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Lack of exponential decay in viscoelastic materials with voids
Systems & Control Letters 63 (2014) 39–42
Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Lack of exponential decay in viscoelastic materials with voids✩ Ti-Jun Xiao ∗ , Yuming Zhang Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
article
info
Article history: Received 23 July 2013 Received in revised form 8 November 2013 Accepted 12 November 2013 Available online 1 December 2013
abstract We study the stability of solutions to a coupled evolution system associated with an isotropic porous and centrosymmetric viscoelastic solid with porous dissipation (a porous elastic system with history). With the help of the method of the semigroup theory and some novel observations, we prove successfully that the condition of equal wave-speed propagation is still necessary for exponential stability of the system in the case of Dirichlet–Dirichlet boundary conditions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Porous elastic systems with history Semigroup method Exponential stability
1. Introduction
The initial and boundary conditions we consider here are
Since the work of Goodman and Cowin [1], in which a continuum theory of granular materials with interstitial voids was established, there have appeared many significant studies for various elastic materials with voids covering applications to many fields, such as biology material, soils sciences, petroleum industry, etc. The present paper concerns an isotropic porous and centrosymmetric viscoelastic solid with porous dissipation, whose motion is described by the following coupled evolution system
ρ u = µuxx + bϕx in (0, π ) × R+ , tt J ϕtt = xx − bux − ξ ϕ + g (0)ϕxx δϕ ∞ + g ′ (s)ϕ (t − s) ds in (0, π ) × R+ .
(1.1)
0
Here, the variables u and ϕ represent, respectively, the displacement of the solid elastic material and the volume fraction, g (·) is the memory kernel function, and ρ, J are positive constants; also, the constitutive coefficients µ, δ, ξ , b satisfy
µξ − b2 ≥ 0,
to ensure the nonnegativity of the internal energy. For more related information, we refer to [2–4] and references therein.
✩ The work was supported partly by the NSF of China (11371095), the Ministry of Education, the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Nonlinear Mathematical Modeling and Methods Laboratory of the Ministry of Education. ∗ Corresponding author. Tel.: +86 2165642341. E-mail address: [email protected] (T.-J. Xiao).
0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.11.006
ut (x, 0) = u1 (x),
ϕ(x, −s) = ϕ0 (x, s) (s ≥ 0),
ϕt (x, 0) = ϕ1 (x),
(1.2)
and u(0, t ) = u(π , t ) = ϕ(0, t ) = ϕ(π , t ) = 0,
t ≥ 0,
(1.3)
respectively. Moreover, the kernel function g ∈ C 2 (0, +∞) ∩ C [0, +∞), satisfies the following assumptions
xx
µ, δ, ξ > 0,
u(x, 0) = u0 (x),
g (+∞) = 0,
g ′ (s) < 0
g (s) ≥ c0 |g (s)| for s > 0, ′′
′
for s > 0,
(1.4) (1.5)
where c0 > 0 is a constant. Thus, we can see that g (s) > 0, g ′ ∈ L1 (0, +∞), and the function |g ′ (s)| is of exponential decay type. If
µ = ξ = b,
(1.6)
then (1.1) can be regarded as a system of Timoshenko type, where the variables u and ϕ represent, respectively, the transverse displacement of a beam and the rotation angle of a filament. For a Timoshenko system with memory (but having null history), Ammar-Khodja, Benabdallah, Muñoz Rivera, Rack [5] showed that whether the dissipation given by the memory effect is strong enough to bring the whole system to an exponential decay is essentially related to the condition of equal wave-speed propagation: Jµ
ρ
− δ − g (0) = 0.
(1.7)
40
T.-J. Xiao, Y. Zhang / Systems & Control Letters 63 (2014) 39–42
For some Timoshenko systems with past history, it was also proved that the condition (1.7) implies an exponential decay of the systems (see, e.g., [6,7]). Therefore, for the special case of (1.6), the solutions to (1.1)–(1.3) are exponentially stable if (1.7) holds. Very recently, for the case of Dirichlet–Neumann boundary conditions (instead of Dirichlet–Dirichlet ones in (1.3)) and the kernel function g satisfying (1.4), (1.5) and g ′′ (s) ≤ k1 |g ′ (s)| for s ≥ s1 ,
(1.8)
where s1 and k1 are positive constants, Pamplona, Muñoz Rivera, Quintanilla [4] showed that the semigroup associated with the porous viscoelastic system is exponentially stable if and only if (1.7) holds. Is the condition (1.7) necessary for exponential stability in the case of Dirichlet–Dirichlet boundary conditions? One expects that the answer is yes, yet proving it appears to be a quite tricky job. Actually, the basic question remains open, because so far, one only knows from [7, Section 4] (and [4]) that it is true for the mixed-type boundary conditions, while [7] shows the sufficiency in the case of Dirichlet–Dirichlet conditions. Our purpose in this paper is to solve the problem by giving a proof of the necessity. The rest of this paper is organized as follows. In Section 2, by introducing an auxiliary variable accounting for the past history of ϕ , we rephrase (1.1)–(1.3) as an abstract operator differential system, which is governed by a semigroup, following the approach of Dafermos [8] (see also [4,7,9]). Section 3 is devoted to presenting and proving our main result with respect to the necessity. 2. Semigroup setting
x ∈ [0, π], (t , s) ∈ R+ × R+ ,
W :=
(R , +
H01
π
0
∞
∞
g (s)ψ(·, s) ds ∈ H (0, π ), ψs ∈ W , ψ(0) = 0 . ′
2
0
Set
⟨U , U ⟩H = ∗
π
ρv v¯ ∗ + µux u¯ ∗x + J φ φ¯ ∗ + δϕx ϕ¯ x∗ + ξ ϕ ϕ¯ ∗
0
+ b(ux ϕ¯ + u¯ x ϕ) + ∗
∗
∗
0
for U = (u, v, ϕ, φ, ψ)T , U ∗ = (u∗ , v ∗ , ϕ ∗ , φ ∗ , ψ ∗ )T .
(2.2)
By hypothesis, we have µξ − b ≥ 0. Take ξ1 ∈ (0, ξ ] such that µξ1 − b2 = 0. Then, 2
⟨U , U ⟩H ≥
π
0
2 1 1 ρ|v|2 + J |φ|2 + δ|ϕx |2 + µ 2 ux − ξ12 ϕ
+ (ξ − ξ1 )|ϕ| + 2
∞
|g (s) ∥ ψx | ds dx. ′
2
(2.3)
0
This enables us to see clearly that the above ⟨U , U ∗ ⟩H defines an inner product on H , and the associated norm ∥ · ∥H is equivalent to the usual one. On the basis of the conditions (1.4) and (1.5), and arguing similarly as in [4, page 687], we infer that 0 ∈ ϱ(A) (the resolvent set of A), and hence, A generates a C0 semigroup T (t ) of contractions on (H , ∥ · ∥H ), according to the Lumer–Phillips theorem. Thus, the Cauchy t ≥ 0.
In view of this, one needs only to study the properties of T (t ), for understanding the behaviors of the system (1.1)–(1.3).
Proof. It is well known (see, e.g., [10–12] or [13] for a more general version) that a C0 semigroup of contractions on a Hilbert space, with the generator G, is exponentially stable if and only if
U (t ) = (u, ut , ϕ, ϕt , ψ t )T . Then we can transform (1.1)–(1.3) into a first order Cauchy problem in H as follows U (t ) = AU (t ),
(2.1)
U (0) = (u0 , u1 , ϕ0 (·, 0), ϕ1 , ϕ0 (·, 0) − ϕ0 (·, −s))T .
(i) ϱ(G) ⊃ iR, and (ii) sup{∥(iβ I − G)−1 ∥; β ∈ R} < ∞. Accordingly, we only need to find a sequence {λn } ⊂ iR, and a sequence {Un } ⊂ D (A) such that lim |λn | = ∞,
n→∞
Here,
J 0
|g (s)|ψx ψ¯ x ds dx, ′
Theorem 3.1. If the condition (1.7) of equal wave-speed propagation does not hold, then the semigroup T (t ) is not exponentially stable.
(obviously a Hilbert space), and
0 µ D2 ρ A= 0 b − D
∞
3. Main result
|g ′ (s) ∥ wx (s)|2 ds dx.
H = H01 (0, π ) × L2 (0, π ) × H01 (0, π ) × L2 (0, π ) × W
dt
δϕ +
(u, v, ϕ, φ, ψ) ∈ H ; u ∈ H 2 (0, π ), v, φ ∈ H01 (0, π ),
0
Set
d
D (A) =
U (t ) = T (t )U (0),
(0, π )),
which is the weighted space with respect to the measure |g ′ | ds endowed with the norm
∥w∥2W =
) with the domain
problem (2.1) is well-posed and the solution is given by
and the history space L2g ′
d dx
Re⟨AU , U ⟩H ≤ 0;
Consider the history variable
ψ t (x, s) = ϕ(x, t ) − ϕ(x, t − s),
(D :=
I 0
0
0
0
0
0
0
I
ρ
0 0
0 b D
1 J
(δ D2 − ξ I ) 0
0 I
0 ∞ 1 ′ 2 − g (s)D ds J
0
−∂s
lim ∥Un ∥H = ∞,
n→∞
but the sequence {Fn }, with Fn := (λn I − A)Un , is a bounded sequence in H . Actually, if we can find such two sequences {λn } and {Un }, then there are two cases: (a) at least one of the λn is in the spectrum of A, (b) all of the λn is in ϱ(A).
T.-J. Xiao, Y. Zhang / Systems & Control Letters 63 (2014) 39–42
41
In the first case, the necessary condition (i) for exponential stability (ϱ(A) ⊃ iR) is violated, and in the second case the necessary condition (ii) for exponential stability (the resolvent operator of A is bounded on the imaginary axis) is violated. Therefore, in either case the desired conclusion (the semigroup T (t ) is not exponentially stable) is obtained. In order to find such two sequences {λn } and {Un }, we observe by the definition of the operator A that choosing
where
U = (u, λu, ϕ, λϕ, (1 − eλs )ϕ)T
with
for
f2n := (ρλ2n + µh2n )An sin(hn x) + bBn hn sin(hn x)
u, ϕ ∈ H 2 (0, π ) ∩ H01 (0, π ),
λn := i
Fn = (0, f2n , 0, f4n , 0)T
gives (3.1)
f4n
where f2 := λ2 ρ u − µuxx − bϕx , ∞
(3.3)
0
Set
c=
Jµ , ρ(δ + g (0))
(3.4)
which is not equal to 1 by hypothesis. We take a sequence hn ⊂ 2N (the set of positive even numbers) such that
chn
lim hn = ∞,
−
2
n→∞
chn 2
≤
1
un = An sin(hn x),
(3.6)
ϕn = Bn cos(hn x) − Bn d˜ n cos(h˜ n x − dn );
∞ 0
g ′ (s)eλn s ds.
1 hn
,
An = A := −
1 b
−
Jµ
ρ
h˜ n := chn ,
dn :=
chn 2
−
chn 2
π,
d˜ n :=
1 cos dn
+ δ + g (0) .
f2n = b sin(hn x) − bc d˜ n sin(h˜ n x − dn ), f4n = gn hn cos(hn x) + ξ ϕn − d˜ n −
Then, noting h˜ n := chn and using (3.10), (3.11) gives
here,
(3.11)
Case (i). There exists a constant C > 0 such that |gn hn | ≤ C for n ∈ N. In this case, we put Bn =
([ ch2n ] is the greatest integer less than ch2n ). We note that such a sequence indeed exists. Then, we construct sequences
,
(3.12)
Jµ
ρ
hn
+ δ c 2 hn + g (0)c 2 hn + gn c 2 hn cos(h˜ n x − dn ).
and An , Bn are some numbers to be specified later. It is complex obvious that dn ∈ 0, π4 by (3.5), and so 1 ≤ d˜ n ≤
here, gn stands for the integral
(3.5)
4
(3.10)
+ gn h2n Bn + bAn hn cos(hn x) + ξ ϕn Jµ − − h2n + δ h˜ 2n + g (0)h˜ 2n ρ + gn h˜ 2n Bn d˜ n cos(h˜ n x − dn );
(3.2)
g ′ (s)(1 − eλs )ϕxx ds.
(3.9)
− bBn d˜ n h˜ n sin(h˜ n x − dn ) = bBn hn sin(hn x) − bBn d˜ n h˜ n sin(h˜ n x − dn ), Jµ := − h2n Bn + δ h2n Bn + g (0)h2n Bn ρ
f4 := λ2 J ϕ − δϕxx + bux + ξ ϕ
+
µ hn . ρ
From (3.1)–(3.3) and (3.6), we know
λ ∈ iR
F := (λI − A)U = (0, f2 , 0, f4 , 0),
(3.13)
From the definition of c (in (3.4)), one has
√ 2;
(3.7)
−
moreover, we have h˜ n π − dn = dn + 2
chn 2
+ δ c 2 + g (0)c 2 = 0,
(3.14)
f4n = gn hn cos(hn x) + ξ ϕn − gn hn d˜ n c 2 cos(h˜ n x − dn )
which indicates cos dn = cos(h˜ n π − dn ). Therefore, it follows that ϕn (0) = ϕn (π ) = 0, and thus we see from (3.6) that un , ϕn ∈ H 2 (0, π ) ∩ H01 (0, π ).
∥Un ∥H ≥ ρ (3.8)
(3.15)
by (3.13). This and (3.12) together indicate that both {f2n } and {f4n } are bounded sequences in L2 (0, π ), because {d˜ n }, {gn hn } are bounded by hypothesis (see also (3.7)). Thus, it follows from (3.9) that {Fn } is bounded in H . On the other hand, using (2.2), (2.3), (3.8) and (3.6) yields that 2
Next, we let
Fn = (λn I − A)Un ,
ρ
and hence
π,
Un = (un , λn un , ϕn , λn ϕn , (1 − eλn s )ϕn )T ,
Jµ
π
π
|λn un | dx = ρ |λn An sin(hn x)|2 dx 0 0 π 1 = µA2 h2n | sin(hn x)|2 dx = π µA2 h2n . 2
0
2
42
T.-J. Xiao, Y. Zhang / Systems & Control Letters 63 (2014) 39–42
∥Un ∥2H → ∞,
as n → ∞,
,
An = −
b
(3.16)
1
−
b|gn |hn
sin(hn x) −
|gn |hn
Jµ
f4n =
|gn |
ρ
≥
+ δ + g (0) =
|gn |hn
|gn |
+
c 2 gn
A
|gn |hn
.
gn
|gn |
π
e− h
π −γ h
s−γ − s +
π
e−π
2
e−π γ −1 h 2
5π 6
π
π
e−s− h
π −γ
h
s−γ − s +
π −γ
sin(hs) ds
sin(hs) ds
ds
h
[t −γ − (t + π )−γ ] dt ,
6
so that
h
∞ 0
≥
δc 2 − + ρ|gn | |gn |
∞
g ′ (s)e−ihs ds ≥ Im h
Jµ
e−π γ h 2
g ′ (s)e−ihs ds
0 5π 6
π
[t −γ − (t + π )−γ ] dt
6
→ +∞ as h → +∞.
cos(h˜ n x − dn ).
|gn |
Furthermore, recalling (3.14) gives f4n =
5π 6h
6h
=
sin(h˜ n x − dn ),
cos(hn x) + ξ ϕn − d˜ n
g (0)c 2
+
bc d˜ n
gn
h
Then, from (3.10), (3.11) we obtain f2n =
s−γ e−s − s +
0
and set
|gn |h2n
π
≥
Case (ii). The sequence {gn hn } is unbounded. In this case, we assume, without loss of generality, that
Bn =
0
by virtue of (3.5).
1
h
≥
as n → ∞,
|gn hn | → ∞,
π
Since (1.7) is not satisfied, we have A ̸= 0. Therefore,
cos(hn x) + ξ ϕn −
c 2 d˜ n gn
|gn |
cos(h˜ n x − dn ).
Thus, we see that f2n , f4n are uniformly bounded in L2 (0, π ), according to (3.7) and (3.16). Hence, {Fn } is bounded in H . Also, noting
Accordingly, the boundedness of the sequence {gn hn } cannot be guaranteed in general, and so we need take both cases into account. (2) The main results in [4] are still true since they can be proved without using [4, Lemma 2.8]. (3) From the proof of Theorem 3.1, we see that once the setting of A generating a C0 semigroup T (t ) of contractions on (H , ∥ · ∥H ) is insured (by the conditions (1.4) and (1.5) in this work), then we can obtain the desired conclusion using only the following properties of g: g ′ ∈ L1 (0, +∞),
|gn | → 0 as n → ∞,
+∞
g ′ (s) ds = −g (0). 0
by the Riemann lemma, enables us to infer
∥Un ∥2H ≥ ρ
π
|λn An sin(hn x)|2 dx =
0
→ ∞,
π µA 2 2|gn |2
as n → ∞.
From the arguments above, we assert that the semigroup is not exponentially stable, and so completes the proof. Remark 3.1. (1) In the proof of Theorem 3.1, we took account into two cases of the sequence {gn hn }. One may wonder whether {gn hn } is always bounded. If so, it is enough to just consider one case. It is claimed in [4, Lemma 2.8] that under some additional assumptions on g , {gn hn } is always bounded. However, the following example (which satisfies the additional conditions posed in [4, Lemma 2.8]) shows that this is not true, and it may happen that
h
∞
g (s)e ′
0
−ihs
ds ,
for h ∈ R, is unbounded, even under the condition (1.8) on g. In what follows, we present our counterexample. Let g ′ (s) = −s−γ e−s , where γ ∈ (0, 1) is a fixed constant. We deduce that for h > 1,
∞
Im
g ′ (s)e−ihs ds =
∞ k=0
−g ′ (s) sin(hs) ds 0
0
=
∞
π + 2kπ h h 2kπ h
π −g ′ (s) + g ′ s + sin(hs) ds h
Acknowledgments The authors would like to thank the associate editor and the referee very much for their very professional and helpful comments and suggestions. References [1] M.A. Goodman, S.C. Cowin, A continuum theory for granular materials, Arch. Ration. Mech. Anal. 44 (1972) 249–266. [2] M. Ciarletta, A. Scalia, On some theorems in the linear theory of viscoelastic materials with voids, J. Elasticity 25 (1991) 149–158. [3] S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, J. Elasticity 13 (1983) 125–147. [4] P.X. Pamplona, J.E. Muñoz Rivera, R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl. 379 (2011) 682–705. [5] F. Ammar-Khodja, A. Benabdallah, J.E. Muñoz Rivera, R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations 194 (2003) 82–115. [6] S.A. Messaoudi, B. Said-Houari, Uniform decay in a Timoshenko-type system with past history, J. Math. Anal. Appl. 360 (2009) 459–475. [7] J.E. Muñoz Rivera, H.D. Fernndez Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (2008) 482–502. [8] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970) 297–308. [9] M. Grasselli, V. Pata, J.E. Muñoz Rivera, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl. 309 (2005) 1–14. [10] F.L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations 1 (1) (1985) 43–56. [11] L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236 (1978) 385–394. [12] Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, in: CRC Research Notes in Mathematics, vol. 398, Chapman & Hall, 1999. [13] T.J. Xiao, J. Liang, The Cauchy Problem for Higher Order Abstract Differential Equations, in: Lect. Notes in Math., vol. 1701, Springer, Berlin, New York, 1998.
Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Lack of exponential decay in viscoelastic materials with voids✩ Ti-Jun Xiao ∗ , Yuming Zhang Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
article
info
Article history: Received 23 July 2013 Received in revised form 8 November 2013 Accepted 12 November 2013 Available online 1 December 2013
abstract We study the stability of solutions to a coupled evolution system associated with an isotropic porous and centrosymmetric viscoelastic solid with porous dissipation (a porous elastic system with history). With the help of the method of the semigroup theory and some novel observations, we prove successfully that the condition of equal wave-speed propagation is still necessary for exponential stability of the system in the case of Dirichlet–Dirichlet boundary conditions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Porous elastic systems with history Semigroup method Exponential stability
1. Introduction
The initial and boundary conditions we consider here are
Since the work of Goodman and Cowin [1], in which a continuum theory of granular materials with interstitial voids was established, there have appeared many significant studies for various elastic materials with voids covering applications to many fields, such as biology material, soils sciences, petroleum industry, etc. The present paper concerns an isotropic porous and centrosymmetric viscoelastic solid with porous dissipation, whose motion is described by the following coupled evolution system
ρ u = µuxx + bϕx in (0, π ) × R+ , tt J ϕtt = xx − bux − ξ ϕ + g (0)ϕxx δϕ ∞ + g ′ (s)ϕ (t − s) ds in (0, π ) × R+ .
(1.1)
0
Here, the variables u and ϕ represent, respectively, the displacement of the solid elastic material and the volume fraction, g (·) is the memory kernel function, and ρ, J are positive constants; also, the constitutive coefficients µ, δ, ξ , b satisfy
µξ − b2 ≥ 0,
to ensure the nonnegativity of the internal energy. For more related information, we refer to [2–4] and references therein.
✩ The work was supported partly by the NSF of China (11371095), the Ministry of Education, the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Nonlinear Mathematical Modeling and Methods Laboratory of the Ministry of Education. ∗ Corresponding author. Tel.: +86 2165642341. E-mail address: [email protected] (T.-J. Xiao).
0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.11.006
ut (x, 0) = u1 (x),
ϕ(x, −s) = ϕ0 (x, s) (s ≥ 0),
ϕt (x, 0) = ϕ1 (x),
(1.2)
and u(0, t ) = u(π , t ) = ϕ(0, t ) = ϕ(π , t ) = 0,
t ≥ 0,
(1.3)
respectively. Moreover, the kernel function g ∈ C 2 (0, +∞) ∩ C [0, +∞), satisfies the following assumptions
xx
µ, δ, ξ > 0,
u(x, 0) = u0 (x),
g (+∞) = 0,
g ′ (s) < 0
g (s) ≥ c0 |g (s)| for s > 0, ′′
′
for s > 0,
(1.4) (1.5)
where c0 > 0 is a constant. Thus, we can see that g (s) > 0, g ′ ∈ L1 (0, +∞), and the function |g ′ (s)| is of exponential decay type. If
µ = ξ = b,
(1.6)
then (1.1) can be regarded as a system of Timoshenko type, where the variables u and ϕ represent, respectively, the transverse displacement of a beam and the rotation angle of a filament. For a Timoshenko system with memory (but having null history), Ammar-Khodja, Benabdallah, Muñoz Rivera, Rack [5] showed that whether the dissipation given by the memory effect is strong enough to bring the whole system to an exponential decay is essentially related to the condition of equal wave-speed propagation: Jµ
ρ
− δ − g (0) = 0.
(1.7)
40
T.-J. Xiao, Y. Zhang / Systems & Control Letters 63 (2014) 39–42
For some Timoshenko systems with past history, it was also proved that the condition (1.7) implies an exponential decay of the systems (see, e.g., [6,7]). Therefore, for the special case of (1.6), the solutions to (1.1)–(1.3) are exponentially stable if (1.7) holds. Very recently, for the case of Dirichlet–Neumann boundary conditions (instead of Dirichlet–Dirichlet ones in (1.3)) and the kernel function g satisfying (1.4), (1.5) and g ′′ (s) ≤ k1 |g ′ (s)| for s ≥ s1 ,
(1.8)
where s1 and k1 are positive constants, Pamplona, Muñoz Rivera, Quintanilla [4] showed that the semigroup associated with the porous viscoelastic system is exponentially stable if and only if (1.7) holds. Is the condition (1.7) necessary for exponential stability in the case of Dirichlet–Dirichlet boundary conditions? One expects that the answer is yes, yet proving it appears to be a quite tricky job. Actually, the basic question remains open, because so far, one only knows from [7, Section 4] (and [4]) that it is true for the mixed-type boundary conditions, while [7] shows the sufficiency in the case of Dirichlet–Dirichlet conditions. Our purpose in this paper is to solve the problem by giving a proof of the necessity. The rest of this paper is organized as follows. In Section 2, by introducing an auxiliary variable accounting for the past history of ϕ , we rephrase (1.1)–(1.3) as an abstract operator differential system, which is governed by a semigroup, following the approach of Dafermos [8] (see also [4,7,9]). Section 3 is devoted to presenting and proving our main result with respect to the necessity. 2. Semigroup setting
x ∈ [0, π], (t , s) ∈ R+ × R+ ,
W :=
(R , +
H01
π
0
∞
∞
g (s)ψ(·, s) ds ∈ H (0, π ), ψs ∈ W , ψ(0) = 0 . ′
2
0
Set
⟨U , U ⟩H = ∗
π
ρv v¯ ∗ + µux u¯ ∗x + J φ φ¯ ∗ + δϕx ϕ¯ x∗ + ξ ϕ ϕ¯ ∗
0
+ b(ux ϕ¯ + u¯ x ϕ) + ∗
∗
∗
0
for U = (u, v, ϕ, φ, ψ)T , U ∗ = (u∗ , v ∗ , ϕ ∗ , φ ∗ , ψ ∗ )T .
(2.2)
By hypothesis, we have µξ − b ≥ 0. Take ξ1 ∈ (0, ξ ] such that µξ1 − b2 = 0. Then, 2
⟨U , U ⟩H ≥
π
0
2 1 1 ρ|v|2 + J |φ|2 + δ|ϕx |2 + µ 2 ux − ξ12 ϕ
+ (ξ − ξ1 )|ϕ| + 2
∞
|g (s) ∥ ψx | ds dx. ′
2
(2.3)
0
This enables us to see clearly that the above ⟨U , U ∗ ⟩H defines an inner product on H , and the associated norm ∥ · ∥H is equivalent to the usual one. On the basis of the conditions (1.4) and (1.5), and arguing similarly as in [4, page 687], we infer that 0 ∈ ϱ(A) (the resolvent set of A), and hence, A generates a C0 semigroup T (t ) of contractions on (H , ∥ · ∥H ), according to the Lumer–Phillips theorem. Thus, the Cauchy t ≥ 0.
In view of this, one needs only to study the properties of T (t ), for understanding the behaviors of the system (1.1)–(1.3).
Proof. It is well known (see, e.g., [10–12] or [13] for a more general version) that a C0 semigroup of contractions on a Hilbert space, with the generator G, is exponentially stable if and only if
U (t ) = (u, ut , ϕ, ϕt , ψ t )T . Then we can transform (1.1)–(1.3) into a first order Cauchy problem in H as follows U (t ) = AU (t ),
(2.1)
U (0) = (u0 , u1 , ϕ0 (·, 0), ϕ1 , ϕ0 (·, 0) − ϕ0 (·, −s))T .
(i) ϱ(G) ⊃ iR, and (ii) sup{∥(iβ I − G)−1 ∥; β ∈ R} < ∞. Accordingly, we only need to find a sequence {λn } ⊂ iR, and a sequence {Un } ⊂ D (A) such that lim |λn | = ∞,
n→∞
Here,
J 0
|g (s)|ψx ψ¯ x ds dx, ′
Theorem 3.1. If the condition (1.7) of equal wave-speed propagation does not hold, then the semigroup T (t ) is not exponentially stable.
(obviously a Hilbert space), and
0 µ D2 ρ A= 0 b − D
∞
3. Main result
|g ′ (s) ∥ wx (s)|2 ds dx.
H = H01 (0, π ) × L2 (0, π ) × H01 (0, π ) × L2 (0, π ) × W
dt
δϕ +
(u, v, ϕ, φ, ψ) ∈ H ; u ∈ H 2 (0, π ), v, φ ∈ H01 (0, π ),
0
Set
d
D (A) =
U (t ) = T (t )U (0),
(0, π )),
which is the weighted space with respect to the measure |g ′ | ds endowed with the norm
∥w∥2W =
) with the domain
problem (2.1) is well-posed and the solution is given by
and the history space L2g ′
d dx
Re⟨AU , U ⟩H ≤ 0;
Consider the history variable
ψ t (x, s) = ϕ(x, t ) − ϕ(x, t − s),
(D :=
I 0
0
0
0
0
0
0
I
ρ
0 0
0 b D
1 J
(δ D2 − ξ I ) 0
0 I
0 ∞ 1 ′ 2 − g (s)D ds J
0
−∂s
lim ∥Un ∥H = ∞,
n→∞
but the sequence {Fn }, with Fn := (λn I − A)Un , is a bounded sequence in H . Actually, if we can find such two sequences {λn } and {Un }, then there are two cases: (a) at least one of the λn is in the spectrum of A, (b) all of the λn is in ϱ(A).
T.-J. Xiao, Y. Zhang / Systems & Control Letters 63 (2014) 39–42
41
In the first case, the necessary condition (i) for exponential stability (ϱ(A) ⊃ iR) is violated, and in the second case the necessary condition (ii) for exponential stability (the resolvent operator of A is bounded on the imaginary axis) is violated. Therefore, in either case the desired conclusion (the semigroup T (t ) is not exponentially stable) is obtained. In order to find such two sequences {λn } and {Un }, we observe by the definition of the operator A that choosing
where
U = (u, λu, ϕ, λϕ, (1 − eλs )ϕ)T
with
for
f2n := (ρλ2n + µh2n )An sin(hn x) + bBn hn sin(hn x)
u, ϕ ∈ H 2 (0, π ) ∩ H01 (0, π ),
λn := i
Fn = (0, f2n , 0, f4n , 0)T
gives (3.1)
f4n
where f2 := λ2 ρ u − µuxx − bϕx , ∞
(3.3)
0
Set
c=
Jµ , ρ(δ + g (0))
(3.4)
which is not equal to 1 by hypothesis. We take a sequence hn ⊂ 2N (the set of positive even numbers) such that
chn
lim hn = ∞,
−
2
n→∞
chn 2
≤
1
un = An sin(hn x),
(3.6)
ϕn = Bn cos(hn x) − Bn d˜ n cos(h˜ n x − dn );
∞ 0
g ′ (s)eλn s ds.
1 hn
,
An = A := −
1 b
−
Jµ
ρ
h˜ n := chn ,
dn :=
chn 2
−
chn 2
π,
d˜ n :=
1 cos dn
+ δ + g (0) .
f2n = b sin(hn x) − bc d˜ n sin(h˜ n x − dn ), f4n = gn hn cos(hn x) + ξ ϕn − d˜ n −
Then, noting h˜ n := chn and using (3.10), (3.11) gives
here,
(3.11)
Case (i). There exists a constant C > 0 such that |gn hn | ≤ C for n ∈ N. In this case, we put Bn =
([ ch2n ] is the greatest integer less than ch2n ). We note that such a sequence indeed exists. Then, we construct sequences
,
(3.12)
Jµ
ρ
hn
+ δ c 2 hn + g (0)c 2 hn + gn c 2 hn cos(h˜ n x − dn ).
and An , Bn are some numbers to be specified later. It is complex obvious that dn ∈ 0, π4 by (3.5), and so 1 ≤ d˜ n ≤
here, gn stands for the integral
(3.5)
4
(3.10)
+ gn h2n Bn + bAn hn cos(hn x) + ξ ϕn Jµ − − h2n + δ h˜ 2n + g (0)h˜ 2n ρ + gn h˜ 2n Bn d˜ n cos(h˜ n x − dn );
(3.2)
g ′ (s)(1 − eλs )ϕxx ds.
(3.9)
− bBn d˜ n h˜ n sin(h˜ n x − dn ) = bBn hn sin(hn x) − bBn d˜ n h˜ n sin(h˜ n x − dn ), Jµ := − h2n Bn + δ h2n Bn + g (0)h2n Bn ρ
f4 := λ2 J ϕ − δϕxx + bux + ξ ϕ
+
µ hn . ρ
From (3.1)–(3.3) and (3.6), we know
λ ∈ iR
F := (λI − A)U = (0, f2 , 0, f4 , 0),
(3.13)
From the definition of c (in (3.4)), one has
√ 2;
(3.7)
−
moreover, we have h˜ n π − dn = dn + 2
chn 2
+ δ c 2 + g (0)c 2 = 0,
(3.14)
f4n = gn hn cos(hn x) + ξ ϕn − gn hn d˜ n c 2 cos(h˜ n x − dn )
which indicates cos dn = cos(h˜ n π − dn ). Therefore, it follows that ϕn (0) = ϕn (π ) = 0, and thus we see from (3.6) that un , ϕn ∈ H 2 (0, π ) ∩ H01 (0, π ).
∥Un ∥H ≥ ρ (3.8)
(3.15)
by (3.13). This and (3.12) together indicate that both {f2n } and {f4n } are bounded sequences in L2 (0, π ), because {d˜ n }, {gn hn } are bounded by hypothesis (see also (3.7)). Thus, it follows from (3.9) that {Fn } is bounded in H . On the other hand, using (2.2), (2.3), (3.8) and (3.6) yields that 2
Next, we let
Fn = (λn I − A)Un ,
ρ
and hence
π,
Un = (un , λn un , ϕn , λn ϕn , (1 − eλn s )ϕn )T ,
Jµ
π
π
|λn un | dx = ρ |λn An sin(hn x)|2 dx 0 0 π 1 = µA2 h2n | sin(hn x)|2 dx = π µA2 h2n . 2
0
2
42
T.-J. Xiao, Y. Zhang / Systems & Control Letters 63 (2014) 39–42
∥Un ∥2H → ∞,
as n → ∞,
,
An = −
b
(3.16)
1
−
b|gn |hn
sin(hn x) −
|gn |hn
Jµ
f4n =
|gn |
ρ
≥
+ δ + g (0) =
|gn |hn
|gn |
+
c 2 gn
A
|gn |hn
.
gn
|gn |
π
e− h
π −γ h
s−γ − s +
π
e−π
2
e−π γ −1 h 2
5π 6
π
π
e−s− h
π −γ
h
s−γ − s +
π −γ
sin(hs) ds
sin(hs) ds
ds
h
[t −γ − (t + π )−γ ] dt ,
6
so that
h
∞ 0
≥
δc 2 − + ρ|gn | |gn |
∞
g ′ (s)e−ihs ds ≥ Im h
Jµ
e−π γ h 2
g ′ (s)e−ihs ds
0 5π 6
π
[t −γ − (t + π )−γ ] dt
6
→ +∞ as h → +∞.
cos(h˜ n x − dn ).
|gn |
Furthermore, recalling (3.14) gives f4n =
5π 6h
6h
=
sin(h˜ n x − dn ),
cos(hn x) + ξ ϕn − d˜ n
g (0)c 2
+
bc d˜ n
gn
h
Then, from (3.10), (3.11) we obtain f2n =
s−γ e−s − s +
0
and set
|gn |h2n
π
≥
Case (ii). The sequence {gn hn } is unbounded. In this case, we assume, without loss of generality, that
Bn =
0
by virtue of (3.5).
1
h
≥
as n → ∞,
|gn hn | → ∞,
π
Since (1.7) is not satisfied, we have A ̸= 0. Therefore,
cos(hn x) + ξ ϕn −
c 2 d˜ n gn
|gn |
cos(h˜ n x − dn ).
Thus, we see that f2n , f4n are uniformly bounded in L2 (0, π ), according to (3.7) and (3.16). Hence, {Fn } is bounded in H . Also, noting
Accordingly, the boundedness of the sequence {gn hn } cannot be guaranteed in general, and so we need take both cases into account. (2) The main results in [4] are still true since they can be proved without using [4, Lemma 2.8]. (3) From the proof of Theorem 3.1, we see that once the setting of A generating a C0 semigroup T (t ) of contractions on (H , ∥ · ∥H ) is insured (by the conditions (1.4) and (1.5) in this work), then we can obtain the desired conclusion using only the following properties of g: g ′ ∈ L1 (0, +∞),
|gn | → 0 as n → ∞,
+∞
g ′ (s) ds = −g (0). 0
by the Riemann lemma, enables us to infer
∥Un ∥2H ≥ ρ
π
|λn An sin(hn x)|2 dx =
0
→ ∞,
π µA 2 2|gn |2
as n → ∞.
From the arguments above, we assert that the semigroup is not exponentially stable, and so completes the proof. Remark 3.1. (1) In the proof of Theorem 3.1, we took account into two cases of the sequence {gn hn }. One may wonder whether {gn hn } is always bounded. If so, it is enough to just consider one case. It is claimed in [4, Lemma 2.8] that under some additional assumptions on g , {gn hn } is always bounded. However, the following example (which satisfies the additional conditions posed in [4, Lemma 2.8]) shows that this is not true, and it may happen that
h
∞
g (s)e ′
0
−ihs
ds ,
for h ∈ R, is unbounded, even under the condition (1.8) on g. In what follows, we present our counterexample. Let g ′ (s) = −s−γ e−s , where γ ∈ (0, 1) is a fixed constant. We deduce that for h > 1,
∞
Im
g ′ (s)e−ihs ds =
∞ k=0
−g ′ (s) sin(hs) ds 0
0
=
∞
π + 2kπ h h 2kπ h
π −g ′ (s) + g ′ s + sin(hs) ds h
Acknowledgments The authors would like to thank the associate editor and the referee very much for their very professional and helpful comments and suggestions. References [1] M.A. Goodman, S.C. Cowin, A continuum theory for granular materials, Arch. Ration. Mech. Anal. 44 (1972) 249–266. [2] M. Ciarletta, A. Scalia, On some theorems in the linear theory of viscoelastic materials with voids, J. Elasticity 25 (1991) 149–158. [3] S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, J. Elasticity 13 (1983) 125–147. [4] P.X. Pamplona, J.E. Muñoz Rivera, R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl. 379 (2011) 682–705. [5] F. Ammar-Khodja, A. Benabdallah, J.E. Muñoz Rivera, R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations 194 (2003) 82–115. [6] S.A. Messaoudi, B. Said-Houari, Uniform decay in a Timoshenko-type system with past history, J. Math. Anal. Appl. 360 (2009) 459–475. [7] J.E. Muñoz Rivera, H.D. Fernndez Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (2008) 482–502. [8] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970) 297–308. [9] M. Grasselli, V. Pata, J.E. Muñoz Rivera, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl. 309 (2005) 1–14. [10] F.L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations 1 (1) (1985) 43–56. [11] L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236 (1978) 385–394. [12] Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, in: CRC Research Notes in Mathematics, vol. 398, Chapman & Hall, 1999. [13] T.J. Xiao, J. Liang, The Cauchy Problem for Higher Order Abstract Differential Equations, in: Lect. Notes in Math., vol. 1701, Springer, Berlin, New York, 1998.