Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 658052, 7 pages
doi:10.1155/2012/658052
Research Article
Existence of Global Attractors for the Nonlinear
Plate Equation with Memory Term
Lifang Niu and Jianwen Zhang
School of Mathematics, Taiyuan University of Technology, Yingze 79 West Street, Shanxi,
Taiyuan 030024, China
Correspondence should be addressed to Jianwen Zhang, niulifang@tyut.edu.cn
Received 21 July 2012; Accepted 6 September 2012
Academic Editor: Sebastian Anita
Copyright q 2012 L. Niu and J. Zhang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A two-dimensional nonlinear plate equation is revisited, which arises from the model of the
viscoelastic thin rectangular plate with four edges supported. We establish that the system is
exponentially decayed if the memory kernel satisfies the condition of the exponential decay.
Furthermore, we show the existence of the global attractor by verifying the condition C.
1. Introduction
In this paper, we investigate the nonlinear plate equation with memory type:
2 ρutt r1 r2 ut φ02 u − N1 β|ux |2 uxx − N2 βuy uyy
∞
φ s ut − sds r1 f x, y ,
2
1.1
in Ω × R ,
0
verifying the initial conditions:
u x, y, 0 u0 x, y, t
1.2
u 0, y, t u 1, y, t ux, 0, t ux, 1, t 0,
uxx 0, y, t uxx 1, y, t uyy x, 0, t uyy x, 1, t,
1.3
and the boundary conditions:
2
Mathematical Problems in Engineering
where x, y ∈ 0, 1, t ∈ R, η, r1 , r2 , N1 , N2 , β are nonnegative constants, Ω 0, 1 × 0, 1 is a
bounded domain with boundary ∂Ω, φ0, φ∞ > 0 and φ s < 0 for every s ∈ R .
The asymptotical behavior of solutions for the nonlinear plate equations had been
studied by many authors 1–10; of those, Santos and Junior 5 studied a kind of plate
equation with memory type. Yang and Zhong 11 studies the plate equation:
utt αxgut Δ2 u λu fu hx,
x ∈ Ω,
1.4
where Ω ⊂ Rn is a bounded domain and proves the existence of a global attractor in the space
H01 Ω × L2 Ω. After Yang and Zhong 11, Yue and Zhong 12 obtained the existence of
a global attractor about some equations similar to 1.4. Xiao 13 discusses the long-time
behavior of the plate equation:
εutt Δ2 u λut βxu fx, u,
x ∈ Ω Rn , t ≥ 0
1.5
on the unbounded domain Rn and show that there exists a compact global attractor for the
above equation satisfying certain initial-boundary data. Wang and Zhang 14 prove that the
two-dimensional nonlinear equation
T
T
2
2
u dx dy uxx − N2 u dx dy uyy 0
ρutt DΔ u εμut − N1 2 Ω x
2 Ω y
2
has a global attractor in space H01 Ω
1.6
L2 Ω.
2. Preliminaries
We denote by H L2 Ω, V H02 Ω endowed with the scalar product and the norm on
dy, u, v H and V ·, ·, | · |, ·, ·, · , respectively, where u, v Ω ux, yvx, ydx
2
Δux,
yΔvx,
ydx
dy.
Define
DA
{v
∈
V,
Av
∈
H},
where
A
Δ
.
For
the operator
Ω
A, we assume that A : DA → H are isomorphism, and there exists α > 0 such that Au, u ≥
αu2 , for all u ∈ V . We also define the power As of A for s ∈ R which operates on the spaces
DAs , and we write V2s DAs , s ∈ R. This is a Hilbert space with the inner product and
norm defined
u, v2s As u, As v,
u2s u, v2s 1/2 ,
∀u, v ∈ DAs ,
2.1
and Ar is an isomorphism from DAs onto DAs−r , for all s, r ∈ R. In particular, DA0 H, DA1/2 V, DA−1/2 V ∗ , DA ⊂ V ⊂ H H ∗ ⊂ V ∗ , where H ∗ , V ∗ are the
dual space, respectively, and each space is dense in the following one and the injections are
continuous. Using the Poincáre inequality we have
v ≥ λ1 |v|,
where λ1 denotes the first eigenvalue of A1/2 .
∀v ∈ V,
2.2
Mathematical Problems in Engineering
3
Let μs −φ s, φ∞ 1 and assume that the memory kernel μ is required to
satisfy the following assumptions:
h1 μ ∈ C1 R L1 R , μ s ≤ 0, for all s ∈ R ;
∞
h2 0 μsds M > 0;
h3 μ s αμs ≤ 0, for all s ∈ R , α > 0.
In view of h1 , let L2μ R , H02 be the Hilbert space of H02 -valued functions on R ,
endowed with the following inner product and the norm:
φ, ψ
and |φ|2μ,V φ, φμ,V ∞
0
μ,V
∞
μs Δφs, Δψs ds
2.3
0
μsφ2 ds. Finally, we introduce the following Hilbert spaces:
H0 V × H × L2μ R , V ,
H1 DA × V × L2μ R , DA.
2.4
We define
ηt x, y, s u x, y, t − u x, y, t − s η x, y, s ,
2.5
and 1.1 is transformed into the system
2 ρutt r1 r2 ut 2 u − N1 β|ux |2 uxx − N2 βuy uyy
∞
μs2 ηds r1 f x, y ,
2.6
0
ηtt ηst ut ,
where the second equation is obtained by differentiating 2.5. The corresponding initialboundary value conditions are then given by
u 0, y, t u 1, y, t ux, 0, t ux, 1, t 0,
uxx 0, y, t uxx 1, y, t uyy x, 0, t uyy x, 1, t,
ηt 0, y, s ηt 1, y, s ηt x, 0, s ηt x, 1, s 0,
t
t
t
t
0, y, s ηxx
1, y, s ηyy
ηxx
x, 0, s ηyy
x, 1, s 0,
u x, y, 0 u1 x, y ,
ut x, y, 0 u2 x, y ,
η0 x, y, s u0 x, y, 0 − u0 x, y, −s ,
where u1 x, y u0 x, y, t, u2 x, y ∂t u0 x, y, t|t0 , x, y ∈ 0, 1, t ≥ 0, s ∈ R .
2.7
4
Mathematical Problems in Engineering
According to the classical Faedo-Galerkin method it is easy to obtain the existence and
uniqueness of solutions and the continuous dependence to the initial value, so we omit it and
only give the following theorem.
Theorem 2.1 see 15, 16. Let h1 hold and fx, y ∈ L2 Ω. Then given any time interval I 0, 1, problems 2.6-2.7 have a unique solution u, ut , η in I with initial data u1 , u2 , η0 ∈ H0 ,
and the mapping u1 , u2 , η0 → ut, ut t, ηst is continuous in H0 .
Thus, it admits to define a C0 semigroup
St : u1 , u2 , η0 −→ ut, ut t, ηst ,
t ∈ R ,
2.8
and they map H0 into themselves.
In addition, the following abstract results will be used in our consideration.
Theorem 2.2 see 17. A C0 semigroup {St}t≥0 in a Banach space X is said to satisfy condition
C which arised by [17] if for any ε > 0 and for any bounded set B of X, there exists tB > 0 and a
finite dimensional subspace X1 of X, such that {P StxX , x ∈ B, t ≥ tB} is bounded and
I − P StxX <X ε,
2.9
where P : X → X1 is a bounded projector.
Lemma 2.3 see 17. Let {St}t≥0 be a C0 semigroup in a Hilbert space M. Then {St}t≥0 has a
global attractor if and only if
1 {St}t≥0 satisfies the condition C;
2 there exists a bounded absorbing subset B of M.
3. Global Attractor Λ in H0
Theorem
3.1. Assume h1 –h3 hold. Then the ball of H0 , B0 BH0 0, R1 , centered at 0 with
R1 c/c0 ρ0 , is a bounded absorbing set in H0 for the semigroup {St}t≥0 .
Proof. We fixed δ and take δ ∈ 0, δ0 , where δ0 minr1 r2 /4ρ, λ21 /2r1 r2 .
First, taking the inner product of the first equation of 2.6 with v ut δu, after
computation we conclude
2 β
β 4
1 d
ρ|v|2 |Δu|2 N1 |ux |2 N2 uy |ux |4 uy 2 dt
2
2
2
4
I1 N1 δ|ux |2 N2 δuy βδ|ux |4 βδuy η, v μ,V r1 f, v ,
3.1
where I1 −δρ|v|2 r1 r2 |v|2 δ|Δu|2 − r1 r2 δ − ρδ2 u, v, and we can easily obtain
I1 ≥
δ
1
|Δu|2 r1 r2 |v|2 .
2
2
3.2
Mathematical Problems in Engineering
5
Combining with the second equation of 2.6 we have
η, v
1 d 2
|η|μ,V η, ηs μ,V δ η, v μ,V .
2 dt
μ,V
3.3
According to h2 –h3 , we conclude
η, ηs
μ,V
1
2
∞
μs
0
d t 2
Δη s ds
ds
2
d
1 ∞ α 2
μ s Δηt s ds ≥ ημ,V ,
2 0
ds
2
∞
δ
μs Δηt s, Δu ds
−
δ η, v μ,V
0
≥ −δ
∞
1/2 ∞
1/2
2
μsΔηt s ds
·
μs|Δu|2 ds
0
≥−
∞
α
4
μsds −
0
δ2
α
∞
3.4
0
μs|Δu|2 ds
0
α 2
Mδ
≥ − ημ,v −
|Δu|2 .
4
α
2
Integrating with 3.4, from 3.3, entails
η, v
μ,V
≥
1 d 2
α 2
Mδ2
η μ,v ημ,v −
|Δu|2 .
2 dt
4
α
3.5
Write Et {ρ|v|2 |Δu|2 N1 |ux |2 N2 |uy |2 β/2|ux |4 β/2|uy |4 |η|2μ,V }, from 2.7,
3.2, and 3.5, we obtain
d
2Mδ
r1 r2 2 α 2
Et δ 1 −
|Δu|2 |v| η μ ,V
dt
α
2
2
4 2r1 2
2
f .
2δN1 |ux | 2δN2 uy δβ|ux |4 δβuy ≤
r2
3.6
2
Take δ small enough, such that 1 − 2Mδ/α > 1/2. Write c 2r1 /r2 |f|2 , c0 min{δ/2, r1 r2 /2ρ, α/2}, thus in line with 3.6, we have
d
Et c0 Et ≤ C.
dt
3.7
By the Gronwall Lemma, we conclude
Et ≤ E0 exp−c0 t c
1 − expc0 ,
c0
∀t ≥ 0.
3.8
6
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Due to 3.8, write ρ0 minρ, 1, we have
2
1
c 1 − expc0 ,
|Δu|2 | |v|2 | ημ,V ≤ E0 exp−c0 t ρ0
c 0 ρ0
Write R1 ∀t ≥ 0.
3.9
c/c0 ρ0 , we end up with
2 lim sup u2 | |v|2 | ημ,V ≤ R21 ,
∀t ≥ 0.
3.10
Theorem 3.2. Suppose f ∈ L2 Ω and conditions h1 –h3 are hold. Then the solution semigroup
{St}t≥0 associated with system 2.6 and 2.7 has a global attractor Λ in H0 , and it attracts all
bounded subsets of H0 , in the norm of H0 .
Proof. Applying Lemma 2.3, we only to prove that the condition C holds in H0 .
Let w
i be an orthonormal basis of DA which consists of eigenvectors of A. It is also
an orthonormal basis of H, V , respectively. The corresponding eigenvalues are denoted by
1 < λ
2 < λ
3 < · · · ,
0<λ
λi −→ ∞,
as i −→ ∞
3.11
i , for all i ∈ N. We write Hm span{w
1 , w
2 , . . . , w
with A
wi λi w
m }. For any u, ut , η ∈ H0 ,
we decompose that u, ut , η u1 , u1t , η1 u2 , u2t , η2 , where u1 , u1t , η1 Pm u, Pm ut , Pm η,
and Pm : H → Hm is the orthogonal projector. Since f ∈ H, for any ε > 0, there exists some
m, such that
I − Pm f < ε .
4
3.12
Taking the scalar product of the first equation of 2.6 in H with v2 u2t Au2 , combining
with the second equation and using the same way with Theorem 3.1, we find
2
d ≤ 2r1 f 2 ≤ r1 ε ,
Et C0 Et
dt
r2
8r2
3.13
{ρ|v2 |2 |Δu2 |2 N1 |u2x |2 N2 |u2y |2 β/2|u2x |4 β/2|u2y |4 |η2 |2 . According
where Et
μ,V
to Theorem 3.1, some M0 and t0 exist, for any t ≥ t0 , such that Et0 ≤ M0 . By the Gronwall
Lemma, we conclude
2
≤ Et0 exp−c0 t − t0 r1 ε
Et
8c0 r2
r1 ε2
≤ M0 exp−c0 t − t0 ,
8c0 r2
3.14
∀t ≥ t0 .
Mathematical Problems in Engineering
7
Take t1 large enough, such that t1 − t0 ≥ 1/c0 ln M0 /ε2 , so we conclude
2
1
r1
1
ε2 ,
|v2 | |Δu2 | η2 μ,V ≤
ρ0
8c0 r2
2
2
for t ≥ t0 .
3.15
Thus S{t}t≥0 satisfies the condition C.
Acknowledgments
Project supported by the Nature Science Foundation of Shanxi province Grant no.
200611005 and Natural Science Foundation of China Grant no. 10772131.
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