Stability properties of an abstract system with applications to linear thermoelastic plates Filippo Dell’Oro, Jaime E. Muñoz Rivera & Vittorino Pata
Journal of Evolution Equations ISSN 1424-3199 Volume 13 Number 4 J. Evol. Equ. (2013) 13:777-794 DOI 10.1007/s00028-013-0202-6
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Author's personal copy J. Evol. Equ. 13 (2013), 777–794 © 2013 Springer Basel 1424-3199/13/040777-18, published online September 24, 2013 DOI 10.1007/s00028-013-0202-6
Journal of Evolution Equations
Stability properties of an abstract system with applications to linear thermoelastic plates ˜ Filippo Dell’Oro, Jaime E. Munoz Rivera and Vittorino Pata
Abstract. This paper deals with an abstract version of the evolution system u tt − ωu tt + (u + θ ) = 0 θt − θ − u t = 0 with ω ≥ 0, ruling the dynamics of linear thermoelastic plates. In particular, we will consider a more general coupling term depending on a real parameter σ . The (exponential and polynomial) decay properties of the related solution semigroup are investigated in dependence of the values of σ.
1. Introduction 1.1. The abstract system Let (H, ·, ·, | · |) be a real Hilbert space, and let A be a strictly positive self-adjoint linear operator A : D(A) H → H. For fixed σ ≤
3 and ω ≥ 0, 2
we consider the abstract hyperbolic-parabolic system u¨ + ω Au¨ + A2 u − Aσ θ = 0, θ˙ + Aθ + Aσ u˙ = 0.
(1.1)
Observe that no restriction σ ≥ 0 is assumed. REMARK 1.1. In fact, some of the results contained in the present paper do not require that the dense embedding D(A) ⊂ H be compact (see Remarks 3.4 and 4.5). Mathematics Subject Classification (2000): 35B35, 35B40, 47D03, 74K20 Keywords: Linear thermoelastic plates, contraction semigroups, exponential decay, polynomial decay.
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The decay properties of the solution semigroup Sω (t) associated with the abstract system (1.1) were analyzed in the recent paper [1], under the assumptions ω = 0 and σ ∈ [0, 1]. There, by means of semigroup techniques, it is shown that S0 (t) is exponentially stable if and only if 21 ≤ σ ≤ 1 and is analytic if and only if σ = 1. Moreover, when 0 ≤ σ < 21 , it is proved that the semigroup decays polynomially to zero as t −1 . The aim of this paper is twofold. First, we analyze the case ω > 0 and we prove that the corresponding semigroup Sω (t) is exponentially stable if and only if σ ≥ 1. Moreover, when 21 ≤ σ < 1, we prove that Sω (t) decays polynomially to zero as t −1/(4−4σ ) , and such a decay rate is optimal. Secondly, we come back to the case ω = 0 and we show that S0 (t) decays polynomially as t −1(1−2σ ) when 0 ≤ σ < 21 . Even in this case, the decay rate is optimal. Incidentally, the proofs of the exponential stability are carried out via explicit energy estimates (see Theorems 3.2 and 3.3), so that they can be exported to the nonlinear case. 1.2. A concrete physical model Given a bounded domain ⊂ R2 with smooth boundary ∂, we consider the evolution system u tt − ωu tt + (u + θ ) = 0, (1.2) θt − θ − u t = 0, in the unknowns u : × [0, ∞) → R and θ : × [0, ∞) → R. Such a system, written in normalized dimensionless form, rules the evolution of a homogeneous linear thermoelastic plate of shape at rest (see [2,12]). Here, the variable u stands for the vertical displacement from equilibrium, while θ represents the relative temperature. Finally, the parameter ω ≥ 0 accounts for rotational inertia, which is proportional to the thickness of the plate, ω = 0, corresponding to the case of a thin plate. The ends of the plate are assumed to be hinged, which translates into the hinged boundary conditions for u, u(t)|∂ = u(t)|∂ = 0, whereas Dirichlet boundary condition for ϑ ϑ(t)|∂ = 0 is taken, expressing the fact that the boundary ∂ is kept at equilibrium temperature for all times. Other boundary conditions for u are also physically meaningful, such as the clamped boundary condition, or conditions of mixed type (see [2]). REMARK 1.2. System (1.2) turns out to be a concrete realization of (1.1), corresponding to the choice σ = 1, H = L 2 () and A = − with domain D(A) = H 2 () ∩ H01 (). In fact, σ = 1 seems to be the only relevant case from the physical viewpoint considered in the literature.
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It is well known that the solution semigroup generated by system (1.2) is exponentially stable for every fixed ω ≥ 0, within different types of boundary conditions. In the absence of rotational inertia, that is, when ω = 0, we refer to [9,11,15–17,19,20]. For ω > 0, see [2,12,18,23] and references therein. Plan of the paper. In Sect. 2, we introduce the notation and we establish the existence of the solution semigroup, while in Sects. 3, 4 and 5, we discuss the main results about exponential and polynomial stability. 2. The dynamical system 2.1. Notation For r ∈ R, we consider the compactly nested family of Hilbert spaces r
Hr = D(A 2 ),
r
r
u, vr = A 2 u, A 2 v,
r
|u|r = |A 2 u|.
For r > 0, it is understood that H−r denotes the completion of the domain, so that H−r is the dual space of Hr . Moreover, the subscript r will be always omitted whenever zero. We will also denote by αn → ∞ the increasing sequence of the (strictly positive) eigenvalues of A, and by wn ∈ H the corresponding normalized eigenvectors. Finally, we define the family of phase spaces H2 × H × H if ω = 0, Hω = 2 1 H ×H ×H if ω > 0, endowed with the Hilbert product norms (u, v, θ ) 2ω = |u|22 + ω|v|21 + |v|2 + |θ |2 . REMARK 2.1. Throughout the paper, the Hölder, Young’s and Poincaré inequalities will be tacitly used in several occasions. 2.2. The contraction semigroup First, we show that system (1.1) generates a contraction semigroup on the space Hω . To this aim, introducing the state vector U (t) = (u(t), v(t), θ (t)), we rewrite system (1.1) as the ODE in Hω d U (t) = Aω U (t), dt where the linear operator Aω is defined as
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⎞ ⎛ ⎞ ⎛ u v Aω ⎝ v ⎠ = ⎝ A1/2 (1 + ω A)−1 A3/2 (Aσ −2 θ − u) ⎠ θ −A(θ + Aσ −1 v) with domains
and
⎧
⎫ 2
⎨ ⎬
σ −2v ∈ H D(Aω ) = U ∈ Hω
A θ − u ∈ H3 , ω > 0, ⎩
θ + Aσ −1 v ∈ H2 ⎭ ⎧
⎫ 2
⎨ ⎬
σ −2v ∈ H D(A0 ) = U ∈ H0
A θ − u ∈ H4 , ω = 0. ⎩
θ + Aσ −1 v ∈ H2 ⎭
REMARK 2.2. By the definition of the domain, if (u, v, θ ) ∈ D(Aω ), it follows that θ ∈ H1 . THEOREM 2.3. The operator Aω is the infinitesimal generator of a contraction semigroup Sω (t) = etAω : Hω → Hω . Proof. The operator Aω is dissipative, for Aω U, U Hω = −|θ |21 ≤ 0, ∀U ∈ D(Aω ).
(2.1)
Next, we show that R(1 − Aω ) = Hω . Then, the claim follows by exploiting the Lumer–Phillips theorem [21]. For F = ( f 1 , f 2 , f 3 ) ∈ Hω , we look for a solution U = (u, v, θ ) ∈ D(Aω ) to the equation U − Aω U = F,
(2.2)
which, componentwise, reads u − v = f1 , v−A
1/2
(2.3) −1
(1 + ω A)
θ + A(θ + A
σ −1
A
3/2
(A
σ −2
θ − u) = f 2 ,
v) = f 3 .
(2.4) (2.5)
Substituting the first equation of the system above into the second one, we obtain (1 + ω A)v + A2 v − Aσ θ = (1 + ω A) f 2 − A2 f 1 , (2.6) θ + Aθ + Aσ v = f 3 . Since f 2 ∈ H and f 1 ∈ H2 , we infer that (1 + ω A) f 2 − A2 f 1 ∈ H−2 .
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Then, we associate with (2.6) the bilinear form on the Hilbert space H2 × H1 , formally obtained by multiplying the system by (v, ˜ θ˜ ) in H2 × H1 , B((v, θ ), (v, ˜ θ˜ )) = v, v ˜ + ωv, v ˜ 1 + v, v ˜ 2 − A 2 θ, Aσ − 2 v ˜ 1
1
+θ, θ˜ + θ, θ˜ 1 + Aσ − 2 v, A 2 θ˜ . 1
1
It is apparent that B is coercive on H2 × H1 . In addition, since σ ≤ 23 , |B((v, θ ), (v, ˜ θ˜ ))| ≤ c |v|2 |v| ˜ 2 + |θ |1 |θ˜ |1 + |θ |1 |v| ˜ 2σ −1 + |θ˜ |1 |v|2σ −1 ≤ c (v, θ ) H2 ×H1 (v, ˜ θ˜ ) H2 ×H1 , for some constant c ≥ 0. Therefore, by means of the Lax–Milgram lemma applied to the bilinear and coercive form B, problem (2.6) admits a unique (weak) solution (v, θ ) ∈ H2 × H1 . In conclusion, in light of (2.3)–(2.5), the vector U = (v + f 1 , v, θ ) ∈ D(Aω )
solves equation (2.2).
REMARK 2.4. The operator Aω does not generate a contraction semigroup whenever σ =
3 + ε, ε > 0. 2
Otherwise, its inverse (1 − Aω )−1 would map the whole space Hω onto D(Aω ). But, from the second equation of (2.6), we learn that v ∈ H2+2ε , and thus, according to the definition of D(Aω ), the operator (1 − Aω )−1 cannot be surjective. 3. Exponential stability We now analyze the exponential stability of the semigroup Sω (t). Recall that Sω (t) is said to be exponentially stable if there are C ≥ 1 and κ > 0 such that1 Sω (t) L(Hω ) ≤ Ce−κt , ∀t ≥ 0. The forthcoming theorems are based on the following abstract lemma [22] (see also [7,8] for the statement used here). LEMMA 3.1. Let A be the infinitesimal generator of a contraction semigroup S(t) on a complex Hilbert space X . Then, the following are equivalent: 1 L(H ) denotes the space of bounded linear operators on H . ω ω
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(i) S(t) is exponentially stable. (ii) There exists ε > 0 such that inf iλx − Ax X ≥ ε x X , ∀x ∈ D(A).
λ∈R
(iii) The imaginary axis iR is contained in the resolvent set of the operator A and sup (iλ − A)−1 L(X ) < ∞.
λ∈R
Throughout the section, C ≥ 0 will denote a generic constant depending only on the structural parameters of the problem. We begin by considering the case ω > 0. THEOREM 3.2. When ω > 0, the semigroup Sω (t) is exponentially stable if and only if σ ≥ 1. Proof. Let us prove the exponential stability for σ ≥ 1. First, for any initial datum U ∈ D(Aω ), setting (u(t), u(t), ˙ θ (t)) = Sω (t)U, we define the energy Eω (t) =
1 1 2 2 Sω (t)U 2ω = |u(t)|22 + ω|u(t)| ˙ ˙ + |θ (t)|2 . 1 + |u(t)| 2 2
Multiplying in H the first equation of (1.1) by u˙ and the second one by θ , we deduce the identity d Eω + |θ |21 = 0. dt
(3.1)
Moreover, we consider the functionals
ω (t) = u(t), ˙ u(t) + ωu(t), ˙ u(t)1 , ˙ θ (t)−σ + ωu(t), ˙ θ (t)1−σ . ω (t) = u(t), We compute d
ω + |u|22 = |u| ˙ 2 + ω|u| ˙ 21 + u, θ σ dt 1 ≤ |u|22 + |u| ˙ 2 + ω|u| ˙ 21 + C|θ |21 , 4
(3.2)
and d ω + |u| ˙ 2 + ω|u| ˙ 21 = |θ |2 − u, θ 2−σ − u, ˙ θ 1−σ − ωu, ˙ θ 2−σ dt ≤ |θ |2 + |u|3−2σ |θ |1 + |u| ˙ 1−2σ |θ |1 + ω|u| ˙ 3−2σ |θ |1 1 2 1 2 ˙ + ω|u| ≤ |u|2 + |u| ˙ 21 + C|θ |21 . (3.3) 8 4
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Finally, we consider the functional ω (t) = Eω (t) + ε ω (t) + 2ω (t) , for ε > 0 small enough such that 1 Eω (t) ≤ ω (t) ≤ CEω (t). 2
(3.4)
By means of (3.1)–(3.3), we have the differential inequality d ε ω + |u|22 + |u| ˙ 2 + ω|u| ˙ 21 + (1 − Cε)|θ |21 ≤ 0, dt 2 and then, possibly reducing ε > 0, we arrive at d ω + κω ≤ 0, dt for some κ > 0. Exploiting the Gronwall lemma, together with (3.4), the claim follows. Next, we prove the lack of exponential stability if σ < 1. The strategy consists in showing that condition (ii) of Lemma 3.1 fails to hold. For every n ∈ N, denoting wn with cn = ωαn + 1, ξn = √ cn we consider the vector Fn = (0, ξn , 0) ∈ Hω with norm Fn ω = 1. We study the equation iλn Un − Aω Un = Fn , for λn ∈ R to be chosen later. Looking for a solution of the form u n = pn wn , vn = qn wn , θn = rn wn , for some pn , qn , rn ∈ C, we write the system2 iλn pn − qn = 0, 1 αn iλn qn + iωλn αn qn + αn2 pn − αnσ rn = √ + ω √ , cn cn σ iλn rn + αn rn + αn qn = 0.
(3.5) (3.6) (3.7)
2 System (3.5)–(3.7) and the following (3.8)–(3.10), as well as the forthcoming systems (4.3)–(4.5) and (4.11)–(4.13), are the versions of (2.3)–(2.5) in the current context.
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Plugging (3.5) and (3.7) into (3.6), we get −λ2n pn − ωλ2n αn pn + αn2 pn +
iλn αn2σ 1 αn pn = √ + ω √ . iλn + αn cn cn
At this point, we fix λn such that −λ2n − ωλ2n αn + αn2 = 0, which implies λn ∼
αn as n → ∞. ω
Thus, pn =
1 + ωαn −2σ αn − iαn1−2σ λ−1 √ n , cn
yielding | pn | ≥ |Im( pn )| ∼ ωαn1−2σ as n → ∞. In conclusion, we have the estimate Un ω ≥ |u n |2 ≥ αn |Im( pn )| ∼ ωαn2−2σ → ∞,
and we are done. Now, we state the corresponding theorem for ω = 0.
THEOREM 3.3. The semigroup S0 (t) is exponentially stable if and only if σ ≥ 21 . Proof. When σ ≥ 21 , the exponential stability of S0 (t) has been proved in [1], by means of semigroups techniques. However, recasting the proof of Theorem 3.2 with ω = 0, is it easy to obtain a proof via explicit energy estimates. The details are left to the reader. The proof of the lack of exponential stability is also contained in [1]. Nonetheless, since it will be needed later, we report here a little argument. Analogously to the case ω > 0, we show that condition (ii) of Lemma 3.1 fails to hold. For every n ∈ N, we set Fn = (0, wn , 0) ∈ H0 , which satisfies by construction Fn 0 = 1. We claim that the equation iαn Un − A0 Un = Fn
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has a unique solution Un = (u n , vn , θn ) ∈ D(A0 ) such that lim Un 0 = ∞.
n→∞
Looking for a solution of the form u n = pn wn , vn = qn wn , θn = rn wn , for some pn , qn , rn ∈ C, we have the system iαn pn − qn = 0,
(3.8)
− λσn rn = 1, iαn rn + λn rn + λσn qn = 0.
(3.9)
iαn qn + λ2n pn
(3.10)
Substituting (3.8) and (3.10) into (3.9), we get qn = αn1−2σ + iαn1−2σ , and, since σ < 21 , we conclude that Un 0 ≥ |qn | ≥ Re(qn ) = αn1−2σ → ∞,
(3.11)
as claimed.
REMARK 3.4. Observe that the proofs of the exponential stability of the semigroup, as well as the proof of the generation of the semigroup, do not require that the dense embedding D(A) ⊂ H be compact. Concerning the proof of the lack of exponential stability, the compactness of the embedding can be relaxed if A is unbounded, by repeating the proof replacing wn with a sequence of approximate eigenvectors relative to approximate eigenvalues tending to infinity. REMARK 3.5. In light of condition (iii) of Lemma 3.1, if the semigroup Sω (t) is exponentially stable, then the resolvent operator (iλ − Aω )−1 is bounded on the imaginary axis iR. It is then interesting to study the behavior of the norm (iλ−Aω )−1 as |λ| → ∞. As already said in the introduction, in [1], it is shown that the semigroup S0 (t) is analytic if σ = 1. In this case, (iλ − A0 )−1 decays to zero as |λ| → ∞ and the semigroup is differentiable. However, as pointed out in [2], the rotational inertia parameter ω is known to change the character of dynamics, as the system (1.1) is hyperbolic when ω > 0. Therefore, in the latter situation, analyticity is out of reach (see, e.g., [13] and [14, Chapter 3]). 4. Semiuniform and polynomial stability In this section, we study the asymptotic properties of Sω (t), particularly when lack of exponential stability occurs. We will prove that the so-called semiuniform stability [5] lim Sω (t)A−1 ω L(Hω ) = 0
t→∞
(4.1)
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holds for every value of the coupling parameter σ ≤ 23 . Since D(Aω ) is dense in Hω , it is immediate to see that (4.1) implies the stability of the semigroup Sω (t), namely lim Sω (t)U ω = 0, ∀U ∈ Hω .
t→∞
Moreover, we will obtain sharp estimates for the rate of decay when σ ∈ [ 21 , 1) (if ω > 0), or σ ∈ [0, 21 ) (if ω = 0). In particular, we will prove that the optimal decay rate is different in the two cases. In what follows, it is understood that Aω and Sω (t) will denote the complexifications of Aω and Sω (t), respectively, acting on the complexification of the space Hω . 4.1. Semiuniform stability We begin by analyzing some properties of the spectrum σ (Aω ) of the infinitesimal generator Aω , showing that it consists entirely of isolated eigenvalues. Indeed, arguing as in the proof of Theorem 2.3, it is a standard matter to prove that the operator Aω is invertible and A−1 ω ∈ L(Hω ). Moreover, since D(Aω ) is compactly embedded into −1 Hω , the inverse Aω is compact. Thus, the claim is a consequence of Lemma 4.1 below (see [10], Theorem 6.29). LEMMA 4.1. (Kato) Let A : D(A) ⊂ X → X a closed linear operator acting on a complex Banach space X . If A is invertible and the inverse operator A−1 is compact, then the spectrum of A consists entirely of isolated eigenvalues. At this point, we recall an abstract result needed in the course of the investigation (see [4, Theorem 4.4.14]; cf. also [3]). LEMMA 4.2. Let S(t) = et A : X → X be a contraction semigroup on a complex Hilbert space X . Suppose that 0 ∈ ρ(A), where ρ(A) denotes the resolvent set of the operator A. Then, the following are equivalent: (i) iR ⊂ ρ(A). (ii) limt→∞ S(t)A−1 L(X ) = 0. If (i)–(ii) occur, for any (strictly) decreasing continuous function h : [0, ∞) → (0, ∞) vanishing at infinity and satisfying S(t)A−1 L(X ) ≤ h(t), there exist positive constants C and λ0 such that 1 −1 −1 , ∀|λ| ≥ λ0 . (iλ − A) L(X ) ≤ Ch 2|λ| We now define the subsets of the complex plane: C− = {z ∈ C : Re z < 0}, C− 0 = {z ∈ C : Re z ≤ 0}. PROPOSITION 4.3. We have the inclusion σ (Aω ) ⊂ C− .
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Proof. The dissipativity of the operator Aω yields σ (Aω ) ⊂ C− 0 . We are left to prove that iR ⊂ ρ(Aω ), where ρ(Aω ) denotes the resolvent set of the operator Aω . In light of the above discussion on the spectrum of Aω , it is enough to show that no eigenvalues of Aω lie on the imaginary axis iR. To this aim, consider λ ∈ R such that iλ is an eigenvalue. Accordingly, λ = 0 (since Aω is invertible) and there exists a nonnull vector U = (u, v, θ ) ∈ D(Aω ) such that iλU − Aω U = 0. Componentwise, we draw the set of equations iλu − v = 0, iλv + iλω Av + A2 u − Aσ θ = 0, iλθ + Aθ + Aσ v = 0. By means of equality (2.1), we have the identity 0 = Re (iλ − Aω )U, U Hω = |θ |21 , and thus, θ = 0. Hence, the third equation of the system above entails v = 0, and then, from the first one, we obtain u = 0. The proof is finished. In light of Lemma 4.2, we have the following corollary. COROLLARY 4.4. The semigroup Sω (t) is semiuniformly stable (and therefore stable) for every σ ≤ 23 . REMARK 4.5. Due to Theorems 3.2 and 3.3, Proposition 4.3 trivially holds without assuming the compactness of the dense embedding D(A) ⊂ H when σ ≥ 1 (if ω > 0), or σ ≥ 21 (if ω = 0). Actually, arguing as in the proofs of the subsequent Theorems 4.6 and 4.7, we can show that iλ is not an approximate eigenvalue of Aω when σ ≥ 21 (if ω > 0), or σ ≥ 0 (if ω = 0), and so iλ ∈ σ (Aω ). For the remaining cases, we are not able to remove the assumption D(A) H. 4.2. Statement of the results about polynomial stability Regarding the case when ω > 0, the result reads as follows. THEOREM 4.6. Let ω > 0. If σ ∈ [ 21 , 1), the semigroup Sω (t) decays polynomially as t −1/(4−4σ ) ; that is, there exists a universal constant C > 0 such that Sω (t)U ω ≤
C Aω U ω , (1 + t)1/(4−4σ )
for every U ∈ D(Aω ) and every t ≥ 0. For ω = 0, we have
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THEOREM 4.7. Let ω = 0. If σ ∈ [0, 21 ), the semigroup S0 (t) decays polynomially as t −1/(1−2σ ) , that is, there exists a universal constant C > 0 such that S0 (t)U 0 ≤
C A0 U 0 , (1 + t)1/(1−2σ )
for every U ∈ D(A0 ) and every t ≥ 0. The proofs of Theorems 4.6 and 4.7 will make use of the recent abstract result [6, Theorem 2.4]. LEMMA 4.8. (Borichev–Tomilov) Let S(t) = et A be a contraction semigroup on a complex Hilbert space X . Suppose that iR ⊂ ρ(A). Then, for every fixed α > 0, (iλ − A)−1 L(X ) = O(|λ|α ) as |λ| → ∞ if and only if S(t)A−1 L(X ) = O(t −1/α ) as t → ∞. The remaining part of the section is devoted to the proofs of Theorems 4.6 and 4.7. 4.3. Proof of Theorem 4.6 Since we know from Proposition 4.3 that iR ⊂ ρ(Aω ), the resolvent equation iλU − Aω U = F
(4.2)
has a unique solution U = (u, v, θ ) for any fixed F = ( f 1 , f 2 , f 3 ) ∈ Hω . Such an equation, written in components, reads iλu − v = f 1 ,
(4.3) σ
iλv + iλω Av + A u − A θ = f 2 + ω A f 2 , 2
σ
iλθ + Aθ + A v = f 3 .
(4.4) (4.5)
First, we show that U ω ≤ C|λ|4−4σ F ω
(4.6)
for |λ| ≥ 1. As before, C ≥ 0 stands for a generic constant depending only on the structural quantities of the problem. In light of dissipativity property (2.1), a multiplication in Hω of (4.2) and U gives |θ |21 = Re(iλ − Aω )U, U Hω = ReF, U Hω ≤ F ω U ω .
(4.7)
This inequality will be tacitly used several times in what follows. Multiplying in H equation (4.4) by u, we obtain |u|22 = −iλv, u − iλω Av, u + Aσ θ, u + f 2 + ω A f 2 , u,
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and exploiting (4.3), we get |u|22 ≤ C|v|21 + C F ω U ω .
(4.8)
Next, a multiplication of (4.5) by A1−σ v, with the aid of (4.4), yields |v|21 = −iλθ, A1−σ v − Aθ, A1−σ v + f 3 , A1−σ v = θ, A1−σ (1 + ω A)−1 (−A2 u + Aσ θ + f 2 + ω A f 2 ) − Aθ, A1−σ v + f 3 , A1−σ v, leading to the estimate |v|21 ≤ C|θ |1 |u|3−2σ + |v|3−2σ + C F ω U ω .
(4.9)
We now need to control the right-hand side of (4.9). Taking the norm in H3−2σ of Eq. (4.3), we get |v|3−2σ ≤ |λ||u|3−2σ + C F ω , and hence, from inequalities (4.9) and (4.7), we arrive at |v|21 ≤ C|λ||θ |1 |u|3−2σ + C|θ |1 F ω + C F ω U ω 1/2 2 ≤ C|λ||u|3−2σ F 1/2 ω U ω + C F ω U ω + C F ω ,
(4.10)
for every |λ| ≥ 1. By interpolation and (4.3), −1 |λ||u|3−2σ ≤ |λ||u|2−2σ |u|2σ 2 1
2σ −1 |v|1 + | f 1 |1 ≤ C|λ|2−2σ U 2−2σ ω ≤ C|λ|2−2σ U ω + C|λ|2−2σ F ω2σ −1 U 2−2σ . ω
We observe that −1 F 2σ U 2−2σ ≤ C F ω + U ω . ω ω Indeed, this is trivially true if σ = 21 , and it follows from the Young’s inequality with 1 , 2σ1−1 otherwise. Accordingly, conjugate exponents 2−2σ |λ||u|3−2σ ≤ C|λ|2−2σ F ω + U ω . A substitution into (4.10) together with the Young’s inequality yields the following: 1/2 2−2σ 3/2 2 |v|21 ≤ C|λ|2−2σ F 3/2 F 1/2 ω U ω +C|λ| ω U ω +C F ω U ω +C F ω C ≤ ε U 2ω + |λ|4−4σ F ω U ω + F 2ω , ε
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for every ε > 0 small and every |λ| ≥ 1. Collecting (4.7), (4.8) and the inequality above, we obtain U 2ω ≤ C |v|21 + F ω U ω C ≤ Cε U 2ω + |λ|4−4σ F ω U ω + F 2ω . ε Then, fixing ε > 0 sufficiently small, 1 U 2ω ≤ C|λ|4−4σ F ω U ω + F 2ω ≤ U 2ω + C|λ|8−8σ F 2ω , 2 for every |λ| ≥ 1, which is nothing but (4.6). Rewriting (4.2) in the form, U = (iλ − Aω )−1 F, estimate (4.6) readily implies (iλ − Aω )−1 L(Hω ) ≤ C|λ|4−4σ , for every |λ| ≥ 1. Hence, by Lemma 4.8, we are led to −1/(4−4σ ) Sω (t)A−1 ) as t → ∞, ω L(Hω ) = O(t
in turn implying Sω (t)U ω ≤
C Aω U ω , (1 + t)1/(4−4σ )
for every U ∈ D(Aω ) and every t ≥ 0. The proof is finished.
4.4. Proof of Theorem 4.7 The argument is carried out along the lines of the case ω > 0. For this reason, we limit ourselves to sketch the proof. As usual, C ≥ 0 will denote a generic constant depending only on the structural quantities of the problem. For a fixed F = ( f 1 , f 2 , f 3 ) ∈ H0 , recalling that iR ⊂ ρ(A0 ), we consider the resolvent system iλu − v = f 1 ,
(4.11) σ
iλv + A u − A θ = f 2 , 2
σ
iλθ + Aθ + A v = f 3 ,
(4.12) (4.13)
and we show that U 0 ≤ C|λ|1−2σ F 0
(4.14)
for |λ| sufficiently large. The dissipativity property (2.1) entails |θ |21 ≤ F 0 U 0 .
(4.15)
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Again, this inequality will be tacitly used in what follows. Multiplying (4.12) by u and exploiting (4.11), we draw the estimate |u|22 ≤ 2|v|2 + C F 0 U 0 .
(4.16)
A multiplication of (4.13) by A−σ v, with the aid of (4.12), gives |v|2 = |θ |2 + θ, A−σ f 2 − θ, A2−σ u − Aθ, A−σ v + f 3 , A−σ v. Thus, by means of the Young’s inequality, we get |v|2 ≤ C|θ |1 |u|3−2σ + |v|1−2σ + C F 0 U 0 C ≤ ε |u|23−2σ + |v|21−2σ + F 0 U 0 , ε
(4.17)
for every ε > 0 small. Multiplying (4.12) by A1−2σ u, we infer that |u|23−2σ ≤ |λ||v||u|2−4σ + C|θ |1 |u|2−4σ + C F 0 U 0 , while multiplying (4.11) by A1−2σ , we deduce |v|21−2σ ≤ |λ||v||u|2−4σ + C F 0 U 0 . Plugging the two inequalities above in (4.17), we obtain |v|2 ≤ Cε|λ| U 0 |u|2−4σ + Cε|θ |1 |u|2−4σ + ≤ Cε|λ| U 0 |u|2−4σ + Cε|u|22−4σ +
C F 0 U 0 ε
C F 0 U 0 , ε
which implies |v|2 ≤ Cε|λ| U 0 |u|2−4σ +
C F 0 U 0 , ε
for every ε > small and |λ| ≥ 1. By interpolation and (4.11), |λ||u|2−4σ ≤ |λ||u|1−2σ |u|2σ 2 1−2σ ≤ C|λ|1−2σ U 0 + C|λ|1−2σ F 2σ . 0 U 0
Thus, exploiting the estimate 1−2σ F 2σ ≤ C F 0 + U 0 , 0 U 0 we arrive at |λ||u|2−4σ ≤ C|λ|1−2σ U 0 + C|λ|1−2σ F 0 . Substituting the inequality above into (4.18), we infer that |v|2 ≤ Cε|λ|1−2σ U 20 + Cε|λ|1−2σ U 0 F 0 +
C F 0 U 0 . ε
(4.18)
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We now choose ε=
1 . 8C|λ|1−2σ
Since ε has to be taken small, this is possible when |λ| is large enough. So, we get |v|2 ≤
1 1 U 20 + C|λ|1−2σ F 0 U 0 ≤ U 20 + C|λ|2−4σ F 20 8 4
for every |λ| large enough. Therefore, collecting (4.15), (4.16) and the inequality above, we end up with (4.14). In conclusion, the control, (iλ − A0 )−1 L(H0 ) ≤ C|λ|1−2σ , holds for |λ| sufficiently large. Finally, by means of Lemma 4.8, we arrive at S0 (t)U 0 ≤
C A0 U 0 , (1 + t)1/(1−2σ )
for every U ∈ D(A0 ) and every t ≥ 0, and we are finished.
5. Optimality of the decay rate We finally prove that the decay rates predicted by Theorems 4.6 and 4.7 are optimal. THEOREM 5.1. If ω > 0 and σ ∈ [ 21 , 1), then lim sup t 1/(4−4σ ) Sω (t)A−1 ω L(Hω ) > 0. t→∞
If ω = 0 and σ ∈ [0, 21 ), then lim sup t 1/(1−2σ ) S0 (t)A−1 0 L(H0 ) > 0. t→∞
Proof. We begin to consider the case ω > 0. Assume that −1/(4−4σ ) Sω (t)A−1 ), as t → ∞. ω L(Hω ) = o(t
In which case, it is apparent that Sω (t)A−1 ω L(Hω ) ≤ h(t) for some continuous decreasing function h(t) = o(t −1/(4−4σ ) ). Then, by Theorem 4.2, 1 −1 −1 , ∀|λ| ≥ λ0 . (iλ − Aω ) L(Hω ) ≤ Ch 2|λ| Since h −1
1 2|λ|
= o(|λ|4−4σ ), as |λ| → ∞,
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we learn that (iλ − Aω )−1 L(Hω ) = o(|λ|4−4σ ), as |λ| → ∞. We will reach a contradiction by showing that lim sup λ4σ −4 (iλ − Aω )−1 L(Hω ) > 0. λ→∞
To this aim, for every n ∈ N, we study the equation iλn Un − Aω Un = Fn , for λn ∈ R. Choosing λn , Un , and Fn as in the proof of Theorem 3.2, and recalling that, as n → ∞, αn and |Im( pn )| ∼ ωαn1−2σ , λn ∼ ω we infer that −4 −4 λ4σ Un ω ≥ λ4σ αn |Im( pn )| → ω3−2σ . n n
The proof for ω > 0 is finished. The case ω = 0 is similar and left to the reader (the only difference being the use of the proof Theorem 3.3 instead of Theorem 3.2). Acknowledgments The authors are grateful to the anonymous referee for very careful reading and extremely useful comments. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
M.S. Alves, S.F. Arantes and J.E. Muñoz Rivera, Exponential and polynomial decay for an abstract hyperbolic-parabolic system, (preprint). G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29 (1998), 155–182. W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), no. 2, 837–852. W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Birkhäuser, Basel, 2011. C.J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ. 8 (2008), 765–780. A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010), no. 2, 455–478. R.F. Curtain and H.J. Zwart, An introduction to infinite-dimensional linear system theory, SpringerVerlang, New York, 1995. C. Giorgi, M.G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: a semigroup approach, Comm. Appl. Anal. 5 (2001), 121–134. D.B. Henry, A. Perissinito and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. 21 (1993), 65–75.
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[email protected] F. Dell’Oro E-mail:
[email protected] J. E. Muñoz Rivera National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitadinha-Petrópolis, Rio de Janeiro, RJ 25651-070, Brazil and Av. Horácio Macedo, Centro de Tecnologia, Instituto de Matemática-UFRJ, Cidade Universitária-Ilha do Fundão, Rio de Janeiro, RJ 21941-972, Brazil E-mail:
[email protected]
