doi:10.1155/2011/879649
Research Article
Exponential Stability of Two Coupled
Second-Order Evolution Equations
Qian Wan and Ti-Jun Xiao
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Correspondence should be addressed to Ti-Jun Xiao,[email protected]
Received 30 October 2010; Accepted 21 November 2010
Academic Editor: Toka Diagana
Copyrightq2011 Q. Wan and T.-J. Xiao. 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.
By using the multiplier technique, we prove that the energy of a system of two coupled second order evolution equationsone is an integro-differential equationdecays exponentially if the convolution kernelkdecays exponentially. An example is give to illustrate that the result obtained can be applied to concrete partial differential equations.
1. Introduction
Of concern is the exponential stability of two coupled second-order evolution equationsone is an integro-differential equationin Hilbert spaceH
ut Aut αut−t
0
kt−sAusdsβ√Avt 0, 1.1
vt Avt β√Aut 0, 1.2
with initial data
u0 u0, v0 v0, u0 u1, v0 v1. 1.3
HereA : DA ⊂ H → H is a positive self-adjoint linear operator,α 0,β > 0,ktis a nonnegative function on0,∞. Moreover, the fractional powerA1/2is defined as in the well
An interesting and difficult point for it is to stabilize the whole system via the damping effect given by only one equation1.1. We remark that there is very few work concerning the situation when the damping mechanism is given by memory terms; see3, where a coupled Timoshenko beam system is investigated.
On the other hand, the stability of the single integro-differential equation has been studied extensively; see, for instance,4,5.
In this paper, through suitably choosing multipliers for the energy together with other techniques, we obtain the desired exponential decay result for the system1.1–1.3. Nonlinear coupled systems with general decay rates will be discussed in a forthcoming paper. InSection 2, we present our exponential decay theorem and its proof. An application is given inSection 3.
2. Exponential Decay Result
We start with stating our assumptions:
1Ais a self-adjoint linear operator inH, satisfying
Au, u a u 2, u∈DA, 2.1
wherea >0.
2α > 0,β >0 are constants.kt:0,∞ → 0,∞is locally absolutely continuous, satisfying
k0>0, 1−
∞
0
ktdt− 1
aα−β2>0, 2.2
and there exists a positive constantλ, such that
kt
−λkt, for a.e. t 0. 2.3
We define the energy of a mild solutionuof1.1–1.3as
Et Eut: 12ut212vt21−
t
0ksds
2
√Aut2
1
2
t
0
kt−s√Aus−√Aut2ds
1
2
√Avt βut21
2
α−β2 ut 2.
2.4
Theorem 2.1. Let the assumptions be satisfied. Then,
ifor any u0, v0 ∈ D
√
A and u1, v1 ∈ H, problem 1.1–1.3 admits a unique mild
solution on0,∞.
The solution is a classical one, ifu0∈DA, v0∈DAandu1, v1∈D
√
A,
iithere exists a constantC >0such that the energy
EtE0e
1−Ct, ∀t 0, 2.5
for any mild solution of 1.1–1.3.
Proof. We denote
wt
ut
vt , w0t
u0t
v0t
, w1t
u1t
v1t
,
A
Aα β√A
β√A A , Bt
ktA 0 0 0 .
2.6
Then,1.1–1.3becomes
wt Awt−t
0
Bt−swsds0 t∈0,∞,
w0 w0, w0 w1,
2.7
in H : H ×H. From the assumptions, one sees that—A is the generator of a strongly continuous cosine function on H, and B· is bounded from DA into Wloc1,10,∞;H. Therefore, we justify the assertioni cf., e.g.6.
Suppose now thatuis a classical solution of1.1–1.3. We observe
Et −1
2kt
√Aut21
2
t
0
kt−s√Aus−√Aut2ds
0,
2.8
by Assumption2and so
Let
μ:1−
∞
0
ktdt−α−β
2
a , 2.10
and take 1< δ <1aμ/2β2. We have
Et 1
2u
t21
2v
t2μ
4
√Aut2
1 2−
1 2δ
√Avt2
1−δβ2
2a √
Aut2, t∈0,∞.
2.11
Furthermore, we need the following lemmas.
Lemma 2.2. For anyT S 0and for anyε1 > 0, there exist positive numbersD1ε1,D2ε1
such that
T
S
√Avt βut2dt
D1ES D2 T
S
t
0
kt−s√Aus−√Aut2dsdt
G1
T
S
ut2
dtG2
1
ε1
T
S
vt2
dt G3ε1G4
T
S
√Aut2dt,
2.12
for some positive constantsGi i1,2,3,4which only depend onα,β,a, andk.
Proof. At first, let us take the inner product of both sides of1.1with√Avtand integrate overS, T. Then, noticing1.2, we obtain
T
S
ut,√Avtdt−T
S
√
Aut, vtdt−βT
S
√Aut2dt
α T
S
ut,√Avtdt
T
S
t
0
kt−s√Ausds, vt
dt
β T
S
t
0
kt−s√Ausds,√Aut
dtβ
T
S
√Avt2dt0.
For the first item, integrating by parts, we have
The second and the fifth items can be treated similarly. Therefore,
β
Equation2.15 × β/α 2.16yields that
T
S
√Avt βut2dt
−
T
S
ut, ut
dt
T
S
ut2
dt
−αβ
T
S
ut,√Avtdtβ α
T
S
√
Aut, vtdt
−αβ
T
S
t
0
kt−s√Ausds, vt
dt
β2−α α
T
S
1−
t
0
ksds √Aut2dt
β α
T
S
t
0
kt−s√Aus−√Autds, vt
dt
β α
T
Skt
√
Aut, vtdt
α−αβ2 T
S
t
0
kt−s√Aus−√Autds,√Aut
dt
−β2α−α
T
S
√Avt2dt−α−β2 T
S ut
2dt.
2.17
Next, we will estimate all the terms on the right side of2.17. From2.11, we have the following estimate:
T
S
ut,√Avtdt
ut,√AvtT
S
2MES, 2.18
Using Young’s inequality and noting2.8, we get, forε1>0,
The treatment of the other terms ofJis similar, giving
β
Thus, we obtain
where ζ1 is a positive constant, small enough to satisfy1ζ1θ < 1. We thus verify our
integrate overS, T. It follows that
−
Plugging this equation into2.16, we find
whereδ3μ, and forε2 >0 we get the conclusion.
Lemma 2.4. For anyT S 0, there exist positive numbersD5,D6such that
Proof. Taking the inner product of both sides of1.2withvtand integrating over S, T, we see
Combining this equation and2.16gives
T
Lemma 2.5. LetS0>0be fixed. For anyT S S0and for anyε3>0, there exist positive numbers
Just as in the proofs of the above lemmas, using Young’s inequality and noting that
T
Proof ofTheorem 2.1(continued). From Assumption2and2.8, we have
Moreover, by the use of Lemmas2.4and2.5, we have
Let
ε1ε−1, ηε2ε2, ε3ε5. 2.38
Takingεsmall enough gives
p1<
μ
2, p2< η. 2.39
Therefore, there is a constantN1 >0 such that
T
S
√Aut2dt
T
S
√Avt βut2dtN1ES 2.40
by2.36. UsingLemma 2.4and2.34, we deduce that for someN2>0,
T
S
ut2
dt
T
S
vt2
dt
D52D6
1
λ
ES
3 2
α−β2
a
T
S
√Aut2dt
T
S
√Avt βut2dt
N2ES.
2.41
Next, define
Ht:ut2vt2√Aut2√Avt βut2
t
0
kt−s√Aus−√Aut2ds.
2.42
It is easy to see that there existM1, M2>0 such thatM1EtHtM2Et. Therefore,
T
SHtdt
N1N22λ
ES, T
SEtdt
1
M1
T
SHtdt
N1N22/λ
M1 ES.
On the other hand, when 0SS0,
T
SEtdt
S0
S Etdt
T
S0
Etdt
S0−SES
N1N22/λ
M1 ES0
S0 N1NM22/λ 1
ES,
2.44
that is,
T
SEtdt
NES, ∀S 0. 2.45
By a standard approximation argument, we see that2.45 is also true for mild solutions. From this integral inequality, we complete the proofcf., e.g.,7, Theorem 8.1.
3. An Example
Example 3.1. Consider a coupled system of Petrovsky equations with a memory term
∂2
tut, ξ Δ2ut, ξ αut, ξ−
t
0
kt−sΔ2ut, ξdsβΔvt, ξ 0, t 0, ξ∈Ω,
∂2
tvt, ξ Δ2vt, ξ βΔut, ξ 0, t 0, ξ∈Ω,
ut, ξ vt, ξ Δut, ξ Δvt, ξ 0, t 0, ξ∈∂Ω,
u0, ξ u0ξ, v0, ξ v0ξ, ∂tu0, ξ u1ξ, ∂tv0, ξ v1ξ, ξ∈Ω,
3.1
whereΩis a bounded open domain inÊ
N, with sufficiently smooth boundary∂Ωandα, β, k
as in Assumption2. LetHL2Ωwith the usual inner product and norm. Here, we denote
by∂tuthe time derivative ofuand byΔuthe Laplacian ofuwith respect to space variableξ.
DefineA:DA⊂H → Hby
A Δ2, withDA H4Ω∩H2
0Ω. 3.2
Acknowledgments
The authors would like to thank the referees for their comments and suggestions. This work was supported partially by the NSF of China10771202, 11071042, the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics 08DZ2271900and the Specialized Research Fund for the Doctoral Program of Higher Education of China
2007035805.
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