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Lack of exponential decay of a coupled system of wave equations with memory
Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
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Lack of exponential decay of a coupled system of wave equations with memory R.G.C. Almeida, M.L. Santos ∗ Faculdade de Matemática-Programa de Pós-Graduação em Matemática e Estatística, Universidade Federal do Pará, Campus Universitario do Guamá, Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brazil
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info
Article history: Received 10 February 2009 Accepted 19 August 2010
abstract In this work, we consider a coupled system of wave equations with memory only acting in one of the equations of the system. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.
Keywords: Wave equation Coupled system Polynomial decay
1. Introduction This paper is concerned with the stability of the C0 -semigroups associated with the following coupled system of wave equation with memory: utt − ∆u +
∞
∫
g (s)∆u(t − s)ds + αv = 0
in Ω ×]0, ∞[,
(1.1)
0
vtt − ∆v + α u = 0 in Ω ×]0, ∞[, u = v = 0 on Γ ×]0, ∞[, (u(x, 0), v(x, 0)) = (u0 , v0 ), in Ω (ut (x, 0), vt (x, 0)) = (u1 (x), v1 (x)) in Ω
(1.2) (1.3) (1.4) (1.5)
where Ω an open bounded set of Rn with smooth boundary Γ . The above model can be used to describe the evolution of a system consisting of two elastic membranes subject to an ∞ elastic force that attracts one membrane to the other with coefficient α > 0. Note that the term 0 g (s)∆u(t − s)ds, acts on the first membrane as a stabilizer. We will show that the coupled system above is dissipative but the corresponding semigroup is not exponentially stable. In addition, we show that the solution of system (1.1)–(1.5) decays polynomially to zero as time goes to infinity. We proceed as in [1], where the authors present a systematic approach combining a theorem by Gearhart [2] (see also [3,4]) in the theory of semigroups with partial differential equations techniques. To show the polynomial decay rates we use the same method higher order energy introduced by Alabau in [5] and developed by Alabau in [6] and Alabau–Cannarsa–Komornik in [7]. Let us mention some papers on weakly dissipative coupled systems. In [5] Alabau introduced the method used in this paper and she proved that the energy of associated weakly dissipative coupled system decays polynomially with explicit polynomial decay rates, for sufficiently smooth solutions. In [6] Alabau developed the ideas introduced in [5] and she proved
∗
Corresponding author. Tel.: +55 91 32017415. E-mail address: [email protected] (M.L. Santos).
1468-1218/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.08.025
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that under a condition on the operators of each equation and on the boundary feedback operator, the energy of smooth solutions of this system decays polynomially. In [7] Alabau et al. proved the polynomial decay rates for an abstract weakly dissipative coupled system using the same ideas introduced in [5]. In [8] Santos et al. studied the weakly dissipative coupled system of wave equations and they proved that the energy associated the smooth solutions decays polynomially. What we would like to do now is to give a new proof for the non exponential stability of the associated semigroup using the Gearhart Theorem. A new feature presented here is the combination of the contradiction argument with the frequency domain multiplier technique. The main contribution of the present paper is to use the term of memory g instead of the damping ut used in the papers mentioned above. The natural question here is whether the dissipation given by the memory effect is strong enough to produce uniform rate of decay for the solution of the system. The main difficulty is that we have to show that we can estimate ∇ utt for the polynomial decay. To overcome this problem we differentiate Eq. (2.4) and using similar procedure as in (2.1) to obtain (4.12). The problem considered here was not studied in the literature. Other results about the lack of exponential stability of the system with memory can be seen in [9–12] and its references. The rest of this work is organized as follows. In Section 2, we show the existence of solutions for system (1.1)–(1.5). In Section 3 we showed that system (1.1)–(1.5) is not exponentially stable. Finally, in Section 4 we showed that the system is polynomially stable. To show the polynomial decay rates we use the same method higher order energy introduced by Alabau in [5] and developed by Alabau in [6] and Alabau–Cannarsa–Komornik in [7]. 2. Existence and uniqueness of global solution In this section we will study the existence and uniqueness of strong and global solutions for system (1.1)–(1.5) using the semigroup techniques. Following the approach of Dafermos [13] and Fabrizio [14], we consider η = ηt (s), the relative history of u, defined as
η = ηt (s) = u(t ) − u(t − s).
(2.1)
Hence, putting
β0 = 1 −
∞
∫
g (s)ds > 0, 0
system (1.1)–(1.5) turns into the system utt − β0 ∆u −
∞
∫
g (τ )∆η(·, τ )dτ + αv = 0
in Ω × (0, ∞),
(2.2)
0
vtt − ∆v + α u = 0 in Ω × (0, ∞), ηt + ηs − ut = 0, in Ω × (0, ∞), t > 0 u = v = 0 on Γ × (0, ∞), (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)) in Ω , (ut (x, 0), vt (x, 0)) = (u1 (x), v1 (x)) in Ω ,
(2.5)
η = η0 ,
(2.8)
η (0) = lim η (s) = 0,
0
t
(2.3) (2.4) (2.6) (2.7)
t ≥ 0,
t
s→0
having sets u0 = u0 (0) and η0 (s) = u0 (0) − u0 (s), where the third equation is obtained differentiating (2.1) with respect to s. Conditions (2.8) means that the history is considered as an initial value. Concerning the kernel g we consider the following hypotheses: g (t ) > 0,
∃k0 , k1 , k2 > 0 :
−k0 g (t ) ≤ g ′ (t ) ≤ −k1 g (t ),
|g ′′ (t )| ≤ k2 g (t ),
∀t ≥ 0.
(2.9)
(R , (Ω )) be the Hilbert space of (Ω )-value functions on R , endowed with the inner product ∫ ∞ ∫ (f , h)L2g (R+ ,H 1 (Ω )) = g (s) ∇ f (x, s) · ∇ h(x, s)dx ds.
In view of (2.9), let
+
L2g
0
H01
0
H01
+
Ω
To give an accurate formulation of the evolution problem we introduce the product Hilbert spaces
H = (H01 (Ω ) × L2 (Ω ))2 × L2g (R+ ; H01 (Ω )) endowed with the following inner product
⟨U , V ⟩ = β0
∫
+α
∫
Ω
∇ u1 · ∇v1 dx +
∫ Ω
Ω
u2 v2 dx +
(u1 v3 + u3 v1 )dx +
∞
∫
g ( s) 0
∫
∫ Ω
∫ Ω
∇ u3 · ∇v3 dx +
Ω
u4 v4 dx
∇ u5 (x, s) · ∇v5 (x, s)dx ds,
where U = (u1 , u2 , u3 , u4 , u5 )T , V = (v1 , v2 , v3 , v4 , v5 )T ∈ H .
(2.10)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
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Let U = (u, ut , v, vt , η)T and we define the operator A : D(A) ⊂ H → H given by
0
I 0 0 0 I
β0 ∆ A= 0 −α I 0
0
−α I 0
∆ 0
0 0 I 0 0
0
T 0 0
−(·)s
with domain D(A) =
(u, ϕ, v, ψ, η)T ∈ H ;
β0 u −
∞
∫
g (s)η(s)ds ∈ H01 (Ω ) ∩ H 2 (Ω ), 0
ϕ∈
H01
(Ω ), v ∈
H01
(Ω ) ∩ H (Ω ), ψ ∈ 2
H01
(Ω ), η ∈ D(T )
where
Tη=
∞
∫
g (s)∆η(s)ds,
∀η ∈ D(T )
0
with D(T ) = {η ∈ L2g (R+ ; H01 (Ω )); ηs ∈ L2g (R+ ; H01 (Ω )), η(0) = 0} where ηs is the distributional derivative of η with respect to the internal variable s. Therefore, system (2.2)–(2.8) is equivalent to dU dt
= AU
(2.11)
U (0) = U0 ,
(2.12)
with U = (u, ut , v, vt , η) , U0 = (u0 , u1 , v0 , v1 , η0 ) . Using the internal product (2.10), we obtain T
T
∫ (β0 ∆u − αv + Iη)ut dx + ∇vt · ∇v dx Ω Ω Ω ∫ ∫ ∫ ∞ ∫ + (∆v − α u)vt dx + α (ut v + uvt )dx + g (τ ) ∇(ut − ητ (x, τ )) · ∇η(x, τ )dx dτ Ω 0 Ω ∫ Ω∞ ∫ 1 = g ′ (τ ) |∇η|2 dx ds.
⟨AU , U ⟩ = β0
∫
2
∫
∇ ut · ∇ u dx +
Ω
0
Using hypothesis (2.9) we obtain
⟨AU , U ⟩ ≤ −
k1 2
∞
∫
g (τ ) 0
∫ Ω
|∇η|2 dx ds ≤ 0.
Thus, A is a dissipative operator. Under the above notation we can establish the following theorem. Theorem 2.1. Assume that the memory kernel g satisfies the conditions (2.9). The linear operator A is the infinitesimal generator of C0 -semigroup S (t ) = et A of contraction on H . Proof. Since D (A) is dense in H and A is a dissipative operator, to prove Theorem 2.1, it is sufficient to prove that 0 ∈ ρ(A). For this, let us take F = (f1 , f2 , f3 , f4 , f5 )T ∈ H and consider the following equation
− AU = F ,
(2.13)
−ϕ = f1 , −β0 ∆u + αv − T η = f2 , −ψ = f3 , −∆v + α u = f4 , −ϕ + ηs = f5 .
(2.14)
i.e.,
(2.15) (2.16) (2.17) (2.18)
From Eqs. (2.14), (2.16) and (2.18) we conclude that
ϕ, ψ ∈ H01 (Ω ),
η ∈ L2g (R+ ; H01 (Ω )).
It easily follows from Eqs. (2.15) and (2.17) or the standard result on the linear elliptic equations that (2.15) and (2.17) have a unique solution (u, v) ∈ (H01 (Ω ) ∩ H 2 (Ω ))2 . Therefore, 0 ∈ ρ(A) and (u, ϕ, v, ψ, η)T ∈ D (A). The proof is now complete.
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3. Lack of exponential decay In this section we will prove that the resolvent operator is not uniformly bounded. This means that the semigroup S (t ) on H in this case is not exponential stable. To simplify calculations we suppose that the kernel is of the form g (s) = e−µs , s ∈ R+ , with µ > 1. Here we will use necessary and sufficient conditions for C0 -semigroups being exponentially stable in a Hilbert space. This result was obtained by Gearhart [2] and Huang [4], independently (see also [3,15]). Theorem 3.1. Let S (t ) = eAt be a C0 -semigroup of contractions on Hilbert space. Then S (t ) is exponentially stable if and only if
ρ(A) ⊇ {iλ : λ ∈ ℜ} ≡ iℜ and lim ‖(iλI − A)−1 ‖ < ∞
|λ|→∞
hold, where ρ(A) is the resolvent set of A. To study the asymptotic behavior of the semigroup associated to (2.2)–(2.4), let us consider the spectral problem:
−∆wm = λm wm in Ω wm = 0 on Γ ,
(3.1)
where lim λm = +∞ with ‖wm ‖L2 (Ω ) = 1.
m→∞
The following theorem describes the main result of this section. Theorem 3.2. Assume that the kernel is of the form g (s) = e−µs , s ∈ R+ , with µ > 1. The semigroup S (t ) on H is not exponentially stable. Proof. To prove that the semigroup S (t ) on H is not exponentially stable, we will find a sequence of bounded functions Fm = (f1,m , f2,m , f3,m , f4,m , f5,m ) ∈ H for which the corresponding solutions of the resolvent equations is not bounded. This will prove that the resolvent operator is not uniformly bounded. Let us consider the equation iλUm − AUm = Fm . To simplify the notation we will omit the subindex m. The equation reads
iλu − ϕ = f1 , ∫ iλϕ − β0 ∆u + αv − iλv − ψ = f3 , iλψ − ∆v + α u = f4 , iλη − ϕ + ηs = f5 .
∞
g (s)∆η(x, s)ds = f2 , (3.2)
0
Let us consider f1 = f3 = f5 = 0 and f2 = f4 = wm to obtain ϕ = iλu e ψ = iλv . Then, system (3.2) becomes
∫ −λ2 u − β0 ∆u + αv −
∞
g (s)∆η(x, s)ds = wm , 0
(3.3)
−λ v − ∆v + α u = wm , iλη + ηs − iλu = 0. 2
We look for solutions of the form u = aw m ,
v = bwm ,
ϕ = c wm ,
ψ = dwm ,
η(x, s) = γ (s)wm
with a, b, c , d ∈ C and γ (s) depend on λ and will be determined explicitly in what follows. From (3.3), we get a and b satisfy
∫ −λ2 a + β0 aλm + α b + λm 2 −λ b + λm b + α a = 1, γs + iλγ − iλa = 0.
∞
g (s)γ (s)ds = 1, 0
(3.4)
Solving (3.4)3 we get
γ (s) = C e−iλs + a.
(3.5)
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Since η(0) = 0 then C = −a, and (3.5) becomes
γ (s) = a − ae−iλs .
(3.6)
Then, from (3.6) we have ∞
∫
g (s)γ (s)ds =
∞
∫
g (s)(a − ae−iλs )ds = aa0 − a
g (s)e−iλs ds
(3.7)
0
0
0
∞
∫
where ∞
∫
g (s)ds.
a0 = 0
Now, choosing λ =
√ λm , using equations (3.4)1 and (3.4)2 we obtain
1
, α ∫ 1 λm (1 − β0 ) λm ∞ g (s)γ (s)ds + , b= − 2 α α 0 α √ λm c=i , α ∫ λm ( 1 − β 0 ) λm ∞ 1 d = i λm g ( s )γ ( s ) ds + − . α2 α 0 α a=
Recalling that
√
λm wm α
ϕ = c wm = i we get
‖ϕ‖2L2 (Ω ) =
λm . α2
Therefore we have
λm
lim ‖Um ‖2H ≥ lim ‖ϕ‖2L2 (Ω ) = lim 2 = ∞ m→∞ m→∞ α
m→∞
which completes the proof.
4. Polynomial decay Here our attention will be focused on the polynomial decay rates. For this we use the same method higher order energy introduced by Alabau in [5] and developed by Alabau in [6] and Alabau–Cannarsa–Komornik in [7]. To begin, let us introduce the following spaces:
Hn = (D(∆
n+1 2
n
) × D(∆ 2 ))2 × L2g (R+ ; D(∆
n+1 2
)),
n ≥ 0,
and for n ≥ 1
D (A ) =
(u, ϕ, v, ψ, η)T ∈ Hn−1 ;
n
β0 u −
∞
∫
g (s)η(s)ds ∈ D(∆
n+1 2
),
0
n 2
ϕ ∈ D(∆ ), v ∈ D(∆
n+1 2
n 2
), ψ ∈ D(∆ ), η ∈ D(T ) n
(4.1)
where
D(T n ) = η ∈ L2g (R+ ; D(∆
n+1 2
n )); ηs ∈ L2g (R; D(∆ 2 )), η(0) = 0 .
Here m
D(∆ 2 ) = u ∈ H m (Ω ); u = ∆u = · · · = ∆[
m−1 ] 2
u = 0 on Γ
.
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First we consider system (2.2)–(2.8) and the hypotheses over the kernel g (2.9) hold. We shall demonstrate that the energy 1
E (t ) =
∫
2
Ω
u2t + β0 |∇ u|2 + vt2 + |∇v|2 + 2α uv +
∞
∫
g (s)|∇ηt (s)|2 ds dx
(4.2)
0
decays to zero polynomially as time goes to infinity. Let us introduce the second and third-order energy E2 (t ) = E (ut , vt , ηt ),
E3 (t ) = E (utt , vtt , ηtt ).
Lemma 4.1. If u0 ∈ D(∆
n+1 2
),
v0 ∈ D(∆
n+1 2
n
),
n
u1 ∈ D(∆ 2 ),
v1 ∈ D(∆ 2 ),
η0 ∈ L2g (R; D(∆
n+1 2
)),
n≥1
then the energies E (t ), E2 (t ) and E3 (t ) associated to solutions of problem (2.2)–(2.8) satisfies d dt d dt d dt
E (t ) ≤ −
k1
∞
∫ ∫
2
Ω
k1
∫ ∫
E2 (t ) ≤ − E3 (t ) ≤ −
2 k1
(4.3)
0
Ω
∞
g (s)|∇ηtt (s)|2 ds dx,
(4.4)
0
∞
∫ ∫
2
g (s)|∇ηt (s)|2 ds dx,
Ω
g (s)|∇ηttt (s)|2 ds dx.
(4.5)
0
Proof. Multiplying Eq. (2.2) by ut and Eq. (2.3) by vt , respectively, summing the results obtained follows the conclusion of inequality (4.3). The inequalities (4.4) and (4.5) are obtained using the same procedure as in (4.3). In what follows we are assuming that the initial data satisfies the conditions of Lemma 4.1. We introduce the functional F1 (t ) = −
∞
∫
∫ Ω
g (s)ηt (s)ds dx.
ut 0
Then the following lemma holds. Lemma 4.2. For any ϵ1 > 0 there exists a positive constant Cϵ1 = d dt
F1 (t ) ≤ −
where β1 =
∞ 0
β1
∫
2
Ω
u2t dx +
β0 ϵ1 2
∫ Ω
|∇ u|2 dx + Cϵ1
β0 β1 2ϵ1
Ω
αβ1 2ϵ1
> 0 such that ∫ ∞ α C (Ω )ϵ1 |v|2 dx g (s)|∇ηt (s)|2 ds dx +
∫ ∫
+ β1 +
2
0
(4.6)
Ω
g (s)ds.
Proof. Using Eqs. (2.2)–(2.4) and Green’s formula we get d dt
2 ∫ ∞ ∫ ∫ ∞ t dx g ( s )∇η ( x , s ) ds ∇u · g (s)∇ηt (x, s)ds dx + 0 Ω 0 Ω ∫ ∞ ∫ ∞ ∫ ∫ ∫ +α v g (s)ηt (x, s)ds dx − β1 u2t dx + ut g (s)ηst (x, s)ds dx.
F1 (t ) = β0
∫
Ω
Ω
0
Ω
0
Since ∞
∫
∫ Ω
g (s)ηst (x, s)ds dx = −
ut
Ω
0
∞
∫
∫
g ′ (s)ηt (x, s)ds dx
ut 0
and
∫ ∫ Ω
2
∞ 0
g (s)∇ηt (x, s)ds dx ≤ β1
∞
∫ ∫ Ω
g (s)|∇ηt (x, s)|2 ds dx, 0
then using Young inequality and hypotheses (2.9) our conclusion follows.
Let us consider the following functional F2 (t ) = F1 (ut , vt , ηtt ). Then using Lemma 4.2 we obtain d dt
F2 (t ) ≤ −
β1 2
∫ Ω
u2tt dx +
β 0 ϵ1 2
∫ Ω
|∇ ut |2 dx + Cϵ1
∞
∫ ∫ Ω
g (s)|∇ηtt (s)|2 ds dx + 0
α C (Ω )ϵ1 2
∫ Ω
|vt |2 dx. (4.7)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
1029
We define the functional Z1 (t ) as Z1 (t ) =
∫ Ω
ut u dx.
Using (2.2) and Green’s formula we get d dt
Z1 (t ) ≤
∫ Ω
u2t dx −
β0
∫
2
|∇ u|2 dx + Ω
β1 2β0
∞
∫ ∫ Ω
g (s)|∇ηt (x, s)|2 ds dx − α
∫
0
Ω
uv dx.
(4.8)
Let us denote by Z2 (t ) the following functional Z2 (t ) = Z1 (ut , vt , ηtt ). Then, from (4.8) we have d dt
Z2 (t ) ≤
∫ Ω
u2tt
dx −
β0
∫
2
β1 |∇ ut | dx + 2 β0 Ω 2
∞
∫ ∫ Ω
g (s)|∇η (x, s)| ds dx − α t t
2
0
∫ Ω
ut vt dx.
(4.9)
The following lemma shows the dissipative property of v . For this let us consider the following functional
Φ (t ) =
∫ Ω
utt vt dx + β0
∞
∫ ∫
∫ Ω
∇ ut · ∇v dx +
Ω
g (s)∇ηtt (s) · ∇v ds dx. 0
Lemma 4.3. For any ϵ2 > 0 we have
∫ ∫ ∞ β0 t 2 Φ (t ) ≤ −α |vt | dx + + g (s)|∇ηtt | ds dx dt 2ϵ2 2ϵ2 Ω Ω 0 ∫ ∫ ∞ k2 β1 k2 β0 + + + g (s)|∇ηt |2 ds dx 2ϵ2 2ϵ2 2ϵ2 Ω 0 ∫ ∫ 1 (g (0) + 1) β0 α + + + ϵ2 |∇v|2 dx + |utt |2 dx β1 2 β1 2 Ω Ω ∫ ∫ α C (Ω ) g (0) |∇ ut |2 dx + |∇ u|2 dx. + 2ϵ2 Ω 2 Ω ∫
d
2
1
(4.10)
Proof. Using Eqs. (2.2) and (2.3) we get d dt
Φ (t ) = −α
∫ Ω
|vt |2 dx −
Ω
∇ utt · ∇v dx − α
∞
∫ ∫
g (s)∇ηttt ds
+ Ω
∫
∫ Ω
· ∇v dx + β0
0
utt u dx
∫ Ω
∇ utt · ∇v dx.
(4.11)
On the other hand, from (2.1) we deduce
ηtt (0) = 0, ηst (0) = ut (t ) (4.12) t utt = ηttt − ηss . ∞ Recalling that β1 = 0 g (s)ds and using hypotheses (2.9) and (4.12), we obtain ∫ ∫ ∫ ∞ β1 ∇ utt · ∇v dx = g (s)∇ utt ds · ∇v dx Ω Ω 0 ∫ ∫ ∞ t t = g (s)(∇ηtt − ∇ηss )ds · ∇v dx Ω 0 ∫ ∫ ∞ ∫ ∫ ∞ t = g (s)∇ηttt ds · ∇v dx − g (s)∇ηss ds · ∇v dx Ω 0 Ω 0 ∫ ∫ ∞ ∫ ∫ ∫ ∞ = g (s)∇ηttt ds · ∇v dx + g (0) ∇ ut · ∇v dx − g ′′ (s)∇ηt ds · ∇v dx Ω 0 Ω Ω 0 ∫ ∫ ∞ ∫ ≤ g (s)∇ηttt ds · ∇v dx + g (0) ∇ ut · ∇v dx
Ω
Ω
0
∞
∫ ∫ − k2
Ω
g (s)∇ηt ds 0
· ∇v dx.
(4.13)
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R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
Substituting the Eq. (4.13) into (4.11) and using Poincaré and Young inequalities we arrive at d dt
2 ∫ ∫ ∞ 1 g (s)∇ηttt ds dx |vt |2 dx + 2ϵ2 β1 Ω 0 Ω ∫ ∫ ∫ ϵ2 α α C (Ω ) + |∇v|2 dx + |utt |2 dx + |∇ u|2 dx 2β1 Ω 2 Ω 2 Ω 2 ∫ ∫ ∞ ∫ ϵ2 k2 g (s)∇ηt ds dx + |∇v|2 dx + 2ϵ2 β1 Ω 2β1 Ω 0 2 ∫ ∫ ∞ ∫ ∫ k2 g (0) (g (0) + 1) + g (s)∇ηt ds dx + |∇ ut |2 dx + ϵ2 |∇v|2 dx 2ϵ2 Ω 2ϵ2 Ω 2 0 Ω 2 ∫ ∫ ∞ ∫ ϵ2 β0 β0 g (s)∇ηttt ds dx + |∇v|2 dx + 2ϵ2 β1 Ω 2β1 Ω 0 2 ∫ ∫ ∞ ∫ k2 β0 ϵ2 β0 + g (s)∇ηt ds dx + |∇v|2 dx. 2ϵ2 β1 Ω 2β1 Ω 0
Φ (t ) ≤ −α
∫
(4.14)
Noting that ∞
∫ ∫ Ω
g (s)∇ηttt ds
2
∞
∫ ∫
dx ≤ β1
Ω
0
g (s)|∇ηttt |2 ds dx 0
and ∞
∫ ∫ Ω
g (s)∇ηt ds
2
dx ≤ β1
0
our conclusion follows.
∞
∫ ∫ Ω
g (s)|∇ηt |2 ds dx 0
Finally, we define the functional
Ψ (t ) =
∫ Ω
vt v dx.
(4.15)
From Eq. (2.3) we get d dt
Ψ (t ) = −
∫
|∇v|2 dx +
∫
Ω
Ω
|vt |2 dx − α
∫ Ω
u v dx.
(4.16)
Now we are in the position to prove the polynomial rate of decay. Theorem 4.1. Assume that the memory kernel g satisfies the conditions (2.9) and the initial data satisfies the conditions of Lemma 4.1, then the first-order energy E (t ) decays polynomially to zero, that is, there exists a positive constant C , being independent of the initial data, such that E (t ) ≤
C t
(E (0) + E2 (0) + E3 (0)).
Moreover, if U0 := (u0 , u1 , v0 , v1 , η) ∈ D(A2n ), then
‖T (t )‖H ≤
2n Cn −
t n k=0
‖Ak U0 ‖H .
Proof. We define L as
L(t ) = N1 (E (t ) + E2 (t ) + E3 (t )) + N2 (F1 (t ) + F2 (t )) + N3 (Z1 (t ) + Z2 (t )) + N4 Φ (t ) + Ψ (t ) where N1 , N2 , N3 and N4 are positive constants. Then from Lemmas 4.1–4.3 we get d dt
β0 β1 αβ1 k 0 C (Ω ) β1 + β1 + + N2 + N3 2 2ϵ1 2ϵ1 2 2β0 ] ∫ ∫ ∞ k2 β1 k2 β0 + + + N4 g (s)|∇ηt (x, s)|2 ds dx 2ϵ2 2ϵ2 ϵ2 Ω 0
L(t ) ≤ − N1
k1
[
−
(4.17)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
αβ1 k0 C (Ω ) β0 β1 β1 + β1 + + N3 N2 + 2 2ϵ1 2ϵ1 2 2β0 ] ∫ ∫ ∞ k2 β1 k2 β0 t 2 g (s)|∇ηt (x, s)| ds dx + + N4 2ϵ2 2ϵ2 ϵ2 0 Ω [ αβ1 k0 C (Ω ) k1 β0 β1 β1 + β1 + + N3 N1 − N2 + 2 2ϵ1 2ϵ1 2 2β0 ] ∫ ∫ ∞ β1 k2 β0 k2 g (s)|∇ηttt (x, s)|2 ds dx + + N4 2ϵ2 2ϵ2 ϵ2 0 Ω ∫ ∫ β1 α β1 α N2 − N3 − N4 u2t dx − N2 − N3 − N4 u2tt dx 2 2 2 2 Ω Ω ∫ β0 ϵ1 α C (Ω ) β0 |∇ u|2 dx N3 − N2 − N4 2 2 2 Ω ∫ ∫ αϵ1 β0 ϵ1 g (0) β0 |∇ ut |2 dx − α N4 − N3 − N2 − N4 N2 − 1 vt2 dx 2 2 2ϵ2 2 Ω Ω ∫ α C (Ω )ϵ1 1 g (0) + 1 β0 1− N2 − + + ϵ2 N 4 |∇v|2 dx. 2 β1 2 β1 Ω
− + − + − − − −
N1
k1
[
−
Now, choosing ϵ1 , ϵ2 small enough such that 0 < ϵ1 < ϵ2 <
β0 β1 (β1 − 2ϵ1 )
α>
1031
N4 (β0 β1 (β1 − 2ϵ1 ) − 2ϵ1 (ϵ1 (β0 + β1 C (Ω ))))
β1 2
(4.18)
and N4 > 0 large enough we get
>0
is small enough. Then taking
β0 β1 + αβ1 k0 C (Ω ) (ϵ1 + α)β0 ϵ2 + g (0)β1 + β1 + + 2 +α k1 2ϵ1 β1 β0 ϵ2 (β1 − 2ϵ1 ) 2 ] k2 + β1 + 2k2 β0 β12 (ϵ1 ϵ2 α + g (0)β1 ) + N4 , + 2β0 (β1 − 2ϵ1 ) 2ϵ2 (ϵ1 + α)β0 ϵ2 + g (0)β1 2(ϵ1 β0 + β1 C (Ω )ϵ1 ) + α N4 , + α N4 N2 > max β1 β0 (β1 − 2ϵ1 ) β0 β1 ϵ2 (β1 − 2ϵ1 ) [
2
N1 >
and N3 > max
(β0 ϵ1 + C (Ω ))β1 ϵ1 β0 β1 (β1 − 2ϵ1 )
N4 ,
(ϵ1 ϵ2 αβ0 + g (0)β1 ) N4 β0 ϵ2 (β1 − 2ϵ1 )
we get that there is a positive constant ω > 0 such that d dt
L(t ) ≤ −w E (t ).
Therefore t
∫
ω
E (s)ds ≤ L(0) − L(t ),
∀t ≥ 0.
(4.19)
0
On the other hand, it is not difficult to prove that there exists a constant ς > 0 such that
L(0) − L(t ) ≤ ς(E (0) + E2 (0) + E3 (0)),
∀t ≥ 0.
(4.20)
From (4.19) and (4.20) we obtain t
∫
E (s)ds ≤ 0
ς (E (0) + E2 (0) + E3 (0)). ω
Finally, since d dt
{tE (t )} = E (t ) + t
d dt
E (t ) ≤ E (t ),
from (4.21) we get E (t ) ≤
C t
(E (0) + E2 (0) + E3 (0))
(4.21)
1032
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
ς
where C = ω . Finally, if U0 ∈ D(A2n ), then using the same procedure as in [5–7] we obtain (4.17), which completes the proof. Acknowledgements This work has been carried out, thanks to the support of the FAPESPA-PA-Brazil. In addition the second author acknowledges the support of the CNPq (306338/2008-4). The authors are thankful to the referees of this paper for valuable suggestions which improved this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
Z. Liu, S. Zheng, Semigroups associated with dissipative systems, in: CRC Reseach Notes in Mathematics, vol. 398, Chapman & Hall, 1999. L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. AMS 236 (1978) 385–394. A. Wiler, Stability of wave equations with dissipative bounded conditions in boundend domain, Differential Integral Equations 7 (2) (1994) 345–366. F. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential. Equations 1 (1) (1985) 43–56. Fatiha Alabau Boussoira, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris, Sér. I 328 (1999) 1015–1020. Fatiha Alabau Boussoira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim. 41 (2) (2002) 511–541. Fatiha Alabau Boussoira, P. Cannarsa, V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ. 2 (2002) 127–150. M.L. Santos, M.P.C. Rocha, S.C. Gomes, Polynomial stability of a coupled system of waves equations weakly dissipative, Appl. Anal. 86 (2007) 1293–1302. Jaime E. Muñoz Rivera, Maria Grazia Naso, Optimal energy decay rate for a class of weakly dissipative second-order systems with memory, Appl. Math. Lett. (in press). Jaime E. Muñoz Rivera, Maria Grazia Naso, Elena Vuk, Asymptotic behaviour of the energy for electromagnetic systems with memory, Math. Methods Appl. Sci. 27 (2004) 819–841. Jaime E. Muñoz Rivera, Hugo D. Fernandez Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502. Maurizio Grasselli, Jaime E. Muñoz Rivera, Vittorino Pata, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl. 309 (2005) 1–14. C. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970) 297–308. M. Fabrizio, A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Philadelphia, 1992. J. Prüss, On the spectrum of C0 -semigroups, Trans. AMS 28 (1984) 847–857.
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Lack of exponential decay of a coupled system of wave equations with memory R.G.C. Almeida, M.L. Santos ∗ Faculdade de Matemática-Programa de Pós-Graduação em Matemática e Estatística, Universidade Federal do Pará, Campus Universitario do Guamá, Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brazil
article
info
Article history: Received 10 February 2009 Accepted 19 August 2010
abstract In this work, we consider a coupled system of wave equations with memory only acting in one of the equations of the system. We show that the solution of this system has a polynomial rate of decay as time tends to infinity, but does not have exponential decay. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.
Keywords: Wave equation Coupled system Polynomial decay
1. Introduction This paper is concerned with the stability of the C0 -semigroups associated with the following coupled system of wave equation with memory: utt − ∆u +
∞
∫
g (s)∆u(t − s)ds + αv = 0
in Ω ×]0, ∞[,
(1.1)
0
vtt − ∆v + α u = 0 in Ω ×]0, ∞[, u = v = 0 on Γ ×]0, ∞[, (u(x, 0), v(x, 0)) = (u0 , v0 ), in Ω (ut (x, 0), vt (x, 0)) = (u1 (x), v1 (x)) in Ω
(1.2) (1.3) (1.4) (1.5)
where Ω an open bounded set of Rn with smooth boundary Γ . The above model can be used to describe the evolution of a system consisting of two elastic membranes subject to an ∞ elastic force that attracts one membrane to the other with coefficient α > 0. Note that the term 0 g (s)∆u(t − s)ds, acts on the first membrane as a stabilizer. We will show that the coupled system above is dissipative but the corresponding semigroup is not exponentially stable. In addition, we show that the solution of system (1.1)–(1.5) decays polynomially to zero as time goes to infinity. We proceed as in [1], where the authors present a systematic approach combining a theorem by Gearhart [2] (see also [3,4]) in the theory of semigroups with partial differential equations techniques. To show the polynomial decay rates we use the same method higher order energy introduced by Alabau in [5] and developed by Alabau in [6] and Alabau–Cannarsa–Komornik in [7]. Let us mention some papers on weakly dissipative coupled systems. In [5] Alabau introduced the method used in this paper and she proved that the energy of associated weakly dissipative coupled system decays polynomially with explicit polynomial decay rates, for sufficiently smooth solutions. In [6] Alabau developed the ideas introduced in [5] and she proved
∗
Corresponding author. Tel.: +55 91 32017415. E-mail address: [email protected] (M.L. Santos).
1468-1218/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.08.025
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R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
that under a condition on the operators of each equation and on the boundary feedback operator, the energy of smooth solutions of this system decays polynomially. In [7] Alabau et al. proved the polynomial decay rates for an abstract weakly dissipative coupled system using the same ideas introduced in [5]. In [8] Santos et al. studied the weakly dissipative coupled system of wave equations and they proved that the energy associated the smooth solutions decays polynomially. What we would like to do now is to give a new proof for the non exponential stability of the associated semigroup using the Gearhart Theorem. A new feature presented here is the combination of the contradiction argument with the frequency domain multiplier technique. The main contribution of the present paper is to use the term of memory g instead of the damping ut used in the papers mentioned above. The natural question here is whether the dissipation given by the memory effect is strong enough to produce uniform rate of decay for the solution of the system. The main difficulty is that we have to show that we can estimate ∇ utt for the polynomial decay. To overcome this problem we differentiate Eq. (2.4) and using similar procedure as in (2.1) to obtain (4.12). The problem considered here was not studied in the literature. Other results about the lack of exponential stability of the system with memory can be seen in [9–12] and its references. The rest of this work is organized as follows. In Section 2, we show the existence of solutions for system (1.1)–(1.5). In Section 3 we showed that system (1.1)–(1.5) is not exponentially stable. Finally, in Section 4 we showed that the system is polynomially stable. To show the polynomial decay rates we use the same method higher order energy introduced by Alabau in [5] and developed by Alabau in [6] and Alabau–Cannarsa–Komornik in [7]. 2. Existence and uniqueness of global solution In this section we will study the existence and uniqueness of strong and global solutions for system (1.1)–(1.5) using the semigroup techniques. Following the approach of Dafermos [13] and Fabrizio [14], we consider η = ηt (s), the relative history of u, defined as
η = ηt (s) = u(t ) − u(t − s).
(2.1)
Hence, putting
β0 = 1 −
∞
∫
g (s)ds > 0, 0
system (1.1)–(1.5) turns into the system utt − β0 ∆u −
∞
∫
g (τ )∆η(·, τ )dτ + αv = 0
in Ω × (0, ∞),
(2.2)
0
vtt − ∆v + α u = 0 in Ω × (0, ∞), ηt + ηs − ut = 0, in Ω × (0, ∞), t > 0 u = v = 0 on Γ × (0, ∞), (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)) in Ω , (ut (x, 0), vt (x, 0)) = (u1 (x), v1 (x)) in Ω ,
(2.5)
η = η0 ,
(2.8)
η (0) = lim η (s) = 0,
0
t
(2.3) (2.4) (2.6) (2.7)
t ≥ 0,
t
s→0
having sets u0 = u0 (0) and η0 (s) = u0 (0) − u0 (s), where the third equation is obtained differentiating (2.1) with respect to s. Conditions (2.8) means that the history is considered as an initial value. Concerning the kernel g we consider the following hypotheses: g (t ) > 0,
∃k0 , k1 , k2 > 0 :
−k0 g (t ) ≤ g ′ (t ) ≤ −k1 g (t ),
|g ′′ (t )| ≤ k2 g (t ),
∀t ≥ 0.
(2.9)
(R , (Ω )) be the Hilbert space of (Ω )-value functions on R , endowed with the inner product ∫ ∞ ∫ (f , h)L2g (R+ ,H 1 (Ω )) = g (s) ∇ f (x, s) · ∇ h(x, s)dx ds.
In view of (2.9), let
+
L2g
0
H01
0
H01
+
Ω
To give an accurate formulation of the evolution problem we introduce the product Hilbert spaces
H = (H01 (Ω ) × L2 (Ω ))2 × L2g (R+ ; H01 (Ω )) endowed with the following inner product
⟨U , V ⟩ = β0
∫
+α
∫
Ω
∇ u1 · ∇v1 dx +
∫ Ω
Ω
u2 v2 dx +
(u1 v3 + u3 v1 )dx +
∞
∫
g ( s) 0
∫
∫ Ω
∫ Ω
∇ u3 · ∇v3 dx +
Ω
u4 v4 dx
∇ u5 (x, s) · ∇v5 (x, s)dx ds,
where U = (u1 , u2 , u3 , u4 , u5 )T , V = (v1 , v2 , v3 , v4 , v5 )T ∈ H .
(2.10)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
1025
Let U = (u, ut , v, vt , η)T and we define the operator A : D(A) ⊂ H → H given by
0
I 0 0 0 I
β0 ∆ A= 0 −α I 0
0
−α I 0
∆ 0
0 0 I 0 0
0
T 0 0
−(·)s
with domain D(A) =
(u, ϕ, v, ψ, η)T ∈ H ;
β0 u −
∞
∫
g (s)η(s)ds ∈ H01 (Ω ) ∩ H 2 (Ω ), 0
ϕ∈
H01
(Ω ), v ∈
H01
(Ω ) ∩ H (Ω ), ψ ∈ 2
H01
(Ω ), η ∈ D(T )
where
Tη=
∞
∫
g (s)∆η(s)ds,
∀η ∈ D(T )
0
with D(T ) = {η ∈ L2g (R+ ; H01 (Ω )); ηs ∈ L2g (R+ ; H01 (Ω )), η(0) = 0} where ηs is the distributional derivative of η with respect to the internal variable s. Therefore, system (2.2)–(2.8) is equivalent to dU dt
= AU
(2.11)
U (0) = U0 ,
(2.12)
with U = (u, ut , v, vt , η) , U0 = (u0 , u1 , v0 , v1 , η0 ) . Using the internal product (2.10), we obtain T
T
∫ (β0 ∆u − αv + Iη)ut dx + ∇vt · ∇v dx Ω Ω Ω ∫ ∫ ∫ ∞ ∫ + (∆v − α u)vt dx + α (ut v + uvt )dx + g (τ ) ∇(ut − ητ (x, τ )) · ∇η(x, τ )dx dτ Ω 0 Ω ∫ Ω∞ ∫ 1 = g ′ (τ ) |∇η|2 dx ds.
⟨AU , U ⟩ = β0
∫
2
∫
∇ ut · ∇ u dx +
Ω
0
Using hypothesis (2.9) we obtain
⟨AU , U ⟩ ≤ −
k1 2
∞
∫
g (τ ) 0
∫ Ω
|∇η|2 dx ds ≤ 0.
Thus, A is a dissipative operator. Under the above notation we can establish the following theorem. Theorem 2.1. Assume that the memory kernel g satisfies the conditions (2.9). The linear operator A is the infinitesimal generator of C0 -semigroup S (t ) = et A of contraction on H . Proof. Since D (A) is dense in H and A is a dissipative operator, to prove Theorem 2.1, it is sufficient to prove that 0 ∈ ρ(A). For this, let us take F = (f1 , f2 , f3 , f4 , f5 )T ∈ H and consider the following equation
− AU = F ,
(2.13)
−ϕ = f1 , −β0 ∆u + αv − T η = f2 , −ψ = f3 , −∆v + α u = f4 , −ϕ + ηs = f5 .
(2.14)
i.e.,
(2.15) (2.16) (2.17) (2.18)
From Eqs. (2.14), (2.16) and (2.18) we conclude that
ϕ, ψ ∈ H01 (Ω ),
η ∈ L2g (R+ ; H01 (Ω )).
It easily follows from Eqs. (2.15) and (2.17) or the standard result on the linear elliptic equations that (2.15) and (2.17) have a unique solution (u, v) ∈ (H01 (Ω ) ∩ H 2 (Ω ))2 . Therefore, 0 ∈ ρ(A) and (u, ϕ, v, ψ, η)T ∈ D (A). The proof is now complete.
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R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
3. Lack of exponential decay In this section we will prove that the resolvent operator is not uniformly bounded. This means that the semigroup S (t ) on H in this case is not exponential stable. To simplify calculations we suppose that the kernel is of the form g (s) = e−µs , s ∈ R+ , with µ > 1. Here we will use necessary and sufficient conditions for C0 -semigroups being exponentially stable in a Hilbert space. This result was obtained by Gearhart [2] and Huang [4], independently (see also [3,15]). Theorem 3.1. Let S (t ) = eAt be a C0 -semigroup of contractions on Hilbert space. Then S (t ) is exponentially stable if and only if
ρ(A) ⊇ {iλ : λ ∈ ℜ} ≡ iℜ and lim ‖(iλI − A)−1 ‖ < ∞
|λ|→∞
hold, where ρ(A) is the resolvent set of A. To study the asymptotic behavior of the semigroup associated to (2.2)–(2.4), let us consider the spectral problem:
−∆wm = λm wm in Ω wm = 0 on Γ ,
(3.1)
where lim λm = +∞ with ‖wm ‖L2 (Ω ) = 1.
m→∞
The following theorem describes the main result of this section. Theorem 3.2. Assume that the kernel is of the form g (s) = e−µs , s ∈ R+ , with µ > 1. The semigroup S (t ) on H is not exponentially stable. Proof. To prove that the semigroup S (t ) on H is not exponentially stable, we will find a sequence of bounded functions Fm = (f1,m , f2,m , f3,m , f4,m , f5,m ) ∈ H for which the corresponding solutions of the resolvent equations is not bounded. This will prove that the resolvent operator is not uniformly bounded. Let us consider the equation iλUm − AUm = Fm . To simplify the notation we will omit the subindex m. The equation reads
iλu − ϕ = f1 , ∫ iλϕ − β0 ∆u + αv − iλv − ψ = f3 , iλψ − ∆v + α u = f4 , iλη − ϕ + ηs = f5 .
∞
g (s)∆η(x, s)ds = f2 , (3.2)
0
Let us consider f1 = f3 = f5 = 0 and f2 = f4 = wm to obtain ϕ = iλu e ψ = iλv . Then, system (3.2) becomes
∫ −λ2 u − β0 ∆u + αv −
∞
g (s)∆η(x, s)ds = wm , 0
(3.3)
−λ v − ∆v + α u = wm , iλη + ηs − iλu = 0. 2
We look for solutions of the form u = aw m ,
v = bwm ,
ϕ = c wm ,
ψ = dwm ,
η(x, s) = γ (s)wm
with a, b, c , d ∈ C and γ (s) depend on λ and will be determined explicitly in what follows. From (3.3), we get a and b satisfy
∫ −λ2 a + β0 aλm + α b + λm 2 −λ b + λm b + α a = 1, γs + iλγ − iλa = 0.
∞
g (s)γ (s)ds = 1, 0
(3.4)
Solving (3.4)3 we get
γ (s) = C e−iλs + a.
(3.5)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
1027
Since η(0) = 0 then C = −a, and (3.5) becomes
γ (s) = a − ae−iλs .
(3.6)
Then, from (3.6) we have ∞
∫
g (s)γ (s)ds =
∞
∫
g (s)(a − ae−iλs )ds = aa0 − a
g (s)e−iλs ds
(3.7)
0
0
0
∞
∫
where ∞
∫
g (s)ds.
a0 = 0
Now, choosing λ =
√ λm , using equations (3.4)1 and (3.4)2 we obtain
1
, α ∫ 1 λm (1 − β0 ) λm ∞ g (s)γ (s)ds + , b= − 2 α α 0 α √ λm c=i , α ∫ λm ( 1 − β 0 ) λm ∞ 1 d = i λm g ( s )γ ( s ) ds + − . α2 α 0 α a=
Recalling that
√
λm wm α
ϕ = c wm = i we get
‖ϕ‖2L2 (Ω ) =
λm . α2
Therefore we have
λm
lim ‖Um ‖2H ≥ lim ‖ϕ‖2L2 (Ω ) = lim 2 = ∞ m→∞ m→∞ α
m→∞
which completes the proof.
4. Polynomial decay Here our attention will be focused on the polynomial decay rates. For this we use the same method higher order energy introduced by Alabau in [5] and developed by Alabau in [6] and Alabau–Cannarsa–Komornik in [7]. To begin, let us introduce the following spaces:
Hn = (D(∆
n+1 2
n
) × D(∆ 2 ))2 × L2g (R+ ; D(∆
n+1 2
)),
n ≥ 0,
and for n ≥ 1
D (A ) =
(u, ϕ, v, ψ, η)T ∈ Hn−1 ;
n
β0 u −
∞
∫
g (s)η(s)ds ∈ D(∆
n+1 2
),
0
n 2
ϕ ∈ D(∆ ), v ∈ D(∆
n+1 2
n 2
), ψ ∈ D(∆ ), η ∈ D(T ) n
(4.1)
where
D(T n ) = η ∈ L2g (R+ ; D(∆
n+1 2
n )); ηs ∈ L2g (R; D(∆ 2 )), η(0) = 0 .
Here m
D(∆ 2 ) = u ∈ H m (Ω ); u = ∆u = · · · = ∆[
m−1 ] 2
u = 0 on Γ
.
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R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
First we consider system (2.2)–(2.8) and the hypotheses over the kernel g (2.9) hold. We shall demonstrate that the energy 1
E (t ) =
∫
2
Ω
u2t + β0 |∇ u|2 + vt2 + |∇v|2 + 2α uv +
∞
∫
g (s)|∇ηt (s)|2 ds dx
(4.2)
0
decays to zero polynomially as time goes to infinity. Let us introduce the second and third-order energy E2 (t ) = E (ut , vt , ηt ),
E3 (t ) = E (utt , vtt , ηtt ).
Lemma 4.1. If u0 ∈ D(∆
n+1 2
),
v0 ∈ D(∆
n+1 2
n
),
n
u1 ∈ D(∆ 2 ),
v1 ∈ D(∆ 2 ),
η0 ∈ L2g (R; D(∆
n+1 2
)),
n≥1
then the energies E (t ), E2 (t ) and E3 (t ) associated to solutions of problem (2.2)–(2.8) satisfies d dt d dt d dt
E (t ) ≤ −
k1
∞
∫ ∫
2
Ω
k1
∫ ∫
E2 (t ) ≤ − E3 (t ) ≤ −
2 k1
(4.3)
0
Ω
∞
g (s)|∇ηtt (s)|2 ds dx,
(4.4)
0
∞
∫ ∫
2
g (s)|∇ηt (s)|2 ds dx,
Ω
g (s)|∇ηttt (s)|2 ds dx.
(4.5)
0
Proof. Multiplying Eq. (2.2) by ut and Eq. (2.3) by vt , respectively, summing the results obtained follows the conclusion of inequality (4.3). The inequalities (4.4) and (4.5) are obtained using the same procedure as in (4.3). In what follows we are assuming that the initial data satisfies the conditions of Lemma 4.1. We introduce the functional F1 (t ) = −
∞
∫
∫ Ω
g (s)ηt (s)ds dx.
ut 0
Then the following lemma holds. Lemma 4.2. For any ϵ1 > 0 there exists a positive constant Cϵ1 = d dt
F1 (t ) ≤ −
where β1 =
∞ 0
β1
∫
2
Ω
u2t dx +
β0 ϵ1 2
∫ Ω
|∇ u|2 dx + Cϵ1
β0 β1 2ϵ1
Ω
αβ1 2ϵ1
> 0 such that ∫ ∞ α C (Ω )ϵ1 |v|2 dx g (s)|∇ηt (s)|2 ds dx +
∫ ∫
+ β1 +
2
0
(4.6)
Ω
g (s)ds.
Proof. Using Eqs. (2.2)–(2.4) and Green’s formula we get d dt
2 ∫ ∞ ∫ ∫ ∞ t dx g ( s )∇η ( x , s ) ds ∇u · g (s)∇ηt (x, s)ds dx + 0 Ω 0 Ω ∫ ∞ ∫ ∞ ∫ ∫ ∫ +α v g (s)ηt (x, s)ds dx − β1 u2t dx + ut g (s)ηst (x, s)ds dx.
F1 (t ) = β0
∫
Ω
Ω
0
Ω
0
Since ∞
∫
∫ Ω
g (s)ηst (x, s)ds dx = −
ut
Ω
0
∞
∫
∫
g ′ (s)ηt (x, s)ds dx
ut 0
and
∫ ∫ Ω
2
∞ 0
g (s)∇ηt (x, s)ds dx ≤ β1
∞
∫ ∫ Ω
g (s)|∇ηt (x, s)|2 ds dx, 0
then using Young inequality and hypotheses (2.9) our conclusion follows.
Let us consider the following functional F2 (t ) = F1 (ut , vt , ηtt ). Then using Lemma 4.2 we obtain d dt
F2 (t ) ≤ −
β1 2
∫ Ω
u2tt dx +
β 0 ϵ1 2
∫ Ω
|∇ ut |2 dx + Cϵ1
∞
∫ ∫ Ω
g (s)|∇ηtt (s)|2 ds dx + 0
α C (Ω )ϵ1 2
∫ Ω
|vt |2 dx. (4.7)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
1029
We define the functional Z1 (t ) as Z1 (t ) =
∫ Ω
ut u dx.
Using (2.2) and Green’s formula we get d dt
Z1 (t ) ≤
∫ Ω
u2t dx −
β0
∫
2
|∇ u|2 dx + Ω
β1 2β0
∞
∫ ∫ Ω
g (s)|∇ηt (x, s)|2 ds dx − α
∫
0
Ω
uv dx.
(4.8)
Let us denote by Z2 (t ) the following functional Z2 (t ) = Z1 (ut , vt , ηtt ). Then, from (4.8) we have d dt
Z2 (t ) ≤
∫ Ω
u2tt
dx −
β0
∫
2
β1 |∇ ut | dx + 2 β0 Ω 2
∞
∫ ∫ Ω
g (s)|∇η (x, s)| ds dx − α t t
2
0
∫ Ω
ut vt dx.
(4.9)
The following lemma shows the dissipative property of v . For this let us consider the following functional
Φ (t ) =
∫ Ω
utt vt dx + β0
∞
∫ ∫
∫ Ω
∇ ut · ∇v dx +
Ω
g (s)∇ηtt (s) · ∇v ds dx. 0
Lemma 4.3. For any ϵ2 > 0 we have
∫ ∫ ∞ β0 t 2 Φ (t ) ≤ −α |vt | dx + + g (s)|∇ηtt | ds dx dt 2ϵ2 2ϵ2 Ω Ω 0 ∫ ∫ ∞ k2 β1 k2 β0 + + + g (s)|∇ηt |2 ds dx 2ϵ2 2ϵ2 2ϵ2 Ω 0 ∫ ∫ 1 (g (0) + 1) β0 α + + + ϵ2 |∇v|2 dx + |utt |2 dx β1 2 β1 2 Ω Ω ∫ ∫ α C (Ω ) g (0) |∇ ut |2 dx + |∇ u|2 dx. + 2ϵ2 Ω 2 Ω ∫
d
2
1
(4.10)
Proof. Using Eqs. (2.2) and (2.3) we get d dt
Φ (t ) = −α
∫ Ω
|vt |2 dx −
Ω
∇ utt · ∇v dx − α
∞
∫ ∫
g (s)∇ηttt ds
+ Ω
∫
∫ Ω
· ∇v dx + β0
0
utt u dx
∫ Ω
∇ utt · ∇v dx.
(4.11)
On the other hand, from (2.1) we deduce
ηtt (0) = 0, ηst (0) = ut (t ) (4.12) t utt = ηttt − ηss . ∞ Recalling that β1 = 0 g (s)ds and using hypotheses (2.9) and (4.12), we obtain ∫ ∫ ∫ ∞ β1 ∇ utt · ∇v dx = g (s)∇ utt ds · ∇v dx Ω Ω 0 ∫ ∫ ∞ t t = g (s)(∇ηtt − ∇ηss )ds · ∇v dx Ω 0 ∫ ∫ ∞ ∫ ∫ ∞ t = g (s)∇ηttt ds · ∇v dx − g (s)∇ηss ds · ∇v dx Ω 0 Ω 0 ∫ ∫ ∞ ∫ ∫ ∫ ∞ = g (s)∇ηttt ds · ∇v dx + g (0) ∇ ut · ∇v dx − g ′′ (s)∇ηt ds · ∇v dx Ω 0 Ω Ω 0 ∫ ∫ ∞ ∫ ≤ g (s)∇ηttt ds · ∇v dx + g (0) ∇ ut · ∇v dx
Ω
Ω
0
∞
∫ ∫ − k2
Ω
g (s)∇ηt ds 0
· ∇v dx.
(4.13)
1030
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
Substituting the Eq. (4.13) into (4.11) and using Poincaré and Young inequalities we arrive at d dt
2 ∫ ∫ ∞ 1 g (s)∇ηttt ds dx |vt |2 dx + 2ϵ2 β1 Ω 0 Ω ∫ ∫ ∫ ϵ2 α α C (Ω ) + |∇v|2 dx + |utt |2 dx + |∇ u|2 dx 2β1 Ω 2 Ω 2 Ω 2 ∫ ∫ ∞ ∫ ϵ2 k2 g (s)∇ηt ds dx + |∇v|2 dx + 2ϵ2 β1 Ω 2β1 Ω 0 2 ∫ ∫ ∞ ∫ ∫ k2 g (0) (g (0) + 1) + g (s)∇ηt ds dx + |∇ ut |2 dx + ϵ2 |∇v|2 dx 2ϵ2 Ω 2ϵ2 Ω 2 0 Ω 2 ∫ ∫ ∞ ∫ ϵ2 β0 β0 g (s)∇ηttt ds dx + |∇v|2 dx + 2ϵ2 β1 Ω 2β1 Ω 0 2 ∫ ∫ ∞ ∫ k2 β0 ϵ2 β0 + g (s)∇ηt ds dx + |∇v|2 dx. 2ϵ2 β1 Ω 2β1 Ω 0
Φ (t ) ≤ −α
∫
(4.14)
Noting that ∞
∫ ∫ Ω
g (s)∇ηttt ds
2
∞
∫ ∫
dx ≤ β1
Ω
0
g (s)|∇ηttt |2 ds dx 0
and ∞
∫ ∫ Ω
g (s)∇ηt ds
2
dx ≤ β1
0
our conclusion follows.
∞
∫ ∫ Ω
g (s)|∇ηt |2 ds dx 0
Finally, we define the functional
Ψ (t ) =
∫ Ω
vt v dx.
(4.15)
From Eq. (2.3) we get d dt
Ψ (t ) = −
∫
|∇v|2 dx +
∫
Ω
Ω
|vt |2 dx − α
∫ Ω
u v dx.
(4.16)
Now we are in the position to prove the polynomial rate of decay. Theorem 4.1. Assume that the memory kernel g satisfies the conditions (2.9) and the initial data satisfies the conditions of Lemma 4.1, then the first-order energy E (t ) decays polynomially to zero, that is, there exists a positive constant C , being independent of the initial data, such that E (t ) ≤
C t
(E (0) + E2 (0) + E3 (0)).
Moreover, if U0 := (u0 , u1 , v0 , v1 , η) ∈ D(A2n ), then
‖T (t )‖H ≤
2n Cn −
t n k=0
‖Ak U0 ‖H .
Proof. We define L as
L(t ) = N1 (E (t ) + E2 (t ) + E3 (t )) + N2 (F1 (t ) + F2 (t )) + N3 (Z1 (t ) + Z2 (t )) + N4 Φ (t ) + Ψ (t ) where N1 , N2 , N3 and N4 are positive constants. Then from Lemmas 4.1–4.3 we get d dt
β0 β1 αβ1 k 0 C (Ω ) β1 + β1 + + N2 + N3 2 2ϵ1 2ϵ1 2 2β0 ] ∫ ∫ ∞ k2 β1 k2 β0 + + + N4 g (s)|∇ηt (x, s)|2 ds dx 2ϵ2 2ϵ2 ϵ2 Ω 0
L(t ) ≤ − N1
k1
[
−
(4.17)
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
αβ1 k0 C (Ω ) β0 β1 β1 + β1 + + N3 N2 + 2 2ϵ1 2ϵ1 2 2β0 ] ∫ ∫ ∞ k2 β1 k2 β0 t 2 g (s)|∇ηt (x, s)| ds dx + + N4 2ϵ2 2ϵ2 ϵ2 0 Ω [ αβ1 k0 C (Ω ) k1 β0 β1 β1 + β1 + + N3 N1 − N2 + 2 2ϵ1 2ϵ1 2 2β0 ] ∫ ∫ ∞ β1 k2 β0 k2 g (s)|∇ηttt (x, s)|2 ds dx + + N4 2ϵ2 2ϵ2 ϵ2 0 Ω ∫ ∫ β1 α β1 α N2 − N3 − N4 u2t dx − N2 − N3 − N4 u2tt dx 2 2 2 2 Ω Ω ∫ β0 ϵ1 α C (Ω ) β0 |∇ u|2 dx N3 − N2 − N4 2 2 2 Ω ∫ ∫ αϵ1 β0 ϵ1 g (0) β0 |∇ ut |2 dx − α N4 − N3 − N2 − N4 N2 − 1 vt2 dx 2 2 2ϵ2 2 Ω Ω ∫ α C (Ω )ϵ1 1 g (0) + 1 β0 1− N2 − + + ϵ2 N 4 |∇v|2 dx. 2 β1 2 β1 Ω
− + − + − − − −
N1
k1
[
−
Now, choosing ϵ1 , ϵ2 small enough such that 0 < ϵ1 < ϵ2 <
β0 β1 (β1 − 2ϵ1 )
α>
1031
N4 (β0 β1 (β1 − 2ϵ1 ) − 2ϵ1 (ϵ1 (β0 + β1 C (Ω ))))
β1 2
(4.18)
and N4 > 0 large enough we get
>0
is small enough. Then taking
β0 β1 + αβ1 k0 C (Ω ) (ϵ1 + α)β0 ϵ2 + g (0)β1 + β1 + + 2 +α k1 2ϵ1 β1 β0 ϵ2 (β1 − 2ϵ1 ) 2 ] k2 + β1 + 2k2 β0 β12 (ϵ1 ϵ2 α + g (0)β1 ) + N4 , + 2β0 (β1 − 2ϵ1 ) 2ϵ2 (ϵ1 + α)β0 ϵ2 + g (0)β1 2(ϵ1 β0 + β1 C (Ω )ϵ1 ) + α N4 , + α N4 N2 > max β1 β0 (β1 − 2ϵ1 ) β0 β1 ϵ2 (β1 − 2ϵ1 ) [
2
N1 >
and N3 > max
(β0 ϵ1 + C (Ω ))β1 ϵ1 β0 β1 (β1 − 2ϵ1 )
N4 ,
(ϵ1 ϵ2 αβ0 + g (0)β1 ) N4 β0 ϵ2 (β1 − 2ϵ1 )
we get that there is a positive constant ω > 0 such that d dt
L(t ) ≤ −w E (t ).
Therefore t
∫
ω
E (s)ds ≤ L(0) − L(t ),
∀t ≥ 0.
(4.19)
0
On the other hand, it is not difficult to prove that there exists a constant ς > 0 such that
L(0) − L(t ) ≤ ς(E (0) + E2 (0) + E3 (0)),
∀t ≥ 0.
(4.20)
From (4.19) and (4.20) we obtain t
∫
E (s)ds ≤ 0
ς (E (0) + E2 (0) + E3 (0)). ω
Finally, since d dt
{tE (t )} = E (t ) + t
d dt
E (t ) ≤ E (t ),
from (4.21) we get E (t ) ≤
C t
(E (0) + E2 (0) + E3 (0))
(4.21)
1032
R.G.C. Almeida, M.L. Santos / Nonlinear Analysis: Real World Applications 12 (2011) 1023–1032
ς
where C = ω . Finally, if U0 ∈ D(A2n ), then using the same procedure as in [5–7] we obtain (4.17), which completes the proof. Acknowledgements This work has been carried out, thanks to the support of the FAPESPA-PA-Brazil. In addition the second author acknowledges the support of the CNPq (306338/2008-4). The authors are thankful to the referees of this paper for valuable suggestions which improved this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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