ∈ Ω, t > τ ,
(1.1)
0
u(x, t ) = 0,
on ∂ Ω × [τ , +∞),
τ
τ
u(x, τ ) = u0 (x),
ut (x, τ ) = u1 (x),
(1.2) u(x, t ) = uτ (x, t ),
x ∈ Ω, τ ∈ R , +
(1.3)
where Ω ⊆ R (n = 1, 2) is a bounded domain with smooth boundary ∂ Ω , uτ (x, t ) (the past history of u) is a given datum which has to be known for all t ≤ τ , the function g represents the kernel of a memory, σ = σ (x, t ) is a non-autonomous term, called a symbol, and ρ is a real number such that n
1<ρ≤
2 n−2
if n ≥ 3;
ρ > 1 if n = 1, 2.
(1.4)
Our problem is of the form f (ut )utt − 1u − 1utt = 0,
(1.5)
which has several modeling features. In the case f (ut ) is a constant, Eq. (1.5) has been used to model extensional vibrations of thin rods (see Love [1, Chapter 20]). In the case f (ut ) is not a constant, Eq. (1.5) can model materials whose density depends
∗
Corresponding author. Tel.: +86 21 67792089 545; fax: +86 21 67823227. E-mail addresses: [email protected] (Y. Qin), [email protected] (B. Feng), [email protected] (M. Zhang).
0362-546X/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2014.01.006
2
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
on the velocity ut . For instance, a thin rod which possesses a rigid surface and with an interior which can deform slightly. We refer the reader to Fabrizio and Morro [2] for several other related models. Let us recall some results concerning viscoelastic wave equations. In [3], Cavalcanti et al. studied the following equation with Dirichlet boundary conditions
|ut |ρ utt − 1u − 1utt + g ∗ 1u − γ 1ut = 0 (1.6) t where g ∗ 1u = 0 g (t − s)1u(s)ds. They established a global existence result for γ ≥ 0 and an exponential decay of energy for γ > 0, and studied the interaction within the |ut |ρ utt and the memory term g ∗ 1u. Messaoudi and Tatar [4] established, for small initial data, the global existence and uniform stability of solutions to the equation
|ut |ρ utt − 1u − 1utt + g ∗ 1u = b|u|p−2 u,
(1.7)
with Dirichlet boundary condition, where γ ≥ 0, ρ, b > 0, p > 2 are constants. In the case b = 0 in (1.7), Messaoudi and Tatar [5] proved the exponential decay of global solutions to (1.7) without smallness of initial data, considering only the dissipation effect given by the memory. Considering nonlinear dissipation, Han and Wang [6] proved the energy decay for the viscoelastic equation with nonlinear damping
|ut |ρ utt − 1u − 1utt + g ∗ 1u + |ut |m ut = 0,
(1.8)
with Dirichlet boundary condition, where ρ > 0, m > 0 are constants. Then Park and Park [7] established the general decay for the viscoelastic problem with nonlinear weak damping
|ut |ρ utt − 1utt − 1u + g ∗ 1u + h(ut ) = 0,
(1.9)
with the Dirichlet boundary condition, where ρ > 0 is a constant. Recently, Araújo et al. [8] studied the following equation
|ut |ρ utt − 1u − 1utt +
+∞
µ(s)1u(t − s)ds + f (u) = h(x), 0
and proved the global existence, uniqueness and exponential stability, and the global attractor was also established, but they did not establish the uniform attractors for the non-autonomous equation. Moreover, we would like to mention some results in [9–13]. +∞ For problem (1.1)–(1.3) with σ (x, t ) = 0, when 0 g (s)1u(t − s)ds was replaced by g ∗ 1u, Han and Wang [14] established the global existence of weak solutions and the uniform decay estimates for the energy by using the Faedo–Galerkin method and the perturbed energy method, respectively. To the best of our knowledge, there is no result on the existence of uniform attractors for the non-autonomous viscoelastic problem (1.1)–(1.3). Therefore in this paper, we shall establish the existence of uniform attractors for problem (1.1)–(1.3) by establishing uniformly asymptotic compactness of the semiprocess generated by their global solutions. Noting that the symbol σ (x, t ), which is dependent on t, so our estimates are more complicated than [8] and we must use new methods to deal with the symbol σ (x, t ) as the change of time. Therefore we improved the results in [8]. For more results concerning attractors, we can refer to [15–21]. Motivated by [22–24], we shall add a new variable η = ηt (x, s) to the system which corresponds to the relative displacement history. Let us define
η = ηt (x, s) = u(x, t ) − u(x, t − s),
t ≥ τ , (x, s) ∈ Ω × R+ .
(1.10)
A direct computation yields
ηtt (x, s) = −ηst (x, s) + ut (x, t ),
t ≥ τ , ( x, s) ∈ Ω × R + ,
(1.11)
and we can take as initial condition (t = τ )
ητ (x, s) = uτ0 (x) − uτ0 (x, τ − s),
( x, s) ∈ Ω × R + .
(1.12)
Thus, the original memory term can be written as +∞
g (s)1u(t − s)ds = 0
+∞
g (s)ds · 1u −
+∞
0
g (s)1ηt (s)ds,
(1.13)
0
and we get a new system ρ
+∞
g (s)ds 1u − 1utt −
|ut | utt − 1 − 0
+∞
g (s)1ηt (s)ds + ut = σ (x, t ),
(1.14)
0
ηtt + ηst = ut ,
(1.15)
with the boundary conditions u=0
on ∂ Ω × R+ ,
ηt = 0 on ∂ Ω × R+ × R+ ,
(1.16)
and initial conditions u(x, τ ) = uτ0 (x),
ut (x, τ ) = uτ1 (x),
ηt (x, 0) = 0,
ητ (x, s) = uτ0 (x) − u(x, τ − s).
(1.17)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
3
The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statements and the proofs of our main results will be given in Sections 3 and 4, respectively. For convenience, we denote the norm and scalar product in L2 (Ω ) by ∥ · ∥ and (·, ·), respectively. C1 denotes a general positive constant, which may be different in different estimates. 2. Preliminaries and the main result We assume the memory kernel g : R+ → R+ is a bounded C 1 function such that +∞
g (s) < +∞,
g (s)ds > 0
l=1−
(2.1)
0
and suppose that there exist a positive constant ξ2 verifying g ′ (t ) ≤ −ξ2 g (t ),
∀ t ≥ 0,
(2.2)
and the symbol
σ ∈ E1 ≡ L2 (R+ , L2 (Ω )).
(2.3)
In order to consider the relative displacement η as a new variable, one introduces the weighted L -space 2
M=
L2g
(R ; +
H01
(Ω )) = u : R → +
H01
(Ω )
+∞
g (s)∥∇ u(s)∥ ds < +∞ 2
,
0
which is a Hilbert space equipped with the inner product and the norm
(u, v)M =
+∞
g (s)
Ω
0
+∞
∇ u(s)∇v(s)dx ds and ∥u∥2M =
g (s)∥∇ u(s)∥2 ds, 0
respectively. Let H = H01 (Ω ) × H01 (Ω ) × M .
(2.4)
The energy of problem (1.14)–(1.17) is given by F (t ) =
1
ρ+2
l
ρ+2
1
1
∥ut (t )∥ρ+2 + ∥∇ u(t )∥2 + ∥∇ ut (t )∥2 + ∥ηt ∥2M .
(2.5)
2 2 2 First, we give the following existence and uniqueness results.
Theorem 2.1. Let (uτ0 , uτ1 , ητ ) ∈ H (∀ τ ∈ R+ ), Rτ = [τ , +∞), and any fixed σ ∈ E1 . Assume (2.1) and (2.2) hold. Then problem (1.14)–(1.17) admits a unique global solution (u, ut , ηt ) ∈ C ([0, T ], H ) such that u ∈ L∞ (Rτ , H01 (Ω )),
ut ∈ L∞ (Rτ , H01 (Ω )),
utt ∈ L2 (Rτ , H01 (Ω )),
ηt ∈ L∞ (Rτ , M).
(2.6)
We now define the symbol space for (1.14)–(1.17). Observe the following important fact: the properly defined (uniform) attractor A of problem (1.14)–(1.17) with the symbol σ0 must be simultaneously the attractor of each problem (1.14)–(1.17) with the symbol σ (t ) ∈ H+ (σ0 ), which is called the hull of σ0 and defined as
Σ = H+ (σ0 ) = [σ0 (t + h)|h ∈ R+ ]E1 where [ · ]E1 denotes the closure in Banach space E1 .
(2.7)
We note that
σ0 ∈ E1 ⊆ E1 = L2loc (R+ , L2 (Ω )) where σ0 is a translation compact function in E1 in the weak topology, which means that σ0 is compact in E1 . We consider p the Banach space Lloc (R+ , E1 ) of functions σ (s), s ∈ R+ with values in a Banach space E1 that are locally p-power integrable in the Bochner sense. In particular, for any time interval [t1 , t2 ] ⊆ R+ ,
t2 t1
∥σ (s)∥pE1 ds < +∞. p
Let σ (s) ∈ Lloc (R+ , E1 ), consider the quantity
ησ (h) = sup
t ∈R+
t +h
t
∥σ (s)∥pE1 ds.
Lemma 2.1. Let Σ be defined as before and σ0 ∈ E1 , then (1) σ0 is a translation compact in E1 and any σ ∈ Σ = H+ (σ0 ) is also a translation compact in E1 , moreover, H+ (σ ) ⊆ H+ (σ0 ); (2) the set H+ (σ0 ) is bounded in L2 (R+ , L2 (Ω )) such that
ησ (h) ≤ ησ0 (h) < +∞,
for all σ ∈ Σ .
4
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
Proof. See, e.g., Chepyzhov and Vishik [25].
Lemma 2.2. For every τ ∈ R, every non-negative locally summable function φ0 on Rτ ≡ [τ , +∞) and every ν > 0, we have t
sup t ≥τ
τ
φ0 (s)e−ν(t −s) ds ≤
t +1
1
sup 1 − e−ν t ≥τ
φ0 (s)ds
t
for a.a. t ≥ τ . Proof. See, e.g., Chepyzhov, Pata and Vishik [26].
Similar to Theorem 2.1, we have the following existence and uniqueness result. Theorem 2.2. Let Σ = H+ (σ0 ) = [σ0 (t + h)|h ∈ R+ ]E1 , where σ0 ∈ E1 is an arbitrary but fixed symbol function. Assume (2.1) and (2.2) hold. Then for any σ ∈ Σ and for any (uτ0 , uτ1 , ητ ) ∈ H (∀ τ ∈ R+ ), problem (1.14)–(1.17) admits a unique global solution (u, ut , ηt ) ∈ H , which generates a unique semi-process {Uσ (t , τ )}, (t ≥ τ ∈ R+ , σ ∈ Σ ) on H of a two-parameter family of operators such that for any t ≥ τ , τ ∈ R+ , Rτ = [τ , +∞), Uσ (t , τ )(uτ0 , uτ1 , ητ ) = (u, ut , ηt ) ∈ H , u ∈ L∞ (Rτ , H01 (Ω )),
(2.8)
ut ∈ L∞ (Rτ , H01 (Ω )),
utt ∈ L2 (Rτ , H01 (Ω )),
ηt ∈ L∞ (Rτ , M).
(2.9)
Now we give the following result concerning the uniform attractors. Theorem 2.3. Assume that σ ∈ E1 and Σ is defined by (2.7), then the family of processes {Uσ (t , τ )} (σ ∈ Σ , t ≥ τ , τ ∈ R+ ) corresponding to (1.14)–(1.17) has a uniformly (w.r.t. σ ∈ Σ ) compact attractor AΣ . 3. The well-posedness The global existence of solutions is the same as in [8,14,24], so we omit the details here. Next we prove the uniqueness of solutions. We consider two symbols σ1 and σ2 and the corresponding solutions (u, ηt ) and (v, ξ t ) of problem (1.14)–(1.17) with initial data (uτ0 , uτ1 , ητ ) and (v0τ , v1τ , ξ τ ) respectively. Let ω(t ) = u(t ) − v(t ) = Uσ1 (t , τ )uτ0 − Uσ2 (t , τ )v0τ , ζ t (x, s) = ηt (x, s) − ξ t (x, s). Then (ω, ζ t ) verifies
|ut |ρ ωtt + vtt (|ut |ρ − |vt |ρ ) − l1ω − 1ωtt −
+∞
g (s)1ζ (s)ds + ωt = σ1 − σ2 ,
x ∈ Ω, t > τ ,
(3.1)
0
ζtt + ζst = ωt ,
(3.2)
with Dirichlet boundary conditions and initial conditions
ω(x, τ ) = ω0τ ,
ωt (x, τ ) = ω1τ ,
ζ τ = ητ − ξ τ .
(3.3)
The corresponding energy for (3.1)–(3.3) is defined Eω (t ) =
1
2
Ω
l
1
1
2
2
2
|ut |ρ ωt2 dx + ∥∇ω∥2 + ∥∇ωt ∥2 + ∥ζ t ∥2M .
(3.4)
It is easy to see that +∞
|∇ζ t (s)|2 ds dx 2 Ω ds 0 +∞ 1 =− g ′ (s)|∇ζ t (s)|2 ds dx.
(ζst , ζ t )M =
1
2
g (s)
Ω
d
0
ρ
Note that x → |x| is differentiable since ρ > 1. Then 1 d 2 dt
ρ
Ω
|ut | ω
2 t dx
ρ
= Ω
|ut | ωtt ωt dx +
ρ 2
Ω
|ut |ρ−2 ut utt ωt2 dx,
and clearly d dt
Eω (t ) = −∥ωt ∥2 +
+
ρ 2
Ω
1 2
+∞
g ′ (s)∥∇ζ t (s)∥2 ds + 0
|ut |ρ−1 utt ωt2 dx −
Ω
Ω
(σ1 − σ2 )ωt dx
vtt ωt (|ut |ρ − |vt |ρ )dx.
(3.5)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
5
To simplify notations, let us say that the norm of the initial data is bounded by some R > 0. Then given T > τ we use CRT to denote several positive constants which depend on R and T . By Young’s inequality and the interpolation inequalities, we derive
(σ1 − σ2 )ωt dx ≤ ∥σ1 − σ2 ∥ ∥ωt ∥ Ω
≤ ∥σ1 − σ2 ∥2 + CRT Eω (t ), ρ ρ ρ−1 ρ−1 2 ∥ut ∥2(ρ+1) ∥utt ∥ ∥ωt ∥22(ρ+1) | ut | utt ωt dx ≤ 2 2
(3.6)
≤ CRT ∥∇ utt ∥ ∥∇ωt ∥2 , ρ ρ − vtt (|ut | −|vt | )ωt dx ≤ C1 |vtt |(|ut |ρ−1 + |vt |ρ−1 )ωt2 dx
(3.7)
Ω
Ω
Ω
ρ−1
ρ−1
≤ C1 ∥vtt ∥(∥ut ∥2(ρ+1) + ∥vt ∥2(ρ+1) )∥ωt ∥22(ρ+1) ≤ C1 ∥∇vtt ∥ ∥∇ωt ∥2 ,
which, together with (3.5)–(3.7), yields for some C1 > 0 large d Eω (t ) ≤ ∥σ1 − σ2 ∥2 + C1 (1 + ∥∇ utt ∥ + ∥∇vtt ∥)Eω (t ). dt Integrating (3.8) from τ to t and using Hölder’s inequality, we have Eω (t ) ≤ Eω (τ ) +
≤ Eω (τ ) +
t
∥σ1 (s) − σ2 (s)∥2 ds + C1
τ
∥σ1 (s) − σ2 (s)∥ ds + C1 2
τ
t
T
(3.8)
τ
(1 + ∥∇ utt ∥ + ∥∇vtt ∥)Eω (s)ds t
τ
21
(1 + ∥∇ utt ∥ + ∥∇vtt ∥) ds 2
t
Eω (s)ds 2
τ
12
.
(3.9)
Note that T
(1 + ∥∇ utt ∥ + ∥∇vtt ∥)2 ds ≤ CRT ,
τ
then we get for any t ∈ [τ , T ]
Eω (t ) ≤ 2 Eω (τ ) + 2
T
∥σ1 (s) − σ2 (s)∥ ds 2
τ
2
t
+ CRT
τ
Eω2 (s)ds.
(3.10)
Applying Gronwall’s inequality, we see that Eω (t ) ≤ Using
Ω
√
2 Eω (τ ) +
T
τ
CRT ∥σ1 (s) − σ2 (s)∥2 ds exp T , 2
∀ t ∈ [τ , T ].
(3.11)
ρ
|ut |ρ |ωt |2 dx ≤ ∥ut ∥ρ+2 ∥ωt ∥2ρ+2 ≤ CRT ∥∇ωt ∥2 , we know that Eω (t ) is equivalent to the norm of u in H and we get
Eω (τ ) ≤ CRT ∥(ω0τ , ω1τ , ζ τ )∥2H , which, together with (3.11), gives for all τ ≤ t ≤ T
∥u(t ) − v(t )∥2H 1 + ∥ut (t ) − vt (t )∥2H 1 + ∥ηt − ξ t ∥2M 0
0
≤ CRT (∥uτ0 − v0τ ∥2H 1 + ∥uτ1 − v1τ ∥2H 1 + ∥ητ − ξ τ ∥2M + ∥σ1 − σ2 ∥2L2 (τ ,T ;L2 (Ω )) ). 0
0
This shows that solutions of (1.14)–(1.17) depend continuously on the initial data. We complete the proof of Theorem 2.1. 4. Uniform attractors In this section, we shall prove Theorem 2.3. To this end, we shall introduce some basic conceptions and basic lemmas. For more results concerning uniform attractors, we can refer to [15,20,21,27,28]. be a parameter set. The operators {Uσ (t , τ )}(t ≥ τ , τ ∈ R+ , σ ∈ Σ ) are said to be a Let X be a Banach space, and Σ if for any σ ∈ Σ , family of processes in X with symbol space Σ Uσ (t , s)Uσ (s, τ ) = Uσ (t , τ ), Uσ (τ , τ ) = Id (identity),
∀ t ≥ s ≥ τ , τ ∈ R+ ,
∀τ ∈R . +
(4.1) (4.2)
6
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
, we say that a family of processes {Uσ (t , τ )} (t ≥ τ , τ ∈ R+ , σ ∈ Σ ) satisfies Let {T (s)} be the translation semigroup on Σ the translation identity if Uσ (t + s, τ + s) = UT (s)σ (t , τ ),
=Σ , T (s)Σ
, t ≥ τ , τ , s ∈ R+ , ∀σ ∈Σ
(4.3)
∀s∈R . +
(4.4)
By B(X ) we denote the collection of the bounded sets of X , and Rτ = [τ , +∞), τ ∈ R . +
) absorbing set for {Uσ (t , τ )} (σ ∈ Definition 4.1. A bounded set B0 ∈ B(X ) is said to be a bounded uniformly (w.r.t. σ ∈ Σ , t ≥ τ , τ ∈ R+ ) if for any τ ∈ R+ and B ∈ B(X ), there exists a time T0 = T0 (B, τ ) ≥ τ such that Σ
Uσ (t , τ )B ⊆ B0 ,
(4.5)
σ ∈Σ
for all t ≥ T0 . In the following, as usual, (w.r.t.) will represent ‘‘with respect to’’.
) attracting for the family of semi-processes {Uσ (t , τ )} (t ≥ Definition 4.2. A set A ⊆ X is said to be uniformly (w.r.t. σ ∈ Σ ) if for any fixed τ ∈ R+ and any B ∈ B(X ), τ , τ ∈ R+ , σ ∈ Σ lim (sup dist(Uσ (t , τ )B, A)) = 0,
(4.6)
t →+∞
here dist(·, ·) stands for the usual Hausdorff semidistance between two sets in X . In particular, a closed uniformly attracting ) attractor of the family of the semi-process {Uσ (t , τ )} (t ≥ τ , τ ∈ R+ , σ ∈ Σ ) set AΣ is said to be the uniform (w.r.t. σ ∈ Σ if it is contained in any closed uniformly attracting set (minimality property).
be a symbol (or parameter) space. We call a Definition 4.3. Let X be a Banach space and B be a bounded subset of X , Σ ×Σ ) to be a contractive function on B × B if for any sequence {xn }∞ function φ(·, ·; ·, ·), defined on (X × X ) × (Σ n =1 ⊆ B ∞ ∞ ∞ , there is a subsequence {xnk }∞ and any {σn } ⊆ Σ k=1 ⊂ {xn }n=1 and {σnk }k=1 ⊂ {σn }n=1 such that lim lim φ(xnk , xnl ; σnk , σnl ) = 0.
(4.7)
k→∞ l→∞
). We denote the set of all contractive functions on B × B by Contr(B, Σ ) be a family of semi-processes satisfying the translation identities (4.3) and Lemma 4.1. Let {Uσ (t , τ )} (t ≥ τ , τ ∈ R+ , σ ∈ Σ ) absorbing set B0 ⊆ X . Moreover, assuming that for any (4.4) on Banach space X and has a bounded uniformly (w.r.t. σ ∈ Σ ) such that ε > 0, there exist T = T (B0 , ε) > 0 and φT ∈ Contr(B0 , Σ ∥Uσ1 (T , 0)x − Uσ2 (T , 0)y∥ ≤ ε + φT (x, y; σ1 , σ2 ),
, t ≥ τ , τ ∈ R+ . ∀σ ∈Σ
(4.8)
) is uniformly (w.r.t. σ ∈ Σ ) asymptotically compact in X . Then {Uσ (t , τ )} (t ≥ τ , τ ∈ R , σ ∈ Σ +
Proof. This lemma is a version for semi-processes of a result by Khanmamedov [29]. A proof can be found in Sun et al. [27, Theorem 4.2]. Next, we will divide into two subsections to prove Theorem 2.3. 4.1. Uniformly (w.r.t. σ ∈ Σ ) absorbing set in H In this subsection we shall establish the family of processes {Uσ (t , τ )} has a bounded uniformly absorbing set given in the following theorem. Theorem 4.1. Assume that σ ∈ E1 and Σ is defined by (2.7), then the family of processes {Uσ (t , τ )} (σ ∈ Σ , t ≥ τ , τ ∈ R+ ) corresponding to (1.14)–(1.17) has a bounded uniformly (w.r.t. σ ∈ Σ ) absorbing set B in H . Proof. We define F (t ) =
1
ρ+2
l
1
1
2
2
2
ρ+2
∥ut ∥ρ+2 + ∥∇ u∥2 + ∥∇ ut ∥2 + ∥ηt ∥2M .
(4.9)
Multiplying (1.14) by ut , integrating the resulting equation over Ω and Young’s inequality, we derive F ′ (t ) = −∥ut ∥2 − (ηst , ηt )M + (σ , ut ) 1
1
2
2
≤ − ∥ ut ∥ 2 +
0
+∞
g ′ (s)∥∇ηt (s)∥2 ds +
1 2
∥σ ∥2 .
(4.10)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
7
Let 1
F1 (t ) = F (t ) +
+∞
2
∥σ (s)∥2 ds,
for all t ≥ τ .
(4.11)
t
Then (4.11) gives F1′ (t ) ≤ 0, whence from (4.9), for t ≥ τ > 0 F (t ) ≤ F1 (t ) ≤ F1 (τ ) = F (τ ) + l 2
1
1
2
2
1
+∞
1
∥σ (s)∥2 ds = F (τ ) + ∥σ ∥2L2 (Rτ ,L2 (Ω )) ,
2 τ
(4.12)
2
∥∇ u∥2 + ∥∇ ut ∥2 + ∥ηt ∥2M ≤ F (t ) ≤ F1 (t ) ≤ F1 (τ ).
(4.13)
Now we define
Φ (t ) =
1
ρ+1
Ω
|ut |ρ ut udx +
Ω
∇ ut · ∇ udx.
(4.14)
From (1.14), integration by parts and Young’s inequality, we derive for any ε ∈ (0, 1),
Φ ′ (t ) =
l1u +
+∞
g (s)1ηt (s)ds − ut + σ , u
1
+
ρ+1
0
= −l∥∇ u∥ − ∇ u, 2
+∞
g (s)∇η (s)ds t
(|ut |ρ ut , ut ) − (1ut , ut )
− (ut , u) +
0
1
ρ+2
ρ+1
∥ut ∥ρ+2 + ∥∇ ut ∥2 + (σ , u).
(4.15)
Using Young’s inequality, Hölder’s inequality and Poincaré’s inequality, we deduce
| − (ut , u)| ≤ ε∥u∥2 + − ∇ u,
1 4ε
+∞
g (s)∇ηt (s)ds ≤ ε∥∇ u∥2 +
0
≤ ε∥∇ u∥2 + |(σ , u)| ≤ ε∥u∥2 +
1 4ε
1
∥ut ∥2 ≤ ελ2 ∥∇ u∥2 +
∥σ ∥2 ≤ ελ2 ∥∇ u∥2 +
4ε
1
∥ ut ∥ 2 ,
1 4ε
2
+∞
4ε Ω 0 1 − l +∞ 4ε
(4.16) g (s)∇ηt (s)ds
dx
g (s)∥∇ηt (s)∥2 ds,
(4.17)
0
∥σ ∥2 ,
(4.18)
hereinafter we use λ to represent the Poincaré constant. From the expression of F (t ), we get
∥∇ u∥2 =
2 l
F (t ) −
2
ρ+2
l(ρ + 2)
1
1
l
l
∥ut ∥ρ+2 − ∥∇ ut ∥2 − ∥ηt ∥2M ,
which, together with (4.15)–(4.18), yields 1
Φ ′ (t ) ≤ − (l − 2ελ2 − ε)∥∇ u∥2 − 2
1
2
4ε
4ε
∥ηt ∥2M +
2 (l − 2ελ2 − ε) F (t ) − l
1
2 l(ρ + 2)
ρ+2 ∥ut ∥ρ+2
ρ+2
1
1
l
l
∥ut ∥ρ+2 − ∥∇ ut ∥2 − ∥ηt ∥2M
ρ+1
1
+ ∥∇ ut ∥2 + ∥σ ∥2 4η 1 l − 2ελ2 − ε 1 ρ+2 = − (l − 2ελ2 − ε)∥∇ u∥2 − (l − 2ελ2 − ε)F (t ) + + ∥ut ∥ρ+2 2 l(ρ + 2) ρ+1 1 1 1 1−l 1 2 2 2 + 1 + (l − 2ελ − ε) ∥∇ ut ∥ + ∥ηt ∥2M + ∥σ ∥2 + ∥ut ∥2 . (l − 2ελ − ε) + 2 2l 4ε 4ε 4ε +
∥ ut ∥ 2 +
1−l
1
(4.19)
Noting that ∥∇ ut ∥2 ≤ 2F (t ) ≤ 2F1 (τ ) and the embedding theorem H 1 (Ω ) ↩→ L2(ρ+1) (Ω ), we have for any ε ∈ (0, 1),
1 − 2ελ2 − ε l(ρ + 2)
+
1
ρ+1
ρ+2 ρ+1 ∥ut ∥ρ+2 ≤ C1 ∥ut ∥ ∥ut ∥2(ρ+1) ≤ C1 ∥∇ ut ∥ρ+1 ∥ut ∥ ≤ ε∥∇ ut ∥2(ρ+1) + Cε ∥ut ∥2 ρ ≤ 2ρ+1 ε F1 (τ )F (t ) + Cε ∥ut ∥2 ,
8
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
which, together with (4.19) and Poincaré’s inequality, gives 1
Φ ′ (t ) ≤ − (l − 2ελ2 − ε)∥∇ u∥2 −
l − 2ελ2 − ε
2
l
λ + ∥∇ ut ∥2 + 4ε
l − 2ελ − ε
2
2
+ 1 + Cε λ + 2
ρ − 2ρ+1 εF1 (τ ) F (t )
2l
l − 2ελ2 − ε 2l
+
1−l 4ε
∥ηt ∥2M +
1 4η
∥σ ∥2 .
(4.20)
Now we take ε ∈ (0, 1) so small that l − 2ελ2 − ε ≥
l 2
,
l − 2ελ2 − ε l
ρ
− 2ρ+1 ε F1 (τ ) ≥
1 4
.
(4.21)
Hence from (4.20)–(4.21), it follows l
1
4
4
Φ ′ (t ) ≤ − ∥∇ u∥2 −
1
F (t ) + C1 ∥ηt ∥2M + C1 ∥∇ ut ∥2 +
4ε
∥σ ∥2 .
(4.22)
We define the functional
Ψ (t ) =
Ω
1ut −
1
ρ+1
|ut |ρ ut
+∞
g (s)ηt (s)dsdx.
(4.23)
0
It follows from (1.14) that
Ψ (t ) = ′
+∞
−l1u − Ω + 1ut − Ω
g (s)1η (s)ds + ut − σ t
+∞
0
g (s)ηt (s)ds
0
1
ρ+1
+∞
ρ
g (s)ηtt (s)ds
|ut | ut 0
:= I1 + I2 .
(4.24)
From Young’s inequality, Hölder’s inequality and Poincaré’s inequality, we derive for any δ ∈ (0, 1),
+∞ 2 t (−l1u) ≤ δ∥∇ u∥2 + l (1 − l) ∥ηt ∥2 , g ( s )η ( s ) ds M 4δ Ω 0 +∞ 2 g (s)1ηt (s)ds ds ≤ (1 − l)∥ηt ∥2M , − Ω 0 +∞ 2 t ut · ≤ δ∥ut ∥2 + λ (1 − l) ∥ηt ∥2 , g ( s )η ( s ) ds M 4δ Ω 0 +∞ 1 1−l t 2 (−σ ) g (s)ηt (s)ds ≤ ∥σ ∥2 + ∥η ∥M , Ω
2
0
(4.25)
(4.26)
(4.27)
2
which, together with (4.25)–(4.27), gives 1
I1 ≤ δ(∥ut ∥2 + ∥∇ u∥2 ) +
2
∥σ ∥2 +
3(1 − l) 2
+
(λ2 + l2 )(1 − l) ∥ηt ∥2M . 4δ
(4.28)
Note that +∞
g (s)ηtt (s)ds = −
+∞
0
g (s)ηst (s)ds +
+∞
0
g (s)ut (t )ds 0
+∞
g ′ (s)ηt (s)ds + (1 − l)ut ,
= 0
then we have I2 = −(1 − l)∥∇ ut ∥2 −
+
1
ρ+1
Ω
|ut |ρ ut
1−l
ρ+1 +∞ 0
ρ+2
∥ut ∥ρ+2 +
+∞
g ′ (s) 0
g ′ (s)ηt (s)dsdx.
Ω
1ut (t )ηt (s)dxds (4.29)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
9
By Young’s inequality, we derive +∞
g (s) ′
Ω
0
1ut (t )η (s)dxds ≤ − t
+∞
g ′ (s)∥∇ ut (t )∥ ∥∇ηt (s)∥ds
0
1−l
≤
4
∥∇ ut (t )∥2 −
+∞
1 1−l
g ′ (s)∥∇ηt (s)∥2 ds,
(4.30)
0
and for any ε > 0 1
ρ+1
Ω
+∞
|ut |ρ ut
g ′ (s)ηt (s)dsdx ≤ ε∥∇ ut ∥2 − Cε
+∞
g ′ (s)∥∇ηt (s)∥2 ds, 0
0
which, together with (4.29)–(4.30) and taking ε > 0 small enough, yields I2 ≤ −
1−l 2
∥∇ ut ∥2 −
1−l
ρ+1
+∞
ρ+2
∥ut ∥ρ+2 − C1
g ′ (s)∥∇ηt (s)∥2 ds.
(4.31)
0
Inserting (4.28) and (4.31) into (4.24), we arrive at 1−l
Ψ ′ (t ) ≤ −
1
∥∇ ut ∥2 + δ(∥∇ u∥2 + ∥ut ∥2 ) + C1 ∥ηt ∥2M + ∥σ ∥2
4
2
+∞
g ′ (s)∥∇ηt (s)∥2 ds −
− C1 0
1−l
ρ+2
ρ+1
∥ut ∥ρ+2 .
(4.32)
Set G(t ) = MF (t ) + ϵ Φ (t ) + Ψ (t ),
(4.33)
where M and ϵ are positive constants. Then it follows from (4.10), (4.22), (4.32) and (2.2) that G′ (t ) ≤ −
lϵ ϵ 1−l − δ ∥ ut ∥ 2 − − δ ∥∇ u∥2 − F (t ) − − C1 ϵ ∥∇ ut ∥2
M 2
4
M
+
2
− C1 − C1 ξ2
4
4
+∞
g (s)∥∇η (s)∥ ds + ′
t
2
ϵ 1 1−l ρ+2 + + ∥σ ∥2 − ∥ut ∥ρ+2 . 2 4ε 2 ρ+1
M
0
(4.34)
Now we claim that there exist two constants β1 , β2 > 0 such that
β1 F (t ) ≤ G(t ) ≤ β2 F (t ),
t ≥ 0.
(4.35)
In fact,
|Φ (t )| ≤ ≤
1
ρ+1
ρ+1
1
l
2
2
∥ut ∥2(ρ+1) ∥u∥ + ∥∇ ut ∥2 + ∥∇ u∥2
1
1
2(ρ+1)
2(ρ + 1)
∥ut ∥2(ρ+1) +
ρ
1
2(ρ + 1)
≤ C1 F (τ ) ∥∇ ut ∥ + C1 ∥∇ u∥ + 2
2
l
∥u∥2 + ∥∇ ut ∥2 + ∥∇ u∥2 2
1 2(ρ + 1)
2
2
1
l
2
2
∥u∥ + ∥∇ ut ∥2 + ∥∇ u∥2
≤ C0 F (t ).
(4.36)
Using Hölder’s inequality, Young’s inequality and Poincaré’s inequality, we get
+∞
g ( s) 0
1 1 1ut η (s)dxds ≤ ∥∇ ut ∥2 + ∥ηt ∥M , 2 2 t
Ω
and
ρ+1
+∞
1
g (s) 0
|ut (t )| ut (t )η (s)dxds ≤ C1 F (τ )ρ ∥∇ ut ∥2 + C1 ∥∇ηt (s)∥2 , ρ
Ω
which, along with (4.37), gives
|Ψ (t )| ≤ C1 F (t ). It follows that for some constant C2 > 0
|ϵ Φ (t ) + Ψ (t )| ≤ C2 F (t ).
t
(4.37)
10
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
Then taking M > C2 we get (4.35) with β1 = M − C2 and β2 = M + C2 . For any t ≥ τ , we take ε so small that 1−l 4
− C1 ϵ > 0.
(4.38)
For fixed ε , we choose δ small enough and M so large that M 2
lϵ
− δ > 0,
4
M
− δ > 0,
− C1 − C1 ξ 2 > 0 .
2
Then there exist a constant γ > 0 such that G′ (t ) ≤ −γ F (t ) + C1 ∥σ ∥2 ,
(4.39)
which, together with (4.35), gives G′ (t ) ≤ −
γ G(t ) + C1 ∥σ ∥2 . β2
(4.40)
Integrating (4.40) over [τ , t ] with respect to t and using Lemmas 2.1–2.2, we obtain − βγ (t −τ )
G(t ) ≤ G(τ )e
≤ CB0 e ≤ CB0
2
− βγ (t −τ )
t
+ C1
− γ (t −τ ) e β2
2
τ
1
+ C1
2
− βγ (t −s)
e
1−e 1
+ C1
1−e
− βγ
2
− βγ
∥σ (s)∥2 ds t +1 sup ∥σ (s)∥2 ds t ≥τ
t
ησ (1).
(4.41)
2
Now for any bounded set B0 ⊆ H , for any (uτ0 , uτ1 , ητ ) ∈ B0 , there exists a constant CB0 > 0 such that F (τ ) ≤ CB0 ≤ C1 . Take
R20
= 2 2C1
1−e t0 = τ −
ησ0 (1)
γ β2
− βγ
+1 ,
2
−1
C1 ησ0 (1) + 1
log
C B0 1 − e
− βγ
2
,
then for any t ≥ t0 ≥ 1, we have − βγ (t −τ )
G(t ) ≤ CB0 e
2
1
+ C1 1−
−γ e β2
ησ0 (1) ≤
R20 2
,
which gives
∥(u, ut , ηt )∥H ≤ 2G(t ) = R20 , i.e., B = B(0, R0 ) = {(u, ut , ηt ) ∈ H : ∥(u, ut , ηt )∥2H ≤ R20 } ⊆ H is a uniform absorbing ball for any σ ∈ E1 . The proof is now complete. 4.2. Uniformly (w.r.t. σ ∈ Σ ) asymptotic compactness in H In this subsection, we will prove the uniformly (w.r.t. σ ∈ Σ ) asymptotic compactness in H , which is given in the following theorem. Theorem 4.2. Assume that σ ∈ E1 and Σ is defined by (2.7), then the family of processes {Uσ (t , τ )} (σ ∈ Σ , t ≥ τ , τ ∈ R+ ) corresponding to (1.14)–(1.17) is uniformly (w.r.t. σ ∈ Σ ) asymptotically compact in H . Proof. For any (uτ0i , uτ1i , ηiτ ) ∈ B, i = 1, 2, we consider two symbols σ1 and σ2 and the corresponding solutions u1 and u2 of problem (1.14)–(1.17) with initial data (uτ0i , uτ1i , ηiτ ), i = 1, 2, respectively. Let ω(t ) = u1 (t ) − u2 (t ) = Uσ1 (t , τ )uτ01 − Uσ2 (t , τ )uτ02 , ζ t (x, s) = η1t (x, s) − η2t (x, s). Then (ω, ζ t ) verifies ρ
ρ
ρ
|u1t | ωtt + u2tt (|u1t | − |u2t | ) − l1ω − 1ωtt −
+∞
g (s)1ζ (s)ds + ωt = σ1 − σ2 ,
x ∈ Ω, t > τ ,
(4.42)
0
ζtt + ζst = ωt ,
(4.43)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
11
with Dirichlet boundary conditions and initial conditions
ω(x, τ ) = ω0τ ,
ωt (x, τ ) = ω1τ ,
ζ τ = η1τ − η2τ .
(4.44)
The corresponding energy for (4.42)–(4.44) is defined 1
Eω (t ) =
2
l
1
1
2
2
2
|u1t |ρ ωt2 dx + ∥∇ω∥2 + ∥∇ωt ∥2 + ∥ζ t ∥2M .
Ω
(4.45)
Clearly, d dt
Eω (t ) = −∥ωt ∥ + 2
+
ρ 2
Ω
1
+∞
2
g (s)∥∇ζ (s)∥ ds + ′
t
2
Ω
0
|u1t |ρ−1 u1tt ωt2 dx −
Ω
(σ1 − σ2 )ωt dx
u2tt ωt (|u1t |ρ − |u2t |ρ )dx.
(4.46)
Using Hölder’s inequality, Poincaré’s inequality and Theorem 4.1, we derive
(σ1 − σ2 )ωt dx ≤ ∥σ1 − σ2 ∥ ∥ωt ∥, Ω ρ ρ ρ−1 ρ−1 2 u1tt ωt dx ≤ ∥u1t ∥2(ρ+1) ∥u1tt ∥ ∥ωt ∥2(ρ+1) ∥ωt ∥ |u1t | 2
(4.47)
2
Ω
≤ CB ∥∇ u1t ∥ρ−1 ∥∇ωt ∥ ∥∇ u1tt ∥ ∥ωt ∥ ≤ CB ∥ωt ∥ ∥∇ u1tt ∥, ρ ρ ρ − ≤ C1 ∥u2tt ∥2(ρ+1) (∥u1t ∥ρ | | u (| u −| u )ω dx 2tt 1t 2t t 2(ρ+1) + ∥u2t ∥2(ρ+1) )∥ωt ∥ Ω ≤ C1 ∥∇ u2tt ∥(∥∇ u1t ∥ρ + ∥∇ u2t ∥ρ )∥ωt ∥ ≤ CB ∥∇ u2tt ∥ ∥ωt ∥,
(4.48)
which, combined with (4.46)–(4.48), yields d dt
Eω (t ) ≤ −∥ωt ∥2 +
1 2
+∞
g ′ (s)∥∇ζ t (s)∥2 ds + ∥ωt ∥ ∥σ1 − σ2 ∥ + CB0 (∥∇ u1tt ∥ + ∥∇ u2tt ∥)∥ωt ∥.
(4.49)
0
We define
Φω (t ) =
|u1t |ρ ωt ωdx +
Ω
Ω
∇ωt · ∇ωdx +
1 2
Ω
ω2 dx.
(4.50)
It is very easy to verify
|Φω (t )| ≤
1 2
1
ρ
(∥∇ω∥2 + ∥∇ωt ∥2 ) + ∥u1t ∥2(ρ+1) ∥ωt ∥2(ρ+1) ∥ω∥ + ∥ω∥2 2
≤ CB (∥∇ω∥2 + ∥∇ωt ∥2 ) ≤ CB Eω (t ).
(4.51)
Taking the derivative of Φω (t ), it follows from (4.42) that
Φω′ (t ) = −
Ω
+ Ω
=
4
u2tt (|u1t |ρ − |u2t |ρ )ωdx − l∥∇ω∥2 −
(σ1 − σ2 )ωdx +
Ω
+∞
∇ω Ω
g (s)∇ζ t (s)dsdx
0
(ρ|u1t |ρ−1 u1tt ωt ω + |u1t |ρ ωt2 )dx + ∥∇ωt ∥2
Ai − l∥∇ω∥2 + ∥∇ωt ∥2 .
(4.52)
i=1
Applying Hölder’s inequality, Young’s inequality, Poincaré’s inequality and Theorem 4.1, we get ρ−1
ρ−1
|A1 | ≤ C1 ∥u2tt ∥2(ρ+1) (∥u1t ∥2(ρ+1) + ∥u2t ∥2(ρ+1) )∥ωt ∥2(ρ+1) ∥ω∥ ≤ C1 ∥∇ u2tt ∥(∥∇ u1t ∥ρ−1 + ∥∇ u2t ∥ρ−1 )∥∇ωt ∥ ∥ω∥ ≤ CB ∥∇ u2tt ∥ ∥ω∥,
(4.53)
12
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
+∞ t g (s)∇ζ (s)ds 0 1−l t 2 ≤ ε∥∇ω∥2 + ∥ζ ∥ , ∀ ε ∈ (0, 1), 4ε |A3 | ≤ ∥σ1 − σ2 ∥ ∥ω∥,
|A2 | ≤ ∥∇ω∥
(4.54) (4.55)
ρ−1 C1 ∥u1tt ∥2(ρ+1) ∥u1t ∥2(ρ+1) ∥ωt ∥2(ρ+1) ∥ω∥
|A4 | ≤ ≤ CB ∥∇ u1tt ∥ ∥ω∥ + CB ∥∇ωt ∥2 .
ρ C1 ∥u1t ∥ρ+2 ∥ωt ∥2ρ+2
+
(4.56)
By virtue of (4.45), we have 2
∥∇ω∥2 =
Eω (t ) −
l
1
l
Ω
1
1
l
l
|u1t |ρ ωt2 dx − ∥∇ωt ∥2 − ∥ζ t ∥2M .
(4.57)
Then from (4.52)–(4.57), we can conclude
Φω′ (t ) ≤ −
l−ε 2
∥∇ω∥2 −
l−ε
2
2
l
1
Eω (t ) −
l
1
Ω
1
|u1t |ρ ωt2 dx − ∥∇ωt2 ∥ − ∥ζ t ∥2M l
l
+ CB ∥∇ωt ∥2 + CB ∥ω∥(∥∇ u1tt ∥ + ∥∇ u2tt ∥) + C1 ∥σ1 − σ2 ∥ ∥ω∥ + ≤−
l−ε 2
1−l 4ε
∥ζ t ∥2M
∥∇ω∥2 − (l − ε)Eω (t ) + Cε ∥∇ωt ∥2 + Cε ∥ζ t ∥2M
+ C1 ∥σ1 − σ2 ∥ ∥ω∥ + CB ∥ω∥(∥∇ u1tt ∥ + ∥∇ u2tt ∥).
(4.58)
Now we define
Ψω (t ) =
ρ
Ω
(1ωt − |u1t | ωt )
+∞
g (s)ζ (s)ds dx. t
(4.59)
0
From (4.42) and integration by parts, we derive
Ψω′ (t ) =
u2tt (|u1t |ρ − |u2t |ρ )
Ω
+∞
g (s)ζ t (s)dsdx + l
Ω
0
+∞
g (s)ζ t (s)ds
+ Ω
2
dx + Ω
0
ωt
+∞
g (s)ζ t (s)dsdx
∇ω 0
+∞
g (s)ζ t (s)dsdx 0
+∞ +∞ (σ1 − σ2 ) g (s)ζ t (s)dsdx − ρ |u1t |ρ−1 u1tt ωt g (s)ζ t (s)dsdx Ω 0 Ω 0 +∞ +∞ + 1ωt g (s)ζtt (s)dsdx − |u1t |ρ ωt g (s)ζtt (s)dsdx
−
Ω
=
8
Ω
0
0
Bi .
(4.60)
i=1
Using Hölder’s inequality, Poincaré’s inequality and Theorem 4.1, we derive for any δ ∈ (0, 1), ρ−1
ρ−1
B1 ≤ ∥u2tt ∥2(ρ+1) (∥u1t ∥2(ρ+1) + ∥u2t ∥2(ρ+1) )∥ωt ∥
≤ CB ∥∇ u2tt ∥ ∥ωt ∥
+∞ 0
0
+∞
g (s)ζ t (s)ds
2(ρ+1)
t g (s)∇ζ (s)ds
1
≤ CB (1 − l) 2 ∥∇ u2tt ∥ ∥ωt ∥ ∥ζ t ∥M 1
≤ CB (1 − l) 2 ∥∇ u2tt ∥ ∥ωt ∥(∥η1t ∥M + ∥η2t ∥M ) ≤ CB′ ∥∇ u2tt ∥ ∥ωt ∥, B2 ≤ δ∥∇ω∥2 +
(1 − l)l ∥ζ t ∥2M 4δ
B3 ≤ (1 − l)∥ζ t ∥2M , B4 ≤ δ∥ωt ∥2 +
λ
(4.61)
2
2
4δ
(1 − l)∥ζ t ∥2M ,
(4.62) (4.63) (4.64)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
13
1
B5 ≤ λ(1 − l) 2 ∥σ1 − σ2 ∥ ∥ζ t ∥M ≤ C1 ∥σ1 − σ2 ∥2 + C1 ∥ζ t ∥2M
(4.65)
+∞ ρ−1 t B6 ≤ ∥u1t ∥2(ρ+1) ∥u1tt ∥2(ρ+1) ∥ωt ∥ g ( s )ζ ( s ) ds 0 2(ρ+1) +∞ t ≤ CB ∥∇ u1tt ∥ ∥ωt ∥ g (s)∇ζ (s)ds 0
1
≤ CB (1 − l) 2 ∥∇ u1tt ∥ ∥ωt ∥ ∥ζ t ∥M ≤ CB′ ∥∇ u1tt ∥ ∥ωt ∥.
(4.66)
Note that +∞
g (s)ζtt (s)ds =
+∞
g (s)ds · ωt −
g (s)ζst (s)ds 0
0
0
+∞
= (1 − l)ωt +
+∞
g ′ (s)ζ t (s)ds, 0
then we see that B7 ≤ −(1 − l)∥∇ωt ∥2 −
+∞
g ′ (s)∥∇ωt (t )∥ ∥∇ζ t (s)∥ds 0
≤ −
1−l 2
∥∇ωt ∥2 −
1
2(1 − l)
+∞
g ′ (s)∥∇ζ t (s)∥2 ds,
(4.67)
0
+∞ ρ ′ t |ut |ρ ωt2 dx + ∥u1t ∥2(ρ+1) ∥ωt ∥2(ρ+1) g ( s )ζ ( s ) ds 0 Ω +∞ 2 C1 λ g (0) ≤ −(1 − l) |ut |ρ ωt2 dx + δ∥∇ωt ∥2 − g ′ (s)∥∇ζ t (s)∥2 ds. 4δ Ω 0
B8 ≤ −(1 − l)
(4.68)
Plugging (4.61)–(4.68) into (4.60), we get
Ψω′ (t ) ≤ CB′ (∥∇ u1tt ∥ + ∥∇ u2tt ∥)∥ωt ∥ + δ∥∇ω∥2 + C1 ∥ζ t ∥2M + δ∥ωt ∥2 + C1 ∥σ1 − σ2 ∥2 − +∞
g ′ (s)∥∇ζ t (s)∥2 ds − (1 − l)
− C1
0
Ω
1−l 2
|u1t |ρ ωt2 dx.
− δ ∥∇ωt ∥2 (4.69)
On the other hand, we can get
1 ρ |Ψω (t )| ≤ (1 − l) 2 ∥∇ωt ∥ ∥ζ t ∥M + ∥u1t ∥ρ+2 ∥ωt ∥ρ+2 1 2
1 2
+∞ 0
g (s)ζ t (s)ds
ρ
≤ (1 − l) ∥∇ωt ∥ ∥ζ ∥M + C1 (1 − l) λ∥∇ u1t ∥ ∥∇ωt ∥ ∥ζ t ∥M t
≤ CB (∥∇ωt ∥2 + ∥ζ t ∥2M ) ≤ CB Eω (t ).
(4.70)
Define Gω (t ) = MEω (t ) + ϵ Φω (t ) + Ψω (t ),
(4.71)
which, together with (4.51) and (4.70), yields
(M − CB ϵ − CB )Eω (t ) ≤ Gω (t ) ≤ (M + CB ϵ + CB )Eω (t ).
(4.72)
Now we take ε > 0 so small and M so large that M 2
Eω (t ) ≤ Gω (t ) ≤ 2MEω (t ).
(4.73)
Then for any t ≥ τ , we have G′ω (t ) ≤ −(l − ε)ϵ Eω (t ) − (M − δ)∥ωt ∥2 −
−
1−l 2
M 2
l−ε ξ2 − C1 ξ2 − C1 ∥ζ t ∥2M − − δ ∥∇ω∥2 2
− δ − Cε ϵ ∥∇ωt ∥2 + C (B, M , ε)(∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥)
+ C (B, M , ε)(∥ωt ∥ + ∥ω∥)∥σ1 − σ2 ∥ + C1 ∥σ1 − σ2 ∥2 .
(4.74)
14
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
Now we take δ > 0 and ϵ > 0 so small that 1−l
− δ − Cε ϵ > 0,
2
l−ε 2
− δ > 0.
For fixed ϵ and δ , we choose M so large that M
ξ2 − C1 ξ2 − C1 > 0,
2
M − δ > 0.
Then there exist some constant β > 0 such that G′ω (t ) ≤ −β Eω (t ) + C1 (∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥) + C1 ∥σ1 − σ2 ∥2 + C1 ∥σ1 − σ2 ∥(∥ωt ∥ + ∥ω∥)
≤−
β 2M
Gω (t ) + C1 (∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥) + C1 ∥σ1 − σ2 ∥2 + C1 ∥σ1 − σ2 ∥(∥ωt ∥ + ∥ω∥). (4.75)
Integrating (4.75) over (τ , t ) with respect to t, we derive β
Gω (t ) ≤ Gω (τ )e− 2M (t −τ ) + C1 t
+ C1
β
− 2M (t −s)
e τ
≤ Gω (τ )e− 2M (t −τ ) + C1 + C1
τ
β
e− 2M (t −s) ∥σ1 − σ2 ∥2 ds
(∥∇ u1tt ∥ + ∥∇ u2tt ∥)(∥ωt ∥ + ∥ω∥)ds + C1
β
t
t
t
τ
(∥ωt ∥2 + ∥ω∥2 )ds
21
∥σ1 − σ2 ∥ ds 2
τ
21
t
τ
12
(∥ωt ∥ + ∥ω∥ )ds 2
2
τ
β
e− 2M (t −s) ∥σ1 − σ2 ∥(∥ωt ∥ + ∥ω∥)ds
t
+ C1
t
τ
∥σ1 − σ2 ∥2 ds
.
(4.76)
For any fixed ε ∈ (0, 1), we choose T > τ so large that β
Gω (τ )e− 2M (T −τ ) ≤ ε, which, together with (4.73) and (4.76), gives Eω (t ) ≤ ε + C1 t
C1
τ
t
τ
21
(∥ωt ∥ + ∥ω∥ )ds 2
2
+ C1
τ
t
∥σ1 − σ2 ∥2 ds
21 12 t (∥ωt ∥2 + ∥ω∥2 )ds := ε + φT ((uτ01 , uτ11 , η1τ ), (uτ02 , uτ12 , η2τ ); σ1 , σ2 ). ∥σ1 − σ2 ∥2 ds
(4.77)
τ
It suffices to show φT (·, ·, ·, ·) ∈ Contr(B, Σ ) for each fixed T > τ . From the proof of existence theorem, we can deduce that for any fixed T > τ , and the bound B depends on T ,
Uσ (t , τ )B
(4.78)
σ ∈Σ t ∈[τ ,T ]
is bounded in H . Let (un , unt , ηnt ) be the solutions corresponding to initial data (uτ0n , uτ1n , ηnτ ) ∈ B with respect to symbol σn ∈ Σ , n = 1, 2, . . . . Then from (4.78), we get un → u
⋆ -weakly in L∞ (0, T ; H01 (Ω )),
unt → ut
⋆ -weakly in L (0, T ; ∞
H01
(Ω )).
(4.79) (4.80)
Taking u1 = un , u2 = um , σ1 = σn , σ2 = σm , noting that compact embedding H01 (Ω ) ↩→↩→ L2 (Ω ), passing to a subsequence if necessary, we have un and unt converge strongly in C ([τ , T ]; L2 (Ω )). Therefore we get T
∥unt − umt ∥2 ds → 0,
τ
as m, n → +∞,
(4.81)
T
τ
∥un − um ∥2 ds → 0,
as m, n → +∞.
(4.82)
Y. Qin et al. / Nonlinear Analysis 101 (2014) 1–15
15
On the other hand, by σm , σn ∈ Σ , we see that T
τ
∥σn − σm ∥2 ds → 0,
as m, n → +∞.
(4.83)
Hence it follows from (4.81)–(4.83) that t φT ((un , unt , ηnt ), (um , umt , ηm ); σn , σm ) → 0 as m, n → +∞,
(4.84)
that is, φT ∈ Contr(B, Σ ). Therefore by Lemma 3.1, the semigroup {Uσ (t , τ )}(t ≥ τ > 0, σ ∈ Σ ) is uniformly asymptotically compact and the proof is now complete. Proof of Theorem 2.3. Combining Theorems 4.1–4.2, we can complete the proof of Theorem 2.3.
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