Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 419717, 25 pages
doi:10.1155/2012/419717
Research Article
Asymptotic Energy Estimates for
Nonlinear Petrovsky Plate Model Subject to
Viscoelastic Damping
Xiuli Lin and Fushan Li
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Correspondence should be addressed to Fushan Li, fushan99@163.com
Received 1 July 2012; Accepted 16 September 2012
Academic Editor: Liming Ge
Copyright q 2012 X. Lin and F. Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the nonlinear Petrovsky plate model under the presence of long-time memory. Under
suitable conditions, we show that the energy functional associated with the equation decays
exponentially or polynomially to zero as time goes to infinity.
1. Introduction
Let Ω be a bounded domain of R2 with a sufficiently smooth boundary Γ Γ0 ∪ Γ1 . We will
assume that Γ0 ∩ Γ1 ∅ and Γ0 , Γ1 have positive measures. The vector ν ν1 , ν2 is the unit
exterior normal and τ −ν2 , ν1 represents the tangential direction to Γ. Here, the variable w
represents displacement of a plate occupying the domain Ω. The governing equation is given
by
w t − γΔw t Δ2 wt Δ2
∞
h swt sds
0
div
Cσt∇wt
1.1
in Ω × 0, ∞,
where the notations wt s wt − s, ∂t . The positive constant γ is the proportional to the
thickness of the plate, ht is relaxation function, div stands for scalar divergence of a vector
field, the stress resultant σt is given by
σt : f∇wt,
1.2
2
Abstract and Applied Analysis
and the nonlinear function f : R2 → M is defined as fs 1/2s ⊗ s for all s ∈ R2 . C is a
linear operator defined on M with value on M, the space of 2 × 2 symmetric matrices, and
defined as
Cσ E
μ tr σ I 1 − μ σ
2
d 1−μ
1.3
for any σ ∈ M, where I is the identity matrix and tr σ denotes the trace of σ. Moreover, d > 0
is the density of the shell, E > 0 denotes the Young modulus, and μ 0 < μ < 1/2 is the
Poisson’s ratio.
With 1.1, we associate the boundary conditions on the portion of the boundary Γ0 ,
w ∂ν w 0
on Γ0 × 0, ∞,
1.4
and the boundary conditions on the remaining portion of the boundary Γ1 ,
B1 wt B1
B2 wt − γ∂ν w t B2
∞
∞
h swt sds 0
on Γ1 × 0, ∞,
0
1.5
h sw sds − Cσtν · ∇wt 0
t
on Γ1 × 0, ∞,
0
where
B1 w Δw 1 − μ B1 w,
B2 w ∂ν Δw 1 − μ ∂τ B2 w
1.6
and the boundary operators B1 and B2 are defined by
B1 w 2ν1 ν2 wxy − ν12 wyy − ν22 wxx ,
B2 w ν12 − ν22 wxy ν1 ν2 wyy − wxx .
1.7
With 1.1, we also associate the initial conditions
w 0 w1
w0 w0 ,
in Ω, ws χs, −∞ < s < 0,
1.8
where
3
2
,
χs, χ s ∈ L∞
c −∞, 0; H × H
1.9
∞
the symbol L∞
c denotes the subspace of L such that there exists a constant T such that the
functions vanish as s < −T .
This problem has its origin in the mathematical description of viscoelastic materials. It
is well known that viscoelastic materials exhibit natural damping, which is due to the special
property of these materials, to retain a memory of their past history.From the mathematical
Abstract and Applied Analysis
3
point of view, these damping effects are modeled by integro-differential operators. Therefore,
the dynamics of viscoelastic materials are of great importance and interest as they have wide
applications in natural sciences. From the physical point of view, the problem 1.1 describes
the position wx, y, t of the material particle x, y at time t, which is clamped in the portion
Γ0 of its boundary.
Models of Petrovsky type are of interest in applications in various areas in
mathematical physics, as well as in geophysics and ocean acoustics 1, 2. The Petrovsky
type models without memory were discussed in 3, 4. Messaoudi 3 considered the initialboundary value problem
utt Δ2 u |ut |m−2 ut |u|p−2 u,
ux, t ∂ν ux, t 0,
x ∈ Ω, t > 0,
x ∈ ∂Ω, t ≥ 0,
ut x, 0 u1 x,
ux, 0 u0 x,
1.10
x ∈ Ω,
established an existence result for 1.10, and showed that the solution continues to exist
globally if m ≥ p, however if m < p and the initial energy is negative, the solution blows up
in finite time. Chen and Zhou 4 proved that the solution of 1.10 blows up with positive
initial energy. Moreover, she claimed that the solution blows up in finite time for vanishing
initial energy under the condition m 2 by different method.
The Petrovsky type equations with memory arouse the attention of mathematicians to
study them. Alabau-Boussouira et al. 5 discussed the initial-boundary value problem of
linear Petrovsky equation related to a plate model with memory,
utt Δ2 u −
t
gt − sΔ2 ut, sds 0 in Ω × 0, ∞,
0
u ∂ν u 0
u|t0 u0 ,
1.11
on ∂Ω × 0, ∞,
ut |t0 u1
in Ω,
and showed that the solution decays exponentially or polynomially as t → ∞ if the initial
data is sufficient small. Yang 6 considered the problem in N-dimensional space,
utt Δ2 u λut u|∂Ω 0,
ux, 0 u0 x,
N
∂
σi uxi in Ω × 0, ∞,
∂x
i
i1
∂u 0 on 0, ∞,
∂n ∂Ω
ut x, 0 u1 x,
1.12
x ∈ Ω,
and proved that under rather mild conditions on nonlinear terms and initial data the abovementioned problem admits a global weak solution and the solution decays exponentially to
zero as t → ∞ in the states of large initial data and small initial energy. In particular, in the
4
Abstract and Applied Analysis
case of space dimension N 1, the weak solution is regularized to be a unique generalized
solution. And if the conditions guaranteeing the global existence of weak solutions are not
valid, then under the opposite conditions, the solutions of above-mentioned problem blow up
in finite time. Muñoz Rivera et al. 7 considered the initial-boundary problem for viscoelastic
plate equation,
utt − γΔutt Δ ut −
2
t
gt − τΔ2 uτdτ 0 in Ω × 0, ∞,
0
u x, y, 0 u0 x, y ,
ut x, y, 0 u1 x, y
in Ω,
u ∂ν u 0 on Γ0 × 0, ∞,
∞
gsut sds 0 on Γ1 × 0, ∞,
B1 ut B1
0
B2 ut − γ∂ν u t B2
∞
gsut sds 0
1.13
on Γ1 × 0, ∞,
0
and proved that the first and second order energies associated with its solution decay
exponentially provided the kernel of the convolution also decays exponentially. When the
kernel decays polynomially then the energy also decays polynomially. More precisely if the
kernel g satisfies
gt ≤ −c0 g 11/p t,
g, g 11/p ∈ L1 R
with p > 2,
1.14
then the energy decays as 1/1 tp . On the recently related papers concerning the Petrovsky
type models, the readers can see references 8–12.
In 13–15, Li et al. proved the existence uniqueness, uniform rates of decay, and
limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells
system, respectively. To our best knowledge, we do not find the research report on the
problem 1.1 which is considered in this paper.
Motivated by the above work, we obtain the energy functional associated with the
equation decays exponentially or polynomially to zero as time goes to infinity. The main
contribution of this paper are as follows. a The problem considered in this paper is nonlinear
equation with integral dissipation, to our knowledge this model has not been considered; b
the hypothesis on h and initial data are weaker; c we naturally define the energy by simple
computation and only define simple auxiliary functionals to prove our result by precise priori
estimates.
The outline of this paper is the following. In Section 2, we present some material
needed to be proved. Section 3 contains the statement and the proof of our results.
Abstract and Applied Analysis
5
2. Notations and Preliminaries
In this section, we will prepare some material needed in the proof of our main results. We use
standard Lebesgue space Lp Ω and Sobolev space H m Ω and adopt the following notations
u, v : u, vL2 Ω ,
u, v : u, vL2 Γ1 ,
up : uLp Ω ,
u : uL2 Ω .
2.1
We denote
|At|2 :
a2ij t,
i,j
d
At : A t aij t ,
dt
At · Bt :
aij tbij t,
i,j
aij t, bij t
At, Bt :
i,j
2.2
for any pair of tensors At aij t and Bt bij t. We introduce the following space
WΩ w ∈ H 2 Ω, w ∂ν w 0 on Γ0 .
2.3
Define the bilinear form a·, · as follows
aw, v Ω
wxx vxx wyy vyy μ wxx vyy wyy vxx 2 1 − μ wxy vxy dx dy.
2.4
For simplicity, we denote aw, w by aw. For the relaxation function ht, we assume that
A1 h : 0, ∞ → 0, ∞ is a C2 function satisfying
ht > 0,
h t < 0,
h t ≥ 0.
2.5
A2 Both h∞ : h∞ and h∞ : h ∞ exist, and
h∞ > 0,
h∞ 0,
h0 1.
2.6
Hypotheses A1 assure that the viscoelastic energy defined below is nonincreasing and the
assumption A2 means that the material behaves like an viscoelastic solid at t ∞ cf.
16.
Our results are based on the following existence theorem.
Theorem 2.1. One assumes that ht satisfies conditions (A1 )-(A2 ). For any initial data
w0 , w1 ∈ H 3 Ω × H 2 Ω
2.7
6
Abstract and Applied Analysis
subject to the compatibility conditions satisfied on the boundary Γ,
∇w0 ∇w1 0, w1 w0 0
∞
0
h sχ−sds 0
B1 w on Γ0 ,
2.8
on Γ1 .
0
Then for any T > 0, there exists a unique global solution to 1.1–1.10 satisfying
w, w , w ∈ L∞ 0, T ; H 3 Ω × H 2 Ω × H 1 Ω .
2.9
Proof. We can use the method used in 13, 17 to show the existence and uniqueness as well
as the regularity of the global solution.
At First, the energy of the system must be properly defined. The total energy can be
expected to consist of two parts. One part involves the current kinetic and strain energies,
and the other will involve the past history of strains. To obtain the appropriate expression of
energy functional, using assumption A2 , we rewrite system 1.1 and 1.5 in the form
w t − γΔw t h∞ Δ wt Δ
2
2
∞
h s wt s − wt ds
0
div
Cσt∇wt in Ω × 0, ∞,
∞
h∞ B1 wt B1
h s wt s − wt ds 0
0
h∞ B2 wt − γ∂ν w t B2
∞
2.10
on Γ1 × 0, ∞,
h s wt s − wt ds
2.11
0
Cσtν · ∇wt 0
on Γ1 × 0, ∞.
In order to define the energy functional, we give the following lemma.
Lemma 2.2. Let w and v be the functions in H 4 Ω ∩ W. Then, one has
Ω
Δ w vdx dy aw, v B2 wv − B1 w∂ν vdΓ.
2
Γ1
2.12
Proof. The definition of aw, v gives
Ω
ΔwΔvdx dy aw, v Ω
1 − μ wxx vyy wyy vxx − 2 1 − μ wxy vxy dx dy.
2.13
Abstract and Applied Analysis
7
Using Green’s formula, we see
Ω
Δ w vdx dy Δw∂ν vdΓ ΔwΔvdx dy
∂ν ΔwvdΓ −
2
Γ1
Γ1
Γ1
∂ν ΔwvdΓ −
Ω
Ω
Γ1
Δw∂ν vdΓ aw, v
2.14
Ω
wxx vyy wyy vxx
1 − μ wxx vyy wyy vxx − 2 1 − μ wxy vxy dx dy,
− 2wxy vxy dx dy wxx vy ν2 wyy vx ν1 dΓ
Γ1
−
−
wxxy vy wxyy vx dx dy
Ω
wxy vy ν1 wxy vx ν2 dΓ
Γ1
Ω
2.15
wxxy vy wxyy vx dx dy
Γ1
wxx vy ν2 wyy vx ν1 dΓ
−
Γ1
wxy vy ν1 wxy vx ν2 dΓ.
Using
∂ν v vx ν1 vy ν2 ,
∂τ v −vx ν2 vy ν1 ,
2.16
vx ∂ν vν1 − ∂τ vν2 ,
vy ∂ν vν2 ∂τ vν1 .
2.17
we get
Inserting 2.14 and 2.15 into 2.17 and noting
Γ1
∂τ wvdΓ Γ
∂τ wvdΓ Γ
wvx dx wvy dy Ω
wvyx − wvxy dxdy 0,
2.18
we have
Γ1
we obtain the conclusion.
w∂τ vdΓ −
Γ1
v∂τ wdΓ,
2.19
8
Abstract and Applied Analysis
In order to define the energy function Et of the problem 1.1–1.8, we give the
following computations. Multiplying equation 2.10 by w , integrating the result over Ω and
adding Green’s formula, we get from Lemma 2.2 and 1.4 that
Ω
w tw tdxdy γ
w t h∞ B2 wt − γ∂ν w t − h∞ ∂ν w tB1 wt dΓ
∞
h s Δ2 wt s − Δ2 wt, w t ds
0
Γ1
Ω
∇w t · ∇w tdxdy h∞ a wt, w t
Γ1
Γ1
Cσt∇wt · νdΓ −
Ω
Cσt∇wt · νdΓ −
Ω
2.20
Cσt∇wt · ∇w tdxdy
Cσt · ∇w t ⊗ ∇wtdxdy.
Clearly
t
Δ2 wt s − Δ2 wt, w t − Δ2 wt s − Δ2 wt, w s − w t
t
t
Δ2 w s − Δ2 wt, w s .
2.21
By applying Lemma 2.2, the first term on the right-hand side of 2.21 equals
1
t
− Δ2 wt s − Δ2 wt, w s − w t − ∂t a wt s − wt
2
t
w s − w t B2 wt s − wt
−
Γ1
t
−∂ν w s − w t B1 wt s − wt dΓ.
2.22
In the same way, the second term on the right-hand side of 2.21 equals
1
t
t
Δ2 w s − Δ2 wt, w s − ∂s a wt s − wt
2
t
w sB2 wt s − wt
Γ1
t
−∂ν w s B1 wt s − wt dΓ.
2.23
Abstract and Applied Analysis
9
Therefore, from 2.21–2.23, we have
1
1
Δ2 wt s − Δ2 wt, w t − ∂t a wt s − wt − ∂s a wt s − wt
2
2
w tB2 wt s − wt − ∂ν w tB1 wt s − wt dΓ.
Γ1
2.24
Note
1
−
2
∞
0
1 d
h s∂t a w s − wt ds −
2 dt
t
∞
h sa wt s − wt ds.
2.25
0
and by h∞ 0
−
1
2
∞
0
∞ 1 ∞ h s∂s a wt s − wt ds −h sa wt s − wt 0 h sa wt s − wt ds
2 0
∞
1
h sa wt s − wt ds.
2 0
2.26
Consequently, we conclude form 2.24–2.26 that
∞
0
1 d ∞ h s Δ w s − Δ wt, w t ds −
h sa wt s − wt ds
2 dt 0
1 ∞ h sa wt s − wt ds
2 0
∞
h s
w tB2 wt s − wt
2
t
2
0
Γ1
−∂ν w tB1 wt s − wt dΓds.
2.27
Summing 2.20 and 2.27, using the boundary conditions 2.11 together with the
symmetric of σt yields
∞
1 d
2
2
w t γ∇w t h∞ awt −
h sa wt s − wt ds
2 dt
0
1 ∞ Cσt · ∇w t ⊗ ∇wtdxdy h sa wt s − wt ds 0.
2
0
Ω
2.28
10
Abstract and Applied Analysis
Note the relation
Cσt · ∇w t ⊗ ∇wtdxdy Ω
1 d
4 dt
Ω
Cσt · σtdxdy.
2.29
In fact
Ω
Cσt · ∇w t ⊗ ∇wtdxdy
wx wx wx wy
wx wx μwy wy
1 − μ wx wy
dxdy
·
α
wy wx wy wy
1 − μ wy wx wy wy μwx wx
Ω
3
2
α
wx 3 wx wy wy wy wx wx wx 2 wy wy dxdy
2.30
Ω
with α E/d1 − μ2 . Denote σ : σσ11
21
σ12
σ22
. Hence
σ μσ
1 − μ σ12
,
Cσ α 11 22
1 − μ σ21 σ22 μσ11
σ11 σ22
σ22 μ σ11
σ22 σ22
σ11
Cσt · σt α σ11
1 − μ σ12
σ12 σ21
σ21
2.31
Cσt · σt ,
which implies that
d
dt
Ω
Cσt · σtdxdy
Ω
Cσt · σtdxdy 2
4α
Ω
Cσt · σ tdxdy
2.32
Ω
Cσt · σtdxdy
Ω
3
2
wx 3 wx wy wy wy wx wx wx 2 wy wy dxdy.
From 2.30 and 2.32, we conclude that 2.29 holds. Combining 2.28 and 2.29, we
conclude that
1 d w t2 γ ∇w t2 h∞ a wt
2 dt
∞
1
h sa wt s − wt ds C σt · σtdxdy
−
2 Ω
0
∞
1
−
h sa wt s − wt ds.
2 0
2.33
Abstract and Applied Analysis
11
The relation 2.33 inspires us to define energy functional as the following
Et 1 w t2 γ ∇w t2
2
∞
t
1
1
h∞ a wt −
h sa w s − wt ds C σt · σtdxdy .
2
2 Ω
0
2.34
From 2.33 and 2.34, we obtain the following lemma.
Lemma 2.3. Under the above notations and assumptions (A1 ) and (A2 ), one has that
d
1
Et −
dt
2
∞
h sa wt s − wt ds ≤ 0.
2.35
0
3. Main Results
In this section, c, c, c, and ci denote some positive constants. Here, we need to point out that
δ denotes the small enough different positive constant and Cδ denotes the different positive
constant depending on δ in a different place, respectively. The main goal of this paper is to
show, respectively, that the solution decays exponentially or polynomially to zero as the time
goes to infinity under suitable conditions. Our main results are formulated below.
Theorem 3.1. Let w be the global solution of the problem 1.1–1.8 with the conditions
h t ≤ ch t,
−c3 h t ≤ h t ≤ −c4 h t.
Then the energy functional Et decays exponentially to zero as the time goes to infinity.
To prove our main result, we will give some important preliminaries.
Lemma 3.2. Assume w ∈ WΩ, functional aw and tensor σt are defined as above. Then
2
2
1 c1 ∇2 w ≤ aw ≤ c2 ∇2 w
2 ∇w2Lq ≤ c3 aw, q ≥ 2
3 w2 ≤ c4 aw
4 c5 |σt|2 ≤ Cσt · σt ≤ c6 |σt|2 .
3.1
12
Abstract and Applied Analysis
Proof. 1 In fact
2
2 dx dy
aw wxx 2 wyy 2μwxx wyy 2 1 − μ wxy
Ω
2
2 2 2 1 − μ wxy
dx dy
≤
wxx 2 wyy μ wxx 2 wyy
Ω
3.2
2 2 2 1 − μ wxy
dx dy,
1 μ wxx 2 wyy
Ω
2
2 dxdy
aw wxx 2 wyy 2μwxx wyy 2 1 − μ wxy
Ω
2
2 2 2 1 − μ wxy
dx dy
≥
wxx 2 wyy − μ wxx 2 wyy
Ω
3.3
Ω
2 2 2 1 − μ wxy
dx dy,
1 − μ wxx 2 wyy
2
2 2 2
2
2
wxx 2 wyy 2 wxy dx dy.
∇ w ∇ w, ∇ w Ω
3.4
Combining 3.2–3.4 and noting 0 < μ < 1/2, we finish the desire conclusion 1.
2 By H 1 Ω → Lq Ω q ≥ 2, the definition of WΩ, and Poincaré inquality, we
know
∇wt2Lq Ω ≤ cwt2H 1 Ω ≤ cwt2WΩ ≤ c3 aw.
3.5
3 Using Poincaré inequality, we get
wt ≤ C∗ ∇wt,
3.6
which with conclusion 2 shows the desire result.
4 Denote σ : σσ11
21
σ12
σ22
. Hence
1 − μ σ12
σ11 μσ
22
,
Cσ α
1 − μ σ21 σ22 μσ11
3.7
with α E/d1 − μ2 > 0, and
2
2
2
2
,
σ22
2μσ11 σ22 1 − μ σ12
σ21
Cσ · σ α σ11
3.8
Abstract and Applied Analysis
13
which imply
2
2
2
2
1 − μ σ12
σ22
σ21
α 1 − μ σ11
2
2
2
2
1 − μ σ12
.
≤ Cσ · σ ≤ α 1 μ σ11
σ22
σ21
3.9
From 3.9 and noting 0 < μ < 1/2, we obtain the desire conclusion 4.
The proof is completed.
Lemma 3.3. Suppose w, v ∈ W and aw, v is defined as above, then
aw, v ≤ δaw, w 9
av, v.
4δ
3.10
Proof. In fact, for η > 0 small enough,
|aw, v| ≤
Ω
≤η
wxx vxx wyy vyy μ wxx vyy wyy vxx 2 1 − μ wxy vxy dx dy
Ω
1
4η
2 2 2 1 − μ wxy
dx dy
1 μ wxx 2 wyy
1μ
≤
η
1−μ
2 2 2 1 − μ vxy
dx dy
1 μ vxx 2 vyy
Ω
3.11
Ω
1μ 1
1 − μ 4η
2 2 2 1 − μ wxy
dx dy
1 − μ wxx 2 wyy
Ω
2 2 2 1 − μ vxy
dx dy.
1 − μ vxx 2 vyy
Denote δ 1 μ/1 − μη, then 1 μ/1 − μ1/4η 1 μ/1 − μ2 1/4δ. Owing
to 1 < 1 μ/1 − μ < 3, from 3.3 and 3.11, we obtain the conclusion.
Lemma 3.4. Define the functional
1
ϕt :
2
Ω
wtw t γ∇w t · ∇wt dx dy,
3.12
then
∞
d
1 w t2 γ ∇w t2 − 1 1 − δ ϕt ≤
h sds aw
dt
2
2
0
∞
t
9
h s
a w s − wt ds.
−
Cσt · σtdx dy 8δ
0
Ω
3.13
14
Abstract and Applied Analysis
Proof. Multiplying 1.1 with w and integrating the result over Ω, we obtain
∞
2
2
h swt sds − div Cσt∇wt wdx dy 0,
w t − γΔw t Δ w Δ
Ω
0
3.14
which with Lemma 2.2 and 1.5 implies
1 d
wtw t γ∇w t · ∇wt dxdy
2 dt Ω
1 w t2 γ ∇w t2 1
wtw t γ∇wt · ∇w t dx dy
2
2 Ω
1
1 2
1
2
w t γ ∇w t wt w t − γΔw t dx dy w∂ν w tdΓ
2
2 Ω
2 Γ1
1 w t2 γ ∇w t2
2
∞
1
2
2
w Δ wΔ
h swt sds
−
2 Ω
0
1
w∂ν w tdΓ
− div
Cσt∇wt dxdy 2 Γ1
2
2 1
1
1 w t γ ∇w t
−
Cσt · ∇wt ⊗ ∇wtdx dy − aw
2
2 Ω
2
∞
∞
1
1
t
− a
h sw sds, w −
wtB2 w h swt s dΓ
2
2 Γ1
0
0
∞
1
1
∂ν wtB1 w h swt s dΓ Cσtν · ∇wwdΓ
2 Γ1
2 Γ1
0
1
1 w t2 γ ∇w t2
w∂ν w tdΓ 2 Γ1
2
∞
1
1
1
t
Cσt · ∇wt ⊗ ∇wtdx dy − aw − a
h sw sds, w
−
2 Ω
2
2
0
1 w t2 γ ∇w t2 −
Cσt · σtdx dy
2
Ω
∞
∞
1
1
−
1
h sds aw − a w,
h s wt s − wt ds .
2
2
0
0
3.15
By Lemma 3.3, we know
∞
∞
1
t
1
t
≤ δawt 9
h s
a w s − wt ds.
a wt,
h
w
−
wt
ds
s
s
2
2
8δ 0
0
3.16
Using 3.15-3.16, we get the conclusion.
Abstract and Applied Analysis
15
Lemma 3.5. Suppose w is the solution of 1.1–1.8 and (A1 ), (A2 ), and 3.1 hold, then
∞
∞
∞
1 0 h swt sds h 0wt 0 h swt sds 0 h s
wt s − wtds;
2 h∞ 0;
∞
∞
∞
3 0 h swt sds h 0wt 0 h swt sds 0 h s
wt s − wtds.
Proof. 1 By the formula of integral by part and A2 , we deduce
∞
∞
∞
t
h sw sds h s∂t w sds −
h s∂s wt sds
t
0
0
∞
−h sw s
0 ∞
t
0
∞
h sw sds h 0wt t
0
∞
h swt sds
0
h s wt s − wt ds.
0
3.17
2 Using A2 and 3.1, we infer the conclusion.
3 By h ∞ 0 and the formula of integral by part, we get the conclusion 3.
The proof is completed.
Lemma 3.6. Define the functional
ψt γ
Γ1
∞
∂ν w t h 0wt h swt sds dΓ
Ω
0
w t − γΔw t
h 0wt ∞
∞
1
a
h swt sds .
2
0
h sw sds dx dy
t
3.18
0
then for some δ δ > 0 small enough, there exist Cδ > 0 and T T > 0 as t > T the following
inequality holds
h 0 d
w t2 γ ∇w t2 δ
ψt ≤
dt
2
∞
t
h s
a w s − wt ds.
Cδ
Ω
Cσt · σtdx dy
3.19
0
∞
Proof. Multiplying 1.1 with h 0wt 0 h swt sds and integrating the result over Ω,
we have
∞
w t − γΔw t Δ2 w Δ2
h swt sds − div
Cσt∇wt
Ω
· h 0wt ∞
0
0
h sw sds dxdy 0,
t
3.20
16
Abstract and Applied Analysis
which with Lemma 3.5 gives
d
dt
∞
h swt sds dxdy
w t − γΔw t h 0wt Ω
0
Ω
∞
h swt sds dxdy
w t − γΔw t h 0wt 0
w t − γΔw t h 0w t Ω
∞
dx dy
h sw sds
t
0
∞
−
h sΔ2 wt sds − div
Cσt∇wt
Δ2 w Ω
3.21
0
∞
× h 0wt h swt sds dxdy
0
∞
t
h sw sds dx dy.
w t − γΔw t h 0w t h 0wt Ω
0
By Green formula, we have
Ω
w t − γΔw t
h 0w t h 0wt 2
2 h 0 w t γ ∇w t − γ
∞
h sw sds dx dy
t
0
Γ1
∂ν w t
∞
h swt sds dΓ
× h 0w t h 0wt 0
γ
Ω
Ω
∇w t · h 0∇wt ∞
3.22
h s∇w sds dx dy
t
0
∞
w t h 0w h swt sds dx dy.
0
Clearly
γ
∞
t
∂ν w t h 0w t h 0wt h sw sds dΓ
Γ1
0
∞
∂ν w t h 0wt h swt sds dΓ
d
dt Γ1
0
∞
∂ν w t h 0wt h swt sds dΓ.
−γ
γ
Γ1
0
3.23
Abstract and Applied Analysis
17
Inserting 3.22 and 3.23 into 3.21 yields
d
dt
w t − γΔw t
Ω
h 0wt ∞
h swt sds dx dy
0
d
dt
γ
Γ1
∞
∂ν w t h 0wt h swt sds dΓ
0
−
Ω
Δ2 w ∞
h sΔ2 wt sds − div
Cσt∇wt
0
× h 0wt ∞
2
2 h swt sds dx dy h 0 w t γ ∇w t
3.24
0
γ
Γ1
γ
Ω
Ω
∂ν w t
∞
h s wt s − wt dΓ
0
∞
∇w t · h 0∇wt h s∇wt sds dx dy
0
∞
w t h 0w h swt sds dx dy.
0
Applying Lemma 2.2, Lemma 3.5 and noting aw, v av, w, we have
−
Δ2 w Δ2
Ω
∞
h swt sds − div
Cσt∇wt
0
∞
∞
× h 0wt h swt sds dx dy −a w,
h s wt s − wt ds
0
0
∞
∞
1 d
a
−
h swt sds,
h swt sds
2 dt
0
0
∞
−
Cσt∇wt ·
h s ∇wt s − ∇wt ds dx dy
Ω
0
∞
−
h s wt s − wt ds
B2 w
Γ1
0
−B1 w∂ν
∞
h s wt s − wt ds dΓ
0
∞
∞
t
h sw sds − Cσtν · ∇wt
h s wt s − wt ds dΓ
−
B2
Γ1
−
0
0
∞
∞
B1
h swt sds ∂ν
h s wt s − wt ds dΓ.
Γ1
0
0
3.25
18
Abstract and Applied Analysis
Inserting 3.25 into 3.24 and owing to 1.5 yield
d
d
ψt dt
dt
Ω
∞
t
h sw sds dxdy
w t − γΔw t h 0wt 0
∞
∞
d
1 d
t
a
h sw sds γ
∂ν w t h 0wt h wt sds dΓ
2 dt
dt Γ1
0
0
2
2 h 0 w t γ ∇w t
−
Ω
Cσt∇wt ·
∞
h s ∇wt s − ∇wt ds dxdy
0
∞
t
h s w s − wt ds
− a w,
0
γ
Ω
Ω
∇w t ·
∞
h s ∇wt s − ∇wt ds dxdy
0
w t
∞
h s ∇wt s − ∇wt ds dxdy.
0
3.26
Now, we estimate some terms on the right-hand side of 3.26. The Lemma 3.3 and 3.1 imply
∞
∞
t
t
a w,
h s
a w s − wt ds.
h s w s − wt ds ≤ δaw, w Cδ
0
3.27
0
Using Cauchy inequality, Hölder inequality, and Lemma 3.2 and noting 3.1, we obtain
∞
t
γ
∇w
h
·
∇w
−
∇wt
ds
dx
dy
t
s
s
Ω
0
2
≤ δγ ∇w t Cδ
∞
t
h s
a w s − wt ds,
0
∞
t
w t
h
∇w
−
∇wt
ds
dx
dy
s
s
Ω
0
2
≤ δw t Cδ
∞
0
t
h s
a w s − wt ds.
3.28
Abstract and Applied Analysis
19
Applying Hölder inequality, Lemma 3.2, and Lemma 2.3, we conclude
∞
t
Cσt∇wt ·
h
∇w
−
∇wt
ds
dxdy
s
s
0
Ω
∞
Cσt ·
h s∇wt ⊗ ∇wt s − ∇wt ds dxdy
0
Ω
∞
h s
∇wt2 ∇wt s − ∇wt2 ds
≤δ
Cσt · σtdxdy Cδ
≤δ
Ω
Ω
0
Cσt · σtdxdy awtCδ
∞
t
h s
a w s − wt ds
3.29
0
∞
t
1
h s
a w s − wt ds
Cσt · σtdxdy E0Cδ
≤δ
h
∞
0
Ω
∞
t
h s
a w s − wt ds.
Cσt · σtdxdy Cδ
δ
Ω
0
Combining 3.26–3.29, we conclude that
h 0 d
ψt ≤
w t2 γ ∇w t2
dt
2
∞
t
h s
a w s − wt ds.
δ
Cσt · σtdxdy Cδ
Ω
3.30
0
The proof is completed.
Proof of Theorem 3.1. Let
Ft P Et Qϕt Rψt.
3.31
By Lemma 2.3, Lemma 3.4, and Lemma 3.6 and noting 3.1, we have
∞
1
2
2
h sa wt s − wt ds Q w t γ∇w t
2
0
∞
1
−Q
h sds awt − Q
Cσt · σtdx dy
1−δ
2
0
Ω
9 ∞ t
h 0 w t2 γ ∇w t2
h s a w s − wt ds R
Q
8δ 0
2
∞
t
h s
a w s − wt ds
Rδ
Cσt · σtdxdy RCδ
1
F t ≤ − P
2
Ω
0
2
2 ≤ K1 w t γ ∇w t K2 awt
K3
Ω
Cσt × σtdx dy K4
∞
0
t
h s
a w s − wt ds,
3.32
20
Abstract and Applied Analysis
with
∞
1
1
K2 − Q 1 − δ h sds − Qh∞ − δ,
2
2
0
1
1
K1 Rh 0 Q,
2
2
K3 −Q Rδ,
1
9Q
RCδ .
K4 − P c3 2
8δ
3.33
Fixing R 1, δ 1/2 min{h∞ , −h 0} > 0, Q 1/2δ − h 0 > 0, P > 9Q/4δc3 2Cδ /c3 > 0, we obtain Ki < 0, i 1, 2, 3, 4. Under the conditions, we obtain from 2.34 and
3.32 that there exists a positive α > 0 such that
d
Ft ≤ −αEt.
dt
3.34
Fixing a large enough positive constant P , we prove Ft and Et are equivalent, that is,
M1 Et ≤ Ft ≤ M2 Et
3.35
with M1 , M2 > 0. Indeed, using Cauchy inequality, Lemma 3.2, and 2.34, we have
Rϕt
Q
γ∇w
·
∇wt
dxdy
wtw
t
t
2
Ω
2
2 Q
wt2 γ∇wt2 w t γ ∇w t ≤ cawt cEt ≤ cEt.
4
3.36
≤
Using the definition of ψt and Green’s formula, we infer
ψt Ω
w t
h s wt s − wt ds dxdy
0
∞
Ω
γ∇w t ·
∞
0
∞
1
h s ∇wt s − ∇wt ds dxdy a
h swt sds .
2
0
3.37
Abstract and Applied Analysis
21
Applying Cauchy inequality, Lemma 3.2 and noting 3.1, we conclude that
Ω
w t
∞
0
t
h s w s − wt ds dxdy
Ω
γ∇w t ·
∞
h s ∇wt s − ∇wt ds dx dy
0
2
2
≤ δw t γδ ∇w t
∞
t
h s
a w s − wt ds,
Cδ
0
3.38
∞
2
1
a
h swt sds ≤ C
∇ h0 − h∞ wtds
2
0
∞
t
2
h s w s − wt 0
≤ cawt c
∞
3.39
t
h s
w s − wt ds.
0
Relations 3.37-3.39 imply that
ψt
≤ cEt.
3.40
Fixing a large enough positive constant P , inequalities 3.36 and 3.40 show 3.35 hold.
Relations 3.34 and 3.35 imply that there exists a positive constant β > 0 such that
d
Ft ≤ −βFt,
dt
3.41
which with Gronwall’s inequality gives
Ft ≤ F0e−βt .
3.42
Combining 3.35 and 3.42, we finish the proof.
Theorem 3.7. Assume that w is the global solution of 1.1–1.8 with condition C|h t|11/p ≤
h t, p > 2. Then for some T > 0 there exists a positive constant C, such that
Et ≤
C
1 tp/2
,
∀t ≥ T.
In order to prove Theorem 3.7, we quote the following technical lemma.
3.43
22
Abstract and Applied Analysis
Lemma 3.8. Suppose that w ∈ L∞ 0, ∞; H 2 Ω and g is a continuous function and there exists
0 < θ < 1, such that
∞
g 1−θ sds < ∞.
3.44
0
Then, there exists C > 0 such that
∞
gsa wt s − wt ds
0
≤C
∞
0
×
∞
1/θp1
g 1−θ sds v2L∞ 0, T ; H 2 Ω
g 11/p sa wt s − wt dxdy
3.45
θp/θp1
.
0
Proof. Applying Hölder inequality, we obtain
∞
gsa wt s − wt ds
0
≤
∞
1/θp1
gs
1−θ a wt s − wt ds
0
t
×
θp/θp1
11/p t
gs
a w s − wt ds
3.46
0
≤C
∞
0
×
∞
1/θp1
g 1−θ sds w2L∞ 0, T ; H 2 Ω
θp/θp1
g 11/p sa wt s − wt ds
.
0
This completes the proof of Lemma 3.8.
Proof of Theorem 3.7. From the hypothesis c|h t|11/p ≤ h t, we have |h |−1/p t ≥ c/p.
Integrating the above inequality over 0, t, we get
h t ≤ c7 1 t−p ,
c7 > 0.
3.47
Fixing θ 1/2 in Lemma 3.8, then 1 − θp p/2 > 1, hence we obtain
∞
0
1−θ
h sds ≤ c8
∞
0
1
1 sp/2
ds < ∞.
3.48
Abstract and Applied Analysis
23
Using this estimate into 3.46, noting 2.34 and Lemma 3.2, we get
∞
h sa wt s − wt ds ≤ c9 E2/p2 0
∞
0
p/p2
11/p t
h a w s − wt ds
.
0
3.49
Denote Gt Qϕt ψt. Applying 3.32 and 3.49, we arrive at
d
Gt ≤ −α1 Et α2
dt
∞
h sa wt s − wt ds
0
≤ −α1 Et α2 c9 E
2/p2
0
∞
p/p2
11/p t
h a w s − wt ds
.
3.50
0
Since |Gt| ≤ cEt, applying Young inequality and from Lemma 3.2, 3.50, we deduce
2
d
d
d 2/p
E tGt GtE2/p−1 t Et E2/p t Gt
dt
p
dt
dt
≤ −
d
2 2/p d
cE t Et E2/p t Gt
p
dt
dt
d 12/p
E
t − α1 E12/p t
dt
p/p2 3.51
∞
11/p
h a wt s − wt ds
α2 C2 E2/p2 0E2/p t
≤ − c10
0
d 12/p
E
t − α1 E12/p t α2 C2 E2/p2 0εE12/p t
dt
∞
11/p t
h a w s − wt ds.
α2 C2 E2/p2 0Cε
≤ − c10
0
Since c|h |11/p t ≤ h t, then
∞
0
11/p t
1
h a w s − wt ds ≤
c
∞
h sa wt s − wt ds.
3.52
0
By Lemma 3.2, we have
∞
d
h sa wt s − wt ds −2 Et.
dt
3.53
11/p t
d
h a w s − wt ds ≤ −c11 Et.
dt
3.54
0
Hence, we get
∞
0
24
Abstract and Applied Analysis
Taking ε small enough, from 3.51 and 3.54, we get
d 2/p
d
d
α1
E tGt ≤ −c10 E12/p t − E12/p t − c12 Et.
dt
dt
2
dt
3.55
d 2/p
α1
d
E tGt c10 Et ≤ −c12 Et − E12/p t.
dt
dt
2
3.56
Thus, we have
Let λ > 0 be a positive constant and define
Ht λEt E2/p t
Gt c10 Et.
3.57
Since |Gt| ≤ cEt, d/dtEt ≤ 0, for λ large enough, we get
λ
Et ≤ Ht ≤ 2λEt,
2
∀t ≥ 0.
3.58
From Lemma 3.2 and 3.56, taking λ large enough, we obtain
d
d
d 2/p
Ht λ Et E tGt C0 Et
dt
dt
dt
≤λ
d
α1
d
Et − c12 Et − E12/p t
dt
dt
2
λ − c12 3.59
d
α1
α1
Et − E12/p t ≤ − E12/p t.
dt
2
2
From 3.58 and 3.59, we have
d
Ht ≤ −c13 H 12/p t.
dt
3.60
Applying Gronwall inequality, we get
Ht ≤
c14
1 tp/2
.
3.61
.
3.62
Hence, we obtain
Et ≤
c15
1 tp/2
This completes the proof of Theorem 3.7.
Abstract and Applied Analysis
25
4. Comment
In this paper, the key point is to construct suitable energy functional Et, find the appropriate
auxiliary functionals ϕt and ψt by multiplier method, and prove that Ft and Et
are equivalent and Ft satisfies F t ≤ −cFt by precise apriori estimates. Thanks to
Lemma 3.8 and auxiliary functional Ht, we show the polynomial decay result under
suitable conditions on relaxation h.
Acknowledgment
The authors would like to thank the referee for his/her careful reading and kind
suggestions. The paper was supported by the National Natural Science Foundation of China
11201258, the Natural Science Foundation of Shandong Province of China ZR2011AM008,
ZR2011AQ006, and ZR2012AM010, the STPF of University in Shandong Province of China
J09LA04, and the Science Foundation of Qufu Normal University of China XJ201114.
References
1 M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 3 of Scattering Theory, Academic
Press, New York, NY, USA, 1979.
2 E. Zauderer, Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics, John
Wiley & Sons, New York, NY, USA, 1989.
3 S. A. Messaoudi, “Global existence and nonexistence in a system of Petrovsky,” Journal of Mathematical
Analysis and Applications, vol. 265, no. 2, pp. 296–308, 2002.
4 W. Chen and Y. Zhou, “Global nonexistence for a semilinear Petrovsky equation,” Nonlinear Analysis
A, vol. 70, no. 9, pp. 3203–3208, 2009.
5 F. Alabau-Boussouira, P. Cannarsa, and D. Sforza, “Decay estimates for second order evolution
equations with memory,” Journal of Functional Analysis, vol. 254, no. 5, pp. 1342–1372, 2008.
6 Z. Yang, “Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave
equations with dissipative term,” Journal of Differential Equations, vol. 187, no. 2, pp. 520–540, 2003.
7 J. E. Muñoz Rivera, E. C. Lapa, and R. Barreto, “Decay rates for viscoelastic plates with memory,”
Journal of Elasticity, vol. 44, no. 1, pp. 61–87, 1996.
8 F. Tahamtani and M. Shahrouzi, “Existence and blow up of solutions to a Petrovsky equation with
memory and nonlinear source term,” Boundary Value Problems, vol. 2012, p. 50, 2012.
9 G. Li, Y. Sun, and W. Liu, “Global existence and blow-up of solutions for a strongly damped Petrovsky
system with nonlinear damping,” Applicable Analysis, vol. 91, no. 3, pp. 575–586, 2012.
10 G. Li, Y. Sun, and W. Liu, “Global existence, uniform decay and blow-up of solutions for a system of
Petrovsky equations,” Nonlinear Analysis A, vol. 74, no. 4, pp. 1523–1538, 2011.
11 X. Han and M. Wang, “Asymptotic behavior for Petrovsky equation with localized damping,” Acta
Applicandae Mathematicae, vol. 110, no. 3, pp. 1057–1076, 2010.
12 S.-T. Wu and L.-Y. Tsai, “On global solutions and blow-up of solutions for a nonlinearly damped
Petrovsky system,” Taiwanese Journal of Mathematics, vol. 13, no. 2A, pp. 545–558, 2009.
13 F. S. Li, “Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerrevon Kármán shallow shell equations,” Acta Mathematica Sinica, vol. 25, no. 12, pp. 2133–2156, 2009.
14 F. Li and Y. Bai, “Uniform decay rates for nonlinear viscoelastic Marguerre-von Kármán equations,”
Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 522–535, 2009.
15 F. Li, “Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Kármán shallow shell
system,” Journal of Differential Equations, vol. 249, no. 6, pp. 1241–1257, 2010.
16 J. E. Lagnese, Boundary Stabilization of Thin Plates, vol. 10 of SIAM Studies in Applied Mathematics, SIAM,
Philadelphia, Pa, USA, 1989.
17 I. Lasiecka, “Weak, classical and intermediate solutions to full von Kármán system of dynamic
nonlinear elasticity,” Applicable Analysis, vol. 68, no. 1-2, pp. 121–145, 1998.
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