- Email: [email protected]
Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations
J. Math. Anal. Appl. 285 (2003) 191–211 www.elsevier.com/locate/jmaa
Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations Hugo Leiva ∗ and Zoraida Sivoli School of Mathematics-CDSNS, Georgia Tech, Atlanta, GA 30332, USA Received 11 March 2002 Submitted by P. Smith
Abstract In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions ut t + ∆2 u + α∆θ = f1 (t, u, θ), t 0, x ∈ Ω, θ − β∆θ − α∆ut = f2 (t, u, θ), t 0, x ∈ Ω, t θ = u = ∆u = 0, t 0, x ∈ ∂Ω, where α = 0, β > 0, Ω is a sufficiently regular bounded domain in RN (N 1) and f1e , f2e : R × L2 (Ω)2 → L2 (Ω) define by f e (t, u, θ)(x) = f (t, u(x), θ(x)), x ∈ Ω, are continuous and locally Lipschitz functions. First, we prove that the linear system (f1 = f2 = 0) generates an analytic strongly continuous semigroup which decays exponentially to zero. Second, under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable, and for a large class of functions f1 , f2 this bounded solution is almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution. 2003 Elsevier Inc. All rights reserved. Keywords: Thermoelastic plate equation; Bounded solutions; Exponential stability; Smoothness
* Corresponding author.
E-mail address: [email protected] (H. Leiva). 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-247X(03)00401-3
192
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
1. Introduction In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions ut t + ∆2 u + α∆θ = f1 (t, u, θ ), t 0, x ∈ Ω, (1.1) θ − β∆θ − α∆ut = f2 (t, u, θ ), t 0, x ∈ Ω, t θ = u = ∆u = 0, t 0, x ∈ ∂Ω, where α = 0, β > 0, Ω is a sufficiently regular bounded domain in RN (N 1) and u, θ denote the vertical deflection and the temperature of the plate, respectively. We shall assume the following hypothesis: (H1) f1e , f2e : R × L2 (Ω)2 → L2 (Ω) define by f e (t, u, θ )(x) = f (t, u(x), θ (x)), x ∈ Ω, are continuous and locally Lipschitz functions, i.e., for every ball Bρ in L2 (Ω)2 of radius ρ > 0 there exist constants L1 (ρ), L2 (ρ) > 0 such that for all (u, θ ), (v, η) ∈ Bρ e f (t, u, θ ) − f e (t, v, η) 2 Li (ρ) u − v 2 + θ − η 2 , t ∈ R. (1.2) L L i i L (H2) there exists Lf > 0 such that fi (t, 0, 0) Lf , ∀t ∈ R, i = 1, 2.
(1.3)
Remark 1.1. The hypothesis (H1) can be satisfied in the case that f1 , f2 : R × R2 → R are continuous and globally Lipschitz functions with Lipschitz constants L1 , L2 > 0, i.e., fi (t, u, θ ) − fi (t, v, η) Li |u − v|2 + |θ − η|2 , t, u, v, θ, η ∈ R, i = 1, 2.
(1.4)
The derivation of the unperturbed (fi = 0, i = 1, 2) thermoelastic plate equation wt t + ∆2 w + α∆θ = 0, t 0, x ∈ Ω, (1.5) θ − β∆θ − α∆wt = 0, t 0, x ∈ Ω, t θ = w = ∆w = 0, t 0, x ∈ ∂Ω, can be found in Lagnese [9], where the author discussed stability of various plate models. Kim [7] studied the system (1.5) with the following homogeneous Dirichlet boundary condition ∂w = w = 0, on ∂Ω, θ= ∂η and he proved the exponential decay of the energy. Also, linear thermoelastic plate equations has been studied in [2–4,10,11,17,18] which conform a good reference. One point that makes this work different from others authors works, is that here we study the existence and stability of a bounded solution for the nonlinear thermoelastic plate equation (1.1). First, we prove that the linear system (f1 = f2 = 0) generates an analytic strongly continuous semigroup which decays exponentially to zero. Second, under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable, and for a large class of functions f1 , f2 this bounded solution is
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
193
almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution. Some notation for this work can be found in [1,12–15].
2. Abstract formulation of the problem In this section we choose the space in which this problem will be set as an abstract ordinary differential equation. Let X = L2 (Ω) = L2 (Ω, R) and consider the linear unbounded operator A : D(A) ⊂ X → X defined by Aφ = −∆φ, where D(A) = H 2 (Ω, R) ∩ H01 (Ω, R).
(2.1)
The operator A has the following very well known properties: the spectrum of A consists of only eigenvalues 0 < λ1 < λ2 < · · · < λn → ∞, each one with finite multiplicity γn equal to the dimension of the corresponding eigenspace. Therefore, (a) there exists a complete orthonormal set {φn,k } of eigenvectors of A. (b) For all x ∈ D(A) we have Ax =
∞
λn
n=1
γn
x, φn,k φn,k =
k=1
∞
λn En x,
(2.2)
n=1
where ·, · is the inner product in X and En x =
γn
x, φn,k φn,k .
(2.3)
k=1
So, {En } is a family of complete orthogonal projections in X and x = x ∈ X. (c) −A generates an analytic semigroup {e−At } given by e−At x =
∞
e−λn t En x.
(d) The fractional powered spaces Xr are given by
∞ Xr = D(Ar ) = x ∈ X: (λn )2r En x 2 < ∞ , n=1
x r = A x = r
∞ n=1
n=1 En x,
(2.4)
n=1
with the norm
∞
1/2 2 λ2r n En x
,
x ∈ Xr ,
r 0,
194
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
and Ar x =
∞
λrn En x.
(2.5)
n=1
Also, for r 0 we define Zr = Xr × X × X, which is a Hilbert space with norm given by 2 w v = u 2r + v 2 + θ 2 . θ Zr Hence, Eq. (1.1) can be written as an abstract system of ordinary differential equation in Z1 = X1 × X × X as follows: u = v, (2.6) v = −A2 u + αAθ + f1 (t, u, θ ), θ = −βAθ − αAv + f2 (t, u, θ ). Finally, the system can be written as first order system of ordinary differential equations in the Hilbert space Z1 = X1 × X × X as follows: z = Az + F (t, z),
z ∈ Z1 , t 0,
where F : R × Z1 → Z1 , 0 u z= v , F (t, u, v, θ ) = f1e (t, u, θ ) , θ f2e (t, u, θ ) and
0 0 IX 2 A = −A 0 αA 0 −αA −βA is an unbounded linear operator with domain D(A) = u ∈ H 4 (Ω): u = ∆u = 0 × D(A) × D(A).
(2.7)
(2.8)
From the hypothesis (H1) we get that F is locally Lipschitz functions, i.e., for every ball Bρ in Z1 of radius ρ > 0 there exists constant Lρ such that F (t, z) − F (t, y) Lρ z − y , t ∈ R, z, y ∈ Z1 , (2.9) and from the hypothesis (H2) we obtain the following estimate: F (t, 0) LF = 2µ(Ω)Lf , t ∈ R,
(2.10)
where µ(Ω) is the Lebesgue measure of Ω.
3. The linear thermoelastic plate equation In this section we shall prove that the linear unbounded operator A given by the linear thermoelastic plate equation (1.5) generates an analytic strongly continuous semigroup which decays exponentially to zero. To this end, we will use the following lemma from [16].
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
195
Lemma 3.1. Let Z be a separable Hilbert space and {An }n1 , {Pn }n1 two families of bounded linear operators in Z with {Pn }n1 being a complete family of orthogonal projections such that An Pn = Pn An ,
n = 1, 2, 3, . . . .
(3.1)
Define the following family of linear operators: T (t)z =
∞
t 0.
eAn t Pn z,
(3.2)
n=1
Then: (a) T (t) is a linear bounded operator if eAn t g(t),
n = 1, 2, 3, . . . ,
(3.3)
for some continuous real-valued function g(t). (b) Under the condition (3.3) {T (t)}t 0 is a C0 -semigroup in the Hilbert space Z whose infinitesimal generator A is given by Az =
∞
z ∈ D(A),
An Pn z,
(3.4)
n=1
with
D(A) = z ∈ Z:
∞
An Pn z 2 < ∞ .
(3.5)
n=1
(c) The spectrum σ (A) of A is given by σ (A) =
∞
σ (A¯ n ),
(3.6)
n=1
where A¯ n = An Pn . Theorem 3.1. The operator A given by (2.8) is the infinitesimal generator of an analytic semigroup {T (t)}t 0 given by T (t)z =
∞
eAj t Pj z,
z ∈ Z1 , t 0,
(3.7)
j =1
where {Pj }j 0 is a complete family of orthogonal projections in the Hilbert space Z1 given by
Ej 0 0 (3.8) Pj = 0 Ej 0 , j = 1, 2, . . . , ∞, 0 0 Ej and
0 1 0 2 Aj = Bj Pj , Bj = −λj 0 αλj , j 1. (3.9) 0 −αλj −βλj
196
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
Moreover, the eigenvalues σ1 (j ), σ2 (j ), σ3 (j ) of the matrix Bj are simple and given by σ1 (j ) = −λj ρ1 ,
σ2 (j ) = −λj ρ2 ,
σ3 (j ) = −λj ρ3 ,
where ρi > 0, i = 1, 2, 3, are the roots of the characteristic equation ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0, and this semigroup decays exponentially to zero T (t) Me−µt , t 0,
(3.10)
where
µ = λ1 min Re(ρ): ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0 .
Proof. Let us compute Az:
0 I 0 w 2 Az = −A 0 αA v 0 −αA −βA θ
v = −A2 w + αAθ −αAv − βAθ ∞ j =1 Ej v ∞ 2 = − ∞ j =1 λj Ej w + α j =1 λj Ej θ ∞ ∞ −α j =1 λj Ej v − β j =1 λj Ej θ Ej v ∞ −λ2j Ej w + αλj Ej θ = j =1 −αλj Ej v − βλj Ej θ
∞ 0 1 0 Ej 0 2 0 Ej 0 αλj −λj = 0 0 0 −αλj −βλj j =1 ∞ = Aj Pj z.
0 0 Ej
w v θ
j =1
It is clear that Aj Pj = Pj Aj . Now, we need to check condition (3.3) from Lemma 3.1. To this end, we have to compute the spectrum of the matrix Bj . The characteristic equation of Bj is given by λ3 + βλj λ2 + λ2j (1 + α 2 )λ + βλ3j = 0. Then,
λ λj
3
λ +β λj
2 + λ2j (1 + α 2 )
λ λj
+ β = 0.
Letting λ/λj = −ρ we obtain the equation ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0.
(3.11)
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
197
From Routh–Hurwitz theorem we obtain that the real part of the roots ρ1 , ρ2 , ρ3 of Eq. (3.11) are positive. Therefore, the eigenvalues σ1 (j ), σ2 (j ), σ3 (j ) of Bj are given by σ1 (j ) = −λj ρ1 ,
σ2 (j ) = −λj ρ2 ,
σ3 (j ) = −λj ρ3 .
(3.12)
Since the eigenvalues of Bj are simple, there exists a complete family of complementaries projections {qi (j )}3i=1 in R3 such that Bj = σ1 (j )q1 (j ) + σ1 (j )q2 (j ) + σ1 (j )q3 (j ), eBj t = e−λj ρ1 t q1 (j ) + e−λj ρ2 t q2 (j ) + e−λj ρ3 t q3 (j ), where qi (j ), i = 1, 2, 3, are given by ρ2 ρ3 − 1 1 λ (ρ − ρ ) q1 (j ) = j 3 2 (ρ1 − ρ2 )(ρ1 − ρ3 ) λj α ρ1 ρ3 − 1 1 λ (ρ − ρ ) q2 (j ) = j 3 1 (ρ2 − ρ1 )(ρ2 − ρ3 ) λj α ρ1 ρ2 − 1 1 q3 (j ) = λj (ρ2 − ρ1 ) (ρ3 − ρ1 )(ρ3 − ρ2 ) λj α
ρ2 +ρ3 λj
α λj
λj
λj
λj
λj
ρ2 ρ3 − 1 − α 2 α(ρ2 + ρ3 − β) , −α(ρ2 + ρ3 − β) (ρ3 − β)2 − α 2 ρ1 +ρ3 α ρ1 ρ3 − 1 − α 2 α(ρ1 + ρ3 − β) , −α(ρ1 + ρ3 − β) (ρ3 − β)2 − α 2 ρ1 +ρ2 α ρ1 ρ2 − 1 − α 2 α(ρ1 + ρ2 − β) . −α(ρ1 + ρ2 − β) (ρ2 − β)2 − α 2
Therefore, Aj = σ1 (j )Pj 1 + σ1 (j )Pj 2 + σ1 (j )Pj 3 , eAj t = e−λj ρ1 t Pj 1 + e−λj ρ2 t Pj 2 + e−λj ρ3 t Pj 3 , and Az =
∞ σ1 (j )Pj 1 z + σ2 (j )Pj 2 z + σ3 (j )Pj 3 z ,
(3.13)
j =1
where Pj i = qi (j )Pj is a complete family of orthogonal projections in Z1 . To prove that eAn t Pn : Z1 → Z1 satisfies condition (3.3) from Lemma 3.1, it will be enough to prove for example that e−λn ρ2 t q2 (n)Pn , n = 1, 2, 3, . . . , satisfies the condition. In fact, consider z = (z1 , z2 , z3 )T ∈ Z1 such that z = 1. Then, z1 21 =
∞
λ2j Ej z1 2 1,
j =1
z3 2X =
∞
z2 2X =
∞
Ej z2 2 1 and
j =1
Ej z3 2 1.
j =1
Therefore, λj Ej z1 1, Ej z2 1, Ej z3 1, j = 1, 2, . . . . Then,
198
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
−λ ρ t e j 2 q2 (n)Pn z2 Z
1
e−2λρ2 t = (ρ2 − ρ1 )2 (ρ2 − ρ3 )2 2 3 En z2 + λαn En z3 (ρ1 ρ3 − 1)En z1 + ρ1λ+ρ n × λn (ρ3 − ρ1 )En z1 + (ρ1 ρ3 − 1 − α 2 )En z2 + α(ρ1 + ρ3 − β)En z3 λn αEn z1 + −α(ρ1 + ρ3 − β)En z2 + [(ρ3 − β)2 − α 2 ]En z3 Z1 ∞ 2 ρ1 + ρ3 α = e−2λn ρ2 t λ2j En z2 + En z3 Ej (ρ1 ρ3 − 1)En z1 + λ λj j +e
j =1 ∞ −2λn ρ2
Ej λn (ρ3 − ρ1 )En z1 + (ρ1 ρ3 − 1 − α 2 )Ez2
j =1
+ e−2λn ρ2 t
2 + α(ρ1 + ρ3 − β)Enz3
∞ Ej λn αEn z1 + −α(ρ1 + ρ3 − β)En z2 j =1
2 + (ρ3 − β)2 − α 2 En z3
2 ρ1 + ρ3 α (ρ = e−2λn ρ2 t λ2n ρ − 1)E z + E z + E z n 1 n 2 n 3 1 3 λn λn 2 + e−2λn ρ2 t λn (ρ3 − ρ1 )En z1 + (ρ1 ρ3 − 1 − α 2 )En z2 α(ρ1 + ρ3 − β)En z3 2 + e−2λn ρ2 t λα En z1 + −α(ρ1 + ρ3 − β)En z2 + (ρ3 − β)2 − α 2 En z3 2 e−2λn ρ2 t |ρ1 ρ3 − 1| + ρ1 + ρ3 + α 2 + e−2λn ρ2 t |ρ3 − ρ1 | + |ρ1 ρ3 − 1 − α 2 | + α|ρ1 + ρ3 − β| 2 + e−2λn ρ2 t α + α|ρ1 + ρ3 − β| + (ρ3 − β)2 − α 2 M 2 e−2λn ρ2 t , where M = M(α, β) 1 depending on α and β. Then we have −λ ρ t e n 2 q2 (n)Pn M(α, β)e−λn ρ2 t , t 0, n = 1, 2, . . . . Z 1
In the same way we obtain that −λ ρ t e n 1 q1 (n)Pn M(α, β)e−λn ρ1 t , t 0, n = 1, 2, . . . , Z 1 −λ ρ t e j 3 q3 (n)Pn (α, β)e−λn ρ3 t , t 0, n = 1, 2, . . . . Z 1
Therefore, eAn t Pn Z1 M(α, β)e−µt , were
t 0, n = 1, 2, . . . ,
µ = λ1 min Re(ρ): ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0 .
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
199
Hence, applying Lemma 3.1 we obtain that A generates a strongly continuous semigroup given by (3.7). Next, we prove this semigroup decays exponentially to zero. In fact, ∞ ∞ ∞ T (t)z2 = eAj t Pj z 2 eAj t 2 Pj z 2 M 2 (α, β)e−2µt Pj z 2 j =1
j =1
= M (α, β)e 2
−2µ
Therefore, T (t) M(α, β)e−µt ,
j =1
z . 2
t 0.
To prove the analyticity of {T (t)}t 0 , we shall use Theorem 1.3.4 from [6]. To this end, it will be enough to prove that the operator −A is sectorial. In order to construct the sector we shall consider the following 3 × 3 matrices: 1 1 1 λn ρ2 λn ρ3 , (3.14) K n = λn ρ1 αρ1 αρ2 αρ3 λ λ λ ρ1 −β n ρ2 −β n ρ3 −β n
−a a13 a 11 12 1 −1 (3.15) Kn = −a21 a22 −a23 , a(α, β)λn a31 −a32 a33 where αρ3 ρ2 (ρ2 − ρ3 ) αρ3 ρ1 (ρ1 − ρ3 ) , a12 = , (ρ3 − β)(ρ2 − β) (ρ3 − β)(ρ1 − β) αρ2 ρ1 (ρ1 − ρ2 ) , a13 = (ρ2 − β)(ρ1 − β) αβ(ρ2 − ρ3 ) αβ(ρ1 − ρ3 ) , a22 = , a21 = (ρ3 − β)(ρ2 − β) (ρ3 − β)(ρ1 − β) αβ(ρ1 − ρ2 ) , a23 = (ρ2 − β)(ρ1 − β) a31 = (ρ3 − ρ2 ), a32 = (ρ3 − ρ1 ), a33 = (ρ2 − ρ1 ), αρ1 ρ3 αρ2 ρ1 αρ1 ρ2 αρ3 ρ1 αρ2 ρ3 αρ3 ρ2 + + − − − . a(α, β) = (ρ3 − β) (ρ1 − β) (ρ2 − β) (ρ1 − β) (ρ3 − β) (ρ2 − β)
a11 =
Then, Bn = K −1 n J nKn, with
−λn ρ1 Jn = 0 0
n = 1, 2, 3, . . . ,
0 −λn ρ2 0
(3.16)
. −λn ρ3
Next, we define two linear bounded operators Kn : X × X × X → X1 × X × X, Kn−1 : X1 × X × X → X × X × X,
(3.17)
200
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
−1 −1 as follows: Kn = K −1 n Pn and Kn = K n Pn . Now we will obtain bounds for Kn T 1 and n . Consider z = (z1 , z2 , z3 ) ∈ Z1 = X × X × X, such that z Z1 = 1. Then,
z1 21 =
∞
λ2j Ej z1 2 1,
z2 2X =
j =1
z3 2X =
∞
Ej z2 2 1 and
j =1
∞
Ej z3 2 1.
j =1
Therefore, λj Ej z1 1, Ej z2 1, Ej z3 1, j = 1, 2, . . . . Then, −1 2 K z = n X×X×X =
1 a(α, β)λ2n
a11En z1 − a12 En z2 + a13 En z3 2 −a21 En z1 + a22 En z2 − a23En z3 a31En z1 − a32 En z2 + a33 En z3 X×X
1 a11 En z1 − a12En z2 + a13 En z3 2 a(α, β)λ2n 1 + −a21 En z1 + a22 En z2 − a23 En z3 2 a(α, β)λ2n 1 + a31 En z1 − a32En z2 + a33En z3 2 a(α, β)λ2n Γ12 (α, β) . λ2n
Therefore, −1 Γ1 (α, β) K 1 . n L(X ×X×X,X×X×X) λn
(3.18)
Next, we will find a bound for Kn L(X×X×X,X1 ×X×X) . To this end we consider z = (z1 , z2 , z3 )T ∈ Z = X × X × X, with z Z = 1. Then, zi 2 =
∞
Ej zi 2 1,
i = 1, 2, 3.
j =1
Therefore, Ej zi 1, i = 1, 2, 3, j = 1, 2, . . . , which implies 2 En z1 + En z2 + En z3 λn ρ1 En z1 + λn ρ2 En z2 + λn ρ3 En z3 Kn z 2Xα ×X = αρ λ 1 n En z1 + αρ2 λn En z2 + αρ3 λn En z3 1 ρ1 −β
= λ2n En z1
ρ2 −β
X ×X×X
ρ3 −β
+ En z2 + En z3
2
+ λn ρ1 En z1 + λn ρ2 En z2 + λn ρ3 En z3 2 2 αρ1 λn αρ2 λn αρ3 λn E E E + z + z + z ρ − β n 1 ρ − β n 2 ρ − β n 3
1 2 Γ2 (α, β)λ2n .
2
3
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
201
Hence Kn L(X×X×X,X1 ×X×X) Γ2 (α, β)λn .
(3.19)
Now, the matrix J n can be written as follows: −J n = diag[λn ρ1 , λn ρ2 , λn ρ3 ]
1 0 0 0 = λn ρ1 0 0 0 + λn ρ2 0 0 0 0 0 = λn ρ1 q1 + λn ρ2 q2 + λn ρ1 q1 .
0 0 0 1 0 + λn ρ3 0 0 0 0
0 0 0 0 0 1
π max arg(ρi ) < θ < . i=1,2,3 2
If λ ∈ Sθ , then λ is a value other than λn ρi , i = 1, 2, 3. Therefore, 1 1 (λ + J n )−1 y 2 = q1 y 2 + q2 y 2 2 (λ − λn ρ1 ) (λ − λn ρ2 )2 1 + q3 y 2 . (λ − λn ρ3 )2 Setting
|λ| : λ ∈ Sθ , n 1; i = 1, 2, 3 |λ − λn ρi |
(λ + J n )−1 y 2
Hence, (λ + J n )−1 N , |λ|
N |λ|
2
q1 y 2 + q2 y 2 + q3 y 2 .
λ ∈ Sθ .
Now, if λ ∈ Sθ , then R(λ, −A)z =
∞ ∞ (λ + An )−1 Pn z = Kn (λ + J n )−1 Kn−1 Pn z. n=1
n=1
This implies ∞ 2 2 R(λ, A)z2 Kn 2 Kn−1 (λ + J n )−1 Pn z 2 n=1
Γ1 (η, γ ) Γ2 (η, γ )
2
N |λ|
(3.21)
(3.23)
where
yields
(3.20)
(3.22)
Now, define the sector Sθ as Sθ = λ ∈ C: θ arg(λ) π, λ = 0 ,
N = sup
2 z 2 .
202
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
Therefore, R(λ, −A) R , |λ|
λ ∈ Sθ .
This completes the proof of the theorem. ✷
4. Existence of the bounded solution In this section we shall prove the existence and stability of unique bounded mild solutions of system (2.7). Definition 4.1 (Mild solution). For mild solution z(t) of (2.7) with initial condition z(t0 ) = z0 ∈ Z1 , we understand a function given by t z(t) = T (t − t0 )z0 +
T (t − s)F s, z(s) ds,
t ∈ R.
(4.1)
t0
Remark 4.1. It is easy to prove that any solution of (2.7) is a solution of (4.1). It may be thought that a solution of (4.1) is always a solution of (2.7) but this is not true in general. However, we shall prove in Theorem 5.2 that bounded solutions of (4.1) are solutions of (2.7). We shall consider Zb = Cb (R, Z1 ) the space of bounded and continuous functions defined in R taking values in Z1 . Zb is a Banach space with supremum norm z b = sup z(t)Z : t ∈ R , z ∈ Zb . 1
A ball of radio ρ > 0 and center zero in this space is given by Bρb = z ∈ Zb : z(t) ρ, t ∈ R . The proof of the following lemma is similar to Lemma 3.1 of [15]. Lemma 4.1. Let z be in Zb . Then, z is a mild solution of (2.7) if and only if z is a solution of the following integral equation: t z(t) =
T (t − s)F s, z(s) ds,
t ∈ R.
(4.2)
−∞
The following theorem refers to bounded mild solutions of system (2.7). Theorem 4.1. Suppose that F is locally Lipschitz and there exists ρ > 0 such that 0 < MLF < (µ − MLρ )ρ,
(4.3)
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
203
b where Lρ is the Lipschitz constant of F in the ball B2ρ . Then, Eq. (2.7) has one and only one bounded mild solution zb (t) which belong Bρb . Moreover, this bounded solution is exponentially stable.
Proof. For the existence of such solution, we shall prove that the following operator T : Bρb → Bρb has a unique fixed point in the ball Bρb : t (T z)(t) =
T (t − s)F s, z(s) ds,
t ∈ R.
−∞
In fact, for z ∈ Bρb we have T z(t)
t
−∞
MLρ ρ + MLF . Me−µ(t −s) Lρ z(s) + LF µ
The condition (4.3) implies that Lρ ρ + MLF < µρ
⇔
Lρ ρ + MLF < ρ. µ
Therefore, T z ∈ Bρb for all z ∈ Bρb . Now, we shall see that T is a contraction mapping. In fact, for all z1 , z2 ∈ Bρb we have that t T z1 (t) − T z2 (t) Me−µ(t −s)Lρ z1 (s) − z2 (s) ds −∞
MLρ z1 − z2 b , µ
t ∈ R.
Hence, MLρ T z1 − z2 b , µ The condition (4.3) implies that T z1 − T z2 b
0 < µ − MLρ
⇔
MLρ < µ
z1 , z2 ∈ Bρb .
⇔
MLρ < 1. µ
Therefore, T has a unique fixed point zb in Bρb : t zb (t) = (T zb )(t) =
T (t − s)F s, zb (s) ds ds,
t ∈ R.
−∞
From Lemma 3.1, zb is a bounded solution of Eq. (4.1). Now, we shall prove that zb (·) is exponentially stable. To this end, we consider any other solution z(·) of Eq. (4.1) such that z(0) − zb (0) < ρ/2M. Then, z(0) < 2ρ. As long as z(t) remains less than 2ρ we obtain the following estimate:
204
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
t z(t) − zb (t) T (t) z(0) − zb (0) + T (t − s) F s, z(s) − F s, zb (s) ds 0
Me
−µt
z(0) − zb (0) +
t
Me−µ(t −s) Lρ z(s) − zb (s) ds.
0
Then, e
z(t) − zb (t) M z(0) − zb (0) +
µ(t )
t
Meµs Lρ z(s) − zb (s) ds.
0
Hence, applying the Gronwall’s inequality we obtain z(t) − zb (t) Me(MLρ −µ)t z(0) − zb (0) ,
t ∈ [0, t1).
From (4.3) we get that MLρ − µ < 0. Therefore z(t) − zb (t) ρ/2. Hence, if z(t) < 2ρ on [0, t1 ) with t1 = sup{t > 0: z(t) < 2ρ, t ∈ [0, t1 )}, then either t1 = ∞ or z(t1 ) = 2ρ. But, the second case contradicts the above computation, b then the solution z(t) remains in the ball B2ρ for all t 0. So, z(t) − zb (t) Me(MLρ −µ)t z(0) − zb (0) , t 0. This concludes the proof of the theorem. ✷ Theorem 4.2. Suppose that F globally Lipschitz with a Lipschitz constant L > 0 and µ > ML.
(4.4)
Then, Eq. (2.7) has one and only one bounded mild solution zb (t) on R. Moreover, this bounded solution is the only bounded solution of Eq. (4.1) and is exponentially stable in large. Proof. Condition (4.4) implies that for ρ > 0 big enough we have the following estimate: 0 < MLF < (µ − ML)ρ.
(4.5)
From here, in a similar way we can prove that the following operator is a contraction mapping from Bρb into Bρb : t (T z)(t) =
T (t − s)F s, z(s) ds,
t ∈ R.
−∞
Therefore, T has a unique fixed point zb in Bρb : t zb (t) = (T zb )(t) = −∞
T (t − s)F s, zb (s) ds ds,
t ∈ R.
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
205
From Lemma 3.1, zb is a bounded solution of Eq. (4.1). Since condition (4.5) holds for any ρ > 0 big enough independent of ML < µ, then zb is the unique bounded solution of Eq. (4.1). To prove that zb (t) is exponentially stable in the large, we shall consider any other solution z(t) of (4.1) and the following estimate: t z(t) − zb (t) T (t) z(0) − zb (0) + T (t − s) F s, z(s) − F s, zb (s) ds 0
Me
−µt
z(0) − zb (0) +
t
Me−µ(t −s) Lρ z(s) − zb (s) ds.
0
Then, e
µt
z(t) − zb (t) M z(0) − zb (0) +
t
MLeµs z(s) − zb (s) ds.
0
Hence, applying the Gronwall’s inequality we obtain z(t) − zb (t) Me(ML−µ)t z(0) − zb (0) ,
t 0.
From (4.5) we know that ML − µ < 0 and therefore zb (t) is exponentially stable in the large ✷ Corollary 4.1. If F is periodic in t of period τ (F (t + τ, ξ ) = f (t, ξ )), then the unique bounded solution given by Theorems 4.1 and 4.2 is also periodic of period τ . Proof. Let zb be the unique solution of (4.1) in the ball Bρb . Then, z(t) = zb (t + τ ) is also a solution of Eq. (4.1) lying in the ball Bρb . In fact, consider z0 = zb (0) and t +τ T (t + τ − s)F s.zb (s) ds zb (t + τ ) = T (t + τ )z0 + 0
τ = T (t)T (τ )z0 +
T (t + τ − s)F s.zb (s) ds
0
t +τ + T (t + τ − s)F s.zb (s) ds τ
τ
= T (t) T (τ )z0 + 0
T (τ − s)F s.zb (s) ds
206
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
t +
T (t − s)F s.zb (s + τ ) ds
0
t = T (t)zb (τ ) +
T (t − s)F s.zb (s + τ ) ds.
0
Therefore, t z(t) = T (t)zb (τ ) +
T (t − s)F s.z(s) ds,
0
and by the uniqueness of the fixed point of the contraction mapping T in this ball, we conclude that zb (t) = zb (t + τ ), t ∈ R. ✷ Remark 4.2. Under some condition, the bounded solution given by Theorems 4.1 and 4.2 is almost periodic; for example, we can study the case when the function F has the following form: F (t, z) = g(z) + P (t),
t, ξ ∈ R,
(4.6)
where P ∈ Cb (R, Z1 ) and g : Z1 → Z1 is a locally Lipschitz function. Corollary 4.2. Suppose F has the form (4.6) and g is a globally Lipschitz function with a Lipschitz constant L > 0. Then the bounded solution zb (·, P ) given by Theorem 4.2 depends continuously on P ∈ Cb (R, Z1 ). Proof. Let P1 , P2 ∈ Cb (R, Z1 ) and zb (·, P1 ), zb (·, P2 ) be the bounded functions given by Theorem 4.2. Then t T (t − s) g zb (s, P2 ) − g zb (s, P2 ) ds zb (t, ·, P1 ) − zb (t, ·, P2 ) = −∞
t +
T (t − s) P1 (s) − P2 (s) ds.
−∞
Hence, zb (·, P1 ) − zb (·, P2 ) ML zb (·, P1 ) − zb (·, P2 ) + M P1 − P2 b . b b µ µ Therefore, zb (·, P1 ) − zb (·, P2 ) b
M P1 − P2 b . µ − ML
✷
Lemma 4.2. Suppose F is as in (4.6). Then, if P (t) is almost periodic, then the unique bounded solution of system (4.1) given by Theorems 4.1 and 4.2 is also almost periodic.
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
207
Proof. To prove this lemma, we shall use the following well known fact, due to Bohr (see Hale [5, Appendix]). A function h ∈ C(R; Z1 ) is almost periodic (a.p.) if and only if the hull H (h) of h is compact in the topology of uniform convergence. Here H (h) is the closure of the set of translates of h under the topology of uniform convergence H (h) = {hτ : τ ∈ R},
hτ (t) = h(t + τ ), t ∈ R.
Since the limit of a uniformly convergent sequence of a.p. functions is a.p., then the set Aρ of a.p. functions in the ball Bρb is closed, where ρ is given by Theorem 4.2. Claim. The contraction mapping T given in Theorems 4.1 and 4.2 leaves Aρ invariant. In fact, if z ∈ Aρ , then h(t) = g(z(t)) + P (t) is also an a.p. function. Now, consider the function t t F (t) = (T z)(t) = T (t − s) g z(s) + P (s) ds = T (t − s)h(s) ds, t ∈ R. −∞
−∞
Then, it is enough to establish that H (F ) is compact in the topology of uniform convergence. Let {Fτk } be any sequence in H (F ). Since h is a.p., we can select from {hτk } a Cauchy subsequence {hτkj }, and we have that t +τkj
Fτkj (t) = F (t + τkj ) =
t T (t + τkj − s)h(s) ds =
−∞
T (t − s)h(s + τkj ) ds.
−∞
Hence, Fτ (t) − Fτ (t) kj ki
t
e−µ(t −s) h(s + τkj ) − h(s + τki ) ds
−∞
t hτkj − hτki b −∞
e−µ(t −s) ds =
1 hτ − hτki b . µ kj
Therefore, {Fτkj } is a Cauchy sequence. So, H (F ) is compact in the topology of uniform convergence, F is a.p. and T Aρ ⊂ Aρ . Now, the unique fixed point of T in the ball Bρb lies in Aρ . Hence, the unique bounded solution zb (t) of Eq. (4.1) given in Theorem 4.2 is also almost periodic. ✷ 5. Smoothness of the bounded solution In this section we shall prove that the bounded solution of Eq. (4.1) given by Theorems 4.1 and 4.2 is also solution of the original equation (2.7). That is to say, this bounded solution is a classic solution of Eq. (2.7). To this end, we will use the following theorem from [8].
208
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
Theorem 5.1. Let A on D(A) be a closed operator in the Banach space X and x ∈ C([a, b); X) with b ∞. Suppose that x(t) ∈ D(A), Ax(t) is continuous on [a, b) and that the improper integrals b
b x(s) ds
Ax(s) ds
and
a
a
exist. Then b
b x(s) ds ∈ D(A)
x(s) ds =
and A
a
b
a
Ax(s) ds. a
Theorem 5.2. The bounded Mild solution zb (t) of Eq. (2.7) given by Theorems 4.1 and 4.2 is a classic solution of this equation on R, i.e., zb (t) = Azb (t) + F t, zb (t) , t ∈ R. Proof. Let zb (t) be the only bounded mild solution of (2.7) given by Theorems 4.1 and 4.2. Then, t z(t) =
∞ T (t − s)g(s) ds =
−∞
T (s)g(t − s) ds,
t ∈ R,
0
where g(s) = F (s, zb (s)). Therefore, g ∈ Cb (R, Z1 ) and g(s) g b , s ∈ (−∞, t). Let us put x(s) = T (t − s)g(s), s ∈ (−∞, t). Then x(s) is a continuous function, and since {T (t)}t 0 is analytic, then x(s) ∈ D(A),
for s < t.
Claim. Ax(s) is continuous on (−∞, t) and the improper integral t Ax(s) ds,
t ∈ R,
−∞
exists. From Theorem 3.1, there exists a complete family of orthogonal projections {qi (j )}3i=1 in R3 such that Aj = σ1 (j )q1 (j ) + σ1 (j )q2 (j ) + σ1 (j )q3 (j ), eAj t = e−λj ρ1 t q1 (j ) + e−λj ρ2 t q2 (j ) + e−λj ρ3 t q3 (j ). Hence, Az =
∞ σ1 (j )Pj 1 z + σ2 (j )Pj 2 z + σ3 (j )Pj 3 z j =1
(5.1)
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
209
and T (t)z =
∞ −λ ρ t e j 1 Pj 1 z + e−λj ρ2 t Pj 2 z + e−λj ρ3 t Pj 3 z , j =1
where Pj i = qi (j )Pj is a complete family of orthogonal projections in Z1 . Therefore, Ax(s) =
∞ −λj ρ1 e−λj ρ1 (t −s)Pj 1 g(s) − λj ρ2 e−λj ρ2 (t −s)pj 2 g(s) j =1
So,
− λj ρ3 e−λj ρ3 (t −s)pj 3 g(s) .
Ax(s) max λj |ρi |e−λj Re(ρi )(t −s): i = 1, 2, 3 g b . j 1
Then, using the dominate convergence theorem, we get that Ax(s) is a continuous function on (−∞, t). Now, consider the following improper integrals: t
∞ Ax(s) ds =
−∞
AT (s)g(t − s) ds 0
∞ ∞ = −λj ρ1 e−λj ρ1 s Pj 1 g(t − s) − λj ρ2 e−λj ρ2 s Pj 2 g(t − s) 0 j =1 − λj ρ3 e−λj ρ3 s Pj 3 g(t − s) ds
∞ ∞ = −λj ρ1 e−λj ρ1 s Pj 1 g(t − s) ds j =1
0
∞ −
λj ρ2 e−λj ρ2 s Pj 2 g(t − s) ds
0
∞ −
λj ρ3 e
−λj ρ3 s
Pj 3 g(t − s) ds .
0
On the other hand, we have that ∞ ∞ −λj ρi e−λj ρi s Pj i g(t − s) ds λj |ρi |e−λj Re(ρi )s Pj i g(t − s) ds 0
0
Therefore, the improper integral t Ax(s) ds −∞
|ρi | g b . Re(ρi )
210
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
exists. Now, from Theorem 5.1 we obtain that t
t x(s) ds ∈ D(A)
and A
−∞
t x(s) ds =
−∞
Ax(s) ds,
−∞
i.e., t
t T (t − s)g(s) ds ∈ D(A)
and A
−∞
t T (t − s)g(s) ds =
−∞
AT (t − s)g(s) ds.
−∞
Now, we are ready to prove that zb (t) is a solution of (2.7). In fact, consider zb (t + h) − zb (t) 1 = h h =
t +h t 1 T (t + h − s)g(s) ds − T (t − s)g(s) ds h
−∞
T (h) − I h
−∞
t T (t − s)g(s) ds + −∞
1 h
t +h T (t + h − s)g(s) ds. t
Using the definition of infinitesimal generator of a semigroup and passing to the limit as h → 0+ we get that zb (t) = A
t T (t − s)g(s) ds + T (0)g(t).
−∞
So,
zb (t) = Azb (t) + F t, zb (t) ,
t ∈ R.
✷
References [1] J.M. Alonso, J. Mawhin, R. Ortega, Bounded solutions of second order semilinear evolution equations and applications to the telegraph equation, J. Math. Pures Appl. 78 (1999) 49–63. [2] G. Avalos, I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste Suppl. XXVIII (1997) 1–28. [3] G. Avalos, I. Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions, SIAM J. Control Optim. 38 (2000) 337–383. [4] G. Avalos, I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29 (1998) 155–182. [5] J. Hale, Ordinary Differential Equations, Wiley–Interscience, New York, 1969. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. [7] J.U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (1992) 889. [8] G.E. Ladas, V. Lakshmikantham, Differential Equations in Abstract Spaces, in: Math. in Science and Engineering, Vol. 85, Academic Press, 1972. [9] J. Lagnese, Boundary Stabilization of Thin Plate, in: SIAM Studies in Appl. Math., Vol. 10, 1989. [10] L. de Teresa, E. Zuazua, Controllability of the linear system of thermoelastic plates, Adv. Differential Equations 1 (1996) 369–402.
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
211
[11] L. de Teresa, Approximate controllability of a semilinear heat equation in RN , SIAM J. Control Optim. 36 (1998) 2128–2147. [12] H. Leiva, Stability of a periodic solution for a system of parabolic equations, J. Appl. Anal. 60 (1996) 277–300. [13] H. Leiva, Existence of bounded solutions of a second order system with dissipation, J. Math. Anal. Appl. 237 (1999) 288–302. [14] L. Garcia, H. Leiva, Center manifold and exponentially bounded solutions of a forced Newtonian system with dissipation, Electron. J. Differential Equations Conf. 5 (2000) 69–77. [15] H. Leiva, Existence of bounded solutions of a second order evolution equation and applications, J. Math. Phys. 41 (2000). [16] H. Leiva, A lemma on C0 -semigroups and applications PDEs systems, Report Series of CDSNS 99-353. [17] Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Comp. Appl. Math. 13 (1994) 81–102. [18] J.H. Ortega, E. Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation, SIAM J. Control Optim. 39 (2001) 1585–1614.
Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations Hugo Leiva ∗ and Zoraida Sivoli School of Mathematics-CDSNS, Georgia Tech, Atlanta, GA 30332, USA Received 11 March 2002 Submitted by P. Smith
Abstract In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions ut t + ∆2 u + α∆θ = f1 (t, u, θ), t 0, x ∈ Ω, θ − β∆θ − α∆ut = f2 (t, u, θ), t 0, x ∈ Ω, t θ = u = ∆u = 0, t 0, x ∈ ∂Ω, where α = 0, β > 0, Ω is a sufficiently regular bounded domain in RN (N 1) and f1e , f2e : R × L2 (Ω)2 → L2 (Ω) define by f e (t, u, θ)(x) = f (t, u(x), θ(x)), x ∈ Ω, are continuous and locally Lipschitz functions. First, we prove that the linear system (f1 = f2 = 0) generates an analytic strongly continuous semigroup which decays exponentially to zero. Second, under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable, and for a large class of functions f1 , f2 this bounded solution is almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution. 2003 Elsevier Inc. All rights reserved. Keywords: Thermoelastic plate equation; Bounded solutions; Exponential stability; Smoothness
* Corresponding author.
E-mail address: [email protected] (H. Leiva). 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-247X(03)00401-3
192
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
1. Introduction In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions ut t + ∆2 u + α∆θ = f1 (t, u, θ ), t 0, x ∈ Ω, (1.1) θ − β∆θ − α∆ut = f2 (t, u, θ ), t 0, x ∈ Ω, t θ = u = ∆u = 0, t 0, x ∈ ∂Ω, where α = 0, β > 0, Ω is a sufficiently regular bounded domain in RN (N 1) and u, θ denote the vertical deflection and the temperature of the plate, respectively. We shall assume the following hypothesis: (H1) f1e , f2e : R × L2 (Ω)2 → L2 (Ω) define by f e (t, u, θ )(x) = f (t, u(x), θ (x)), x ∈ Ω, are continuous and locally Lipschitz functions, i.e., for every ball Bρ in L2 (Ω)2 of radius ρ > 0 there exist constants L1 (ρ), L2 (ρ) > 0 such that for all (u, θ ), (v, η) ∈ Bρ e f (t, u, θ ) − f e (t, v, η) 2 Li (ρ) u − v 2 + θ − η 2 , t ∈ R. (1.2) L L i i L (H2) there exists Lf > 0 such that fi (t, 0, 0) Lf , ∀t ∈ R, i = 1, 2.
(1.3)
Remark 1.1. The hypothesis (H1) can be satisfied in the case that f1 , f2 : R × R2 → R are continuous and globally Lipschitz functions with Lipschitz constants L1 , L2 > 0, i.e., fi (t, u, θ ) − fi (t, v, η) Li |u − v|2 + |θ − η|2 , t, u, v, θ, η ∈ R, i = 1, 2.
(1.4)
The derivation of the unperturbed (fi = 0, i = 1, 2) thermoelastic plate equation wt t + ∆2 w + α∆θ = 0, t 0, x ∈ Ω, (1.5) θ − β∆θ − α∆wt = 0, t 0, x ∈ Ω, t θ = w = ∆w = 0, t 0, x ∈ ∂Ω, can be found in Lagnese [9], where the author discussed stability of various plate models. Kim [7] studied the system (1.5) with the following homogeneous Dirichlet boundary condition ∂w = w = 0, on ∂Ω, θ= ∂η and he proved the exponential decay of the energy. Also, linear thermoelastic plate equations has been studied in [2–4,10,11,17,18] which conform a good reference. One point that makes this work different from others authors works, is that here we study the existence and stability of a bounded solution for the nonlinear thermoelastic plate equation (1.1). First, we prove that the linear system (f1 = f2 = 0) generates an analytic strongly continuous semigroup which decays exponentially to zero. Second, under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable, and for a large class of functions f1 , f2 this bounded solution is
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
193
almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution. Some notation for this work can be found in [1,12–15].
2. Abstract formulation of the problem In this section we choose the space in which this problem will be set as an abstract ordinary differential equation. Let X = L2 (Ω) = L2 (Ω, R) and consider the linear unbounded operator A : D(A) ⊂ X → X defined by Aφ = −∆φ, where D(A) = H 2 (Ω, R) ∩ H01 (Ω, R).
(2.1)
The operator A has the following very well known properties: the spectrum of A consists of only eigenvalues 0 < λ1 < λ2 < · · · < λn → ∞, each one with finite multiplicity γn equal to the dimension of the corresponding eigenspace. Therefore, (a) there exists a complete orthonormal set {φn,k } of eigenvectors of A. (b) For all x ∈ D(A) we have Ax =
∞
λn
n=1
γn
x, φn,k φn,k =
k=1
∞
λn En x,
(2.2)
n=1
where ·, · is the inner product in X and En x =
γn
x, φn,k φn,k .
(2.3)
k=1
So, {En } is a family of complete orthogonal projections in X and x = x ∈ X. (c) −A generates an analytic semigroup {e−At } given by e−At x =
∞
e−λn t En x.
(d) The fractional powered spaces Xr are given by
∞ Xr = D(Ar ) = x ∈ X: (λn )2r En x 2 < ∞ , n=1
x r = A x = r
∞ n=1
n=1 En x,
(2.4)
n=1
with the norm
∞
1/2 2 λ2r n En x
,
x ∈ Xr ,
r 0,
194
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
and Ar x =
∞
λrn En x.
(2.5)
n=1
Also, for r 0 we define Zr = Xr × X × X, which is a Hilbert space with norm given by 2 w v = u 2r + v 2 + θ 2 . θ Zr Hence, Eq. (1.1) can be written as an abstract system of ordinary differential equation in Z1 = X1 × X × X as follows: u = v, (2.6) v = −A2 u + αAθ + f1 (t, u, θ ), θ = −βAθ − αAv + f2 (t, u, θ ). Finally, the system can be written as first order system of ordinary differential equations in the Hilbert space Z1 = X1 × X × X as follows: z = Az + F (t, z),
z ∈ Z1 , t 0,
where F : R × Z1 → Z1 , 0 u z= v , F (t, u, v, θ ) = f1e (t, u, θ ) , θ f2e (t, u, θ ) and
0 0 IX 2 A = −A 0 αA 0 −αA −βA is an unbounded linear operator with domain D(A) = u ∈ H 4 (Ω): u = ∆u = 0 × D(A) × D(A).
(2.7)
(2.8)
From the hypothesis (H1) we get that F is locally Lipschitz functions, i.e., for every ball Bρ in Z1 of radius ρ > 0 there exists constant Lρ such that F (t, z) − F (t, y) Lρ z − y , t ∈ R, z, y ∈ Z1 , (2.9) and from the hypothesis (H2) we obtain the following estimate: F (t, 0) LF = 2µ(Ω)Lf , t ∈ R,
(2.10)
where µ(Ω) is the Lebesgue measure of Ω.
3. The linear thermoelastic plate equation In this section we shall prove that the linear unbounded operator A given by the linear thermoelastic plate equation (1.5) generates an analytic strongly continuous semigroup which decays exponentially to zero. To this end, we will use the following lemma from [16].
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
195
Lemma 3.1. Let Z be a separable Hilbert space and {An }n1 , {Pn }n1 two families of bounded linear operators in Z with {Pn }n1 being a complete family of orthogonal projections such that An Pn = Pn An ,
n = 1, 2, 3, . . . .
(3.1)
Define the following family of linear operators: T (t)z =
∞
t 0.
eAn t Pn z,
(3.2)
n=1
Then: (a) T (t) is a linear bounded operator if eAn t g(t),
n = 1, 2, 3, . . . ,
(3.3)
for some continuous real-valued function g(t). (b) Under the condition (3.3) {T (t)}t 0 is a C0 -semigroup in the Hilbert space Z whose infinitesimal generator A is given by Az =
∞
z ∈ D(A),
An Pn z,
(3.4)
n=1
with
D(A) = z ∈ Z:
∞
An Pn z 2 < ∞ .
(3.5)
n=1
(c) The spectrum σ (A) of A is given by σ (A) =
∞
σ (A¯ n ),
(3.6)
n=1
where A¯ n = An Pn . Theorem 3.1. The operator A given by (2.8) is the infinitesimal generator of an analytic semigroup {T (t)}t 0 given by T (t)z =
∞
eAj t Pj z,
z ∈ Z1 , t 0,
(3.7)
j =1
where {Pj }j 0 is a complete family of orthogonal projections in the Hilbert space Z1 given by
Ej 0 0 (3.8) Pj = 0 Ej 0 , j = 1, 2, . . . , ∞, 0 0 Ej and
0 1 0 2 Aj = Bj Pj , Bj = −λj 0 αλj , j 1. (3.9) 0 −αλj −βλj
196
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
Moreover, the eigenvalues σ1 (j ), σ2 (j ), σ3 (j ) of the matrix Bj are simple and given by σ1 (j ) = −λj ρ1 ,
σ2 (j ) = −λj ρ2 ,
σ3 (j ) = −λj ρ3 ,
where ρi > 0, i = 1, 2, 3, are the roots of the characteristic equation ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0, and this semigroup decays exponentially to zero T (t) Me−µt , t 0,
(3.10)
where
µ = λ1 min Re(ρ): ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0 .
Proof. Let us compute Az:
0 I 0 w 2 Az = −A 0 αA v 0 −αA −βA θ
v = −A2 w + αAθ −αAv − βAθ ∞ j =1 Ej v ∞ 2 = − ∞ j =1 λj Ej w + α j =1 λj Ej θ ∞ ∞ −α j =1 λj Ej v − β j =1 λj Ej θ Ej v ∞ −λ2j Ej w + αλj Ej θ = j =1 −αλj Ej v − βλj Ej θ
∞ 0 1 0 Ej 0 2 0 Ej 0 αλj −λj = 0 0 0 −αλj −βλj j =1 ∞ = Aj Pj z.
0 0 Ej
w v θ
j =1
It is clear that Aj Pj = Pj Aj . Now, we need to check condition (3.3) from Lemma 3.1. To this end, we have to compute the spectrum of the matrix Bj . The characteristic equation of Bj is given by λ3 + βλj λ2 + λ2j (1 + α 2 )λ + βλ3j = 0. Then,
λ λj
3
λ +β λj
2 + λ2j (1 + α 2 )
λ λj
+ β = 0.
Letting λ/λj = −ρ we obtain the equation ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0.
(3.11)
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
197
From Routh–Hurwitz theorem we obtain that the real part of the roots ρ1 , ρ2 , ρ3 of Eq. (3.11) are positive. Therefore, the eigenvalues σ1 (j ), σ2 (j ), σ3 (j ) of Bj are given by σ1 (j ) = −λj ρ1 ,
σ2 (j ) = −λj ρ2 ,
σ3 (j ) = −λj ρ3 .
(3.12)
Since the eigenvalues of Bj are simple, there exists a complete family of complementaries projections {qi (j )}3i=1 in R3 such that Bj = σ1 (j )q1 (j ) + σ1 (j )q2 (j ) + σ1 (j )q3 (j ), eBj t = e−λj ρ1 t q1 (j ) + e−λj ρ2 t q2 (j ) + e−λj ρ3 t q3 (j ), where qi (j ), i = 1, 2, 3, are given by ρ2 ρ3 − 1 1 λ (ρ − ρ ) q1 (j ) = j 3 2 (ρ1 − ρ2 )(ρ1 − ρ3 ) λj α ρ1 ρ3 − 1 1 λ (ρ − ρ ) q2 (j ) = j 3 1 (ρ2 − ρ1 )(ρ2 − ρ3 ) λj α ρ1 ρ2 − 1 1 q3 (j ) = λj (ρ2 − ρ1 ) (ρ3 − ρ1 )(ρ3 − ρ2 ) λj α
ρ2 +ρ3 λj
α λj
λj
λj
λj
λj
ρ2 ρ3 − 1 − α 2 α(ρ2 + ρ3 − β) , −α(ρ2 + ρ3 − β) (ρ3 − β)2 − α 2 ρ1 +ρ3 α ρ1 ρ3 − 1 − α 2 α(ρ1 + ρ3 − β) , −α(ρ1 + ρ3 − β) (ρ3 − β)2 − α 2 ρ1 +ρ2 α ρ1 ρ2 − 1 − α 2 α(ρ1 + ρ2 − β) . −α(ρ1 + ρ2 − β) (ρ2 − β)2 − α 2
Therefore, Aj = σ1 (j )Pj 1 + σ1 (j )Pj 2 + σ1 (j )Pj 3 , eAj t = e−λj ρ1 t Pj 1 + e−λj ρ2 t Pj 2 + e−λj ρ3 t Pj 3 , and Az =
∞ σ1 (j )Pj 1 z + σ2 (j )Pj 2 z + σ3 (j )Pj 3 z ,
(3.13)
j =1
where Pj i = qi (j )Pj is a complete family of orthogonal projections in Z1 . To prove that eAn t Pn : Z1 → Z1 satisfies condition (3.3) from Lemma 3.1, it will be enough to prove for example that e−λn ρ2 t q2 (n)Pn , n = 1, 2, 3, . . . , satisfies the condition. In fact, consider z = (z1 , z2 , z3 )T ∈ Z1 such that z = 1. Then, z1 21 =
∞
λ2j Ej z1 2 1,
j =1
z3 2X =
∞
z2 2X =
∞
Ej z2 2 1 and
j =1
Ej z3 2 1.
j =1
Therefore, λj Ej z1 1, Ej z2 1, Ej z3 1, j = 1, 2, . . . . Then,
198
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
−λ ρ t e j 2 q2 (n)Pn z2 Z
1
e−2λρ2 t = (ρ2 − ρ1 )2 (ρ2 − ρ3 )2 2 3 En z2 + λαn En z3 (ρ1 ρ3 − 1)En z1 + ρ1λ+ρ n × λn (ρ3 − ρ1 )En z1 + (ρ1 ρ3 − 1 − α 2 )En z2 + α(ρ1 + ρ3 − β)En z3 λn αEn z1 + −α(ρ1 + ρ3 − β)En z2 + [(ρ3 − β)2 − α 2 ]En z3 Z1 ∞ 2 ρ1 + ρ3 α = e−2λn ρ2 t λ2j En z2 + En z3 Ej (ρ1 ρ3 − 1)En z1 + λ λj j +e
j =1 ∞ −2λn ρ2
Ej λn (ρ3 − ρ1 )En z1 + (ρ1 ρ3 − 1 − α 2 )Ez2
j =1
+ e−2λn ρ2 t
2 + α(ρ1 + ρ3 − β)Enz3
∞ Ej λn αEn z1 + −α(ρ1 + ρ3 − β)En z2 j =1
2 + (ρ3 − β)2 − α 2 En z3
2 ρ1 + ρ3 α (ρ = e−2λn ρ2 t λ2n ρ − 1)E z + E z + E z n 1 n 2 n 3 1 3 λn λn 2 + e−2λn ρ2 t λn (ρ3 − ρ1 )En z1 + (ρ1 ρ3 − 1 − α 2 )En z2 α(ρ1 + ρ3 − β)En z3 2 + e−2λn ρ2 t λα En z1 + −α(ρ1 + ρ3 − β)En z2 + (ρ3 − β)2 − α 2 En z3 2 e−2λn ρ2 t |ρ1 ρ3 − 1| + ρ1 + ρ3 + α 2 + e−2λn ρ2 t |ρ3 − ρ1 | + |ρ1 ρ3 − 1 − α 2 | + α|ρ1 + ρ3 − β| 2 + e−2λn ρ2 t α + α|ρ1 + ρ3 − β| + (ρ3 − β)2 − α 2 M 2 e−2λn ρ2 t , where M = M(α, β) 1 depending on α and β. Then we have −λ ρ t e n 2 q2 (n)Pn M(α, β)e−λn ρ2 t , t 0, n = 1, 2, . . . . Z 1
In the same way we obtain that −λ ρ t e n 1 q1 (n)Pn M(α, β)e−λn ρ1 t , t 0, n = 1, 2, . . . , Z 1 −λ ρ t e j 3 q3 (n)Pn (α, β)e−λn ρ3 t , t 0, n = 1, 2, . . . . Z 1
Therefore, eAn t Pn Z1 M(α, β)e−µt , were
t 0, n = 1, 2, . . . ,
µ = λ1 min Re(ρ): ρ 3 − βρ 2 + (1 + α 2 )ρ − β = 0 .
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
199
Hence, applying Lemma 3.1 we obtain that A generates a strongly continuous semigroup given by (3.7). Next, we prove this semigroup decays exponentially to zero. In fact, ∞ ∞ ∞ T (t)z2 = eAj t Pj z 2 eAj t 2 Pj z 2 M 2 (α, β)e−2µt Pj z 2 j =1
j =1
= M (α, β)e 2
−2µ
Therefore, T (t) M(α, β)e−µt ,
j =1
z . 2
t 0.
To prove the analyticity of {T (t)}t 0 , we shall use Theorem 1.3.4 from [6]. To this end, it will be enough to prove that the operator −A is sectorial. In order to construct the sector we shall consider the following 3 × 3 matrices: 1 1 1 λn ρ2 λn ρ3 , (3.14) K n = λn ρ1 αρ1 αρ2 αρ3 λ λ λ ρ1 −β n ρ2 −β n ρ3 −β n
−a a13 a 11 12 1 −1 (3.15) Kn = −a21 a22 −a23 , a(α, β)λn a31 −a32 a33 where αρ3 ρ2 (ρ2 − ρ3 ) αρ3 ρ1 (ρ1 − ρ3 ) , a12 = , (ρ3 − β)(ρ2 − β) (ρ3 − β)(ρ1 − β) αρ2 ρ1 (ρ1 − ρ2 ) , a13 = (ρ2 − β)(ρ1 − β) αβ(ρ2 − ρ3 ) αβ(ρ1 − ρ3 ) , a22 = , a21 = (ρ3 − β)(ρ2 − β) (ρ3 − β)(ρ1 − β) αβ(ρ1 − ρ2 ) , a23 = (ρ2 − β)(ρ1 − β) a31 = (ρ3 − ρ2 ), a32 = (ρ3 − ρ1 ), a33 = (ρ2 − ρ1 ), αρ1 ρ3 αρ2 ρ1 αρ1 ρ2 αρ3 ρ1 αρ2 ρ3 αρ3 ρ2 + + − − − . a(α, β) = (ρ3 − β) (ρ1 − β) (ρ2 − β) (ρ1 − β) (ρ3 − β) (ρ2 − β)
a11 =
Then, Bn = K −1 n J nKn, with
−λn ρ1 Jn = 0 0
n = 1, 2, 3, . . . ,
0 −λn ρ2 0
(3.16)
. −λn ρ3
Next, we define two linear bounded operators Kn : X × X × X → X1 × X × X, Kn−1 : X1 × X × X → X × X × X,
(3.17)
200
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
−1 −1 as follows: Kn = K −1 n Pn and Kn = K n Pn . Now we will obtain bounds for Kn T 1 and n . Consider z = (z1 , z2 , z3 ) ∈ Z1 = X × X × X, such that z Z1 = 1. Then,
z1 21 =
∞
λ2j Ej z1 2 1,
z2 2X =
j =1
z3 2X =
∞
Ej z2 2 1 and
j =1
∞
Ej z3 2 1.
j =1
Therefore, λj Ej z1 1, Ej z2 1, Ej z3 1, j = 1, 2, . . . . Then, −1 2 K z = n X×X×X =
1 a(α, β)λ2n
a11En z1 − a12 En z2 + a13 En z3 2 −a21 En z1 + a22 En z2 − a23En z3 a31En z1 − a32 En z2 + a33 En z3 X×X
1 a11 En z1 − a12En z2 + a13 En z3 2 a(α, β)λ2n 1 + −a21 En z1 + a22 En z2 − a23 En z3 2 a(α, β)λ2n 1 + a31 En z1 − a32En z2 + a33En z3 2 a(α, β)λ2n Γ12 (α, β) . λ2n
Therefore, −1 Γ1 (α, β) K 1 . n L(X ×X×X,X×X×X) λn
(3.18)
Next, we will find a bound for Kn L(X×X×X,X1 ×X×X) . To this end we consider z = (z1 , z2 , z3 )T ∈ Z = X × X × X, with z Z = 1. Then, zi 2 =
∞
Ej zi 2 1,
i = 1, 2, 3.
j =1
Therefore, Ej zi 1, i = 1, 2, 3, j = 1, 2, . . . , which implies 2 En z1 + En z2 + En z3 λn ρ1 En z1 + λn ρ2 En z2 + λn ρ3 En z3 Kn z 2Xα ×X = αρ λ 1 n En z1 + αρ2 λn En z2 + αρ3 λn En z3 1 ρ1 −β
= λ2n En z1
ρ2 −β
X ×X×X
ρ3 −β
+ En z2 + En z3
2
+ λn ρ1 En z1 + λn ρ2 En z2 + λn ρ3 En z3 2 2 αρ1 λn αρ2 λn αρ3 λn E E E + z + z + z ρ − β n 1 ρ − β n 2 ρ − β n 3
1 2 Γ2 (α, β)λ2n .
2
3
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
201
Hence Kn L(X×X×X,X1 ×X×X) Γ2 (α, β)λn .
(3.19)
Now, the matrix J n can be written as follows: −J n = diag[λn ρ1 , λn ρ2 , λn ρ3 ]
1 0 0 0 = λn ρ1 0 0 0 + λn ρ2 0 0 0 0 0 = λn ρ1 q1 + λn ρ2 q2 + λn ρ1 q1 .
0 0 0 1 0 + λn ρ3 0 0 0 0
0 0 0 0 0 1
π max arg(ρi ) < θ < . i=1,2,3 2
If λ ∈ Sθ , then λ is a value other than λn ρi , i = 1, 2, 3. Therefore, 1 1 (λ + J n )−1 y 2 = q1 y 2 + q2 y 2 2 (λ − λn ρ1 ) (λ − λn ρ2 )2 1 + q3 y 2 . (λ − λn ρ3 )2 Setting
|λ| : λ ∈ Sθ , n 1; i = 1, 2, 3 |λ − λn ρi |
(λ + J n )−1 y 2
Hence, (λ + J n )−1 N , |λ|
N |λ|
2
q1 y 2 + q2 y 2 + q3 y 2 .
λ ∈ Sθ .
Now, if λ ∈ Sθ , then R(λ, −A)z =
∞ ∞ (λ + An )−1 Pn z = Kn (λ + J n )−1 Kn−1 Pn z. n=1
n=1
This implies ∞ 2 2 R(λ, A)z2 Kn 2 Kn−1 (λ + J n )−1 Pn z 2 n=1
Γ1 (η, γ ) Γ2 (η, γ )
2
N |λ|
(3.21)
(3.23)
where
yields
(3.20)
(3.22)
Now, define the sector Sθ as Sθ = λ ∈ C: θ arg(λ) π, λ = 0 ,
N = sup
2 z 2 .
202
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
Therefore, R(λ, −A) R , |λ|
λ ∈ Sθ .
This completes the proof of the theorem. ✷
4. Existence of the bounded solution In this section we shall prove the existence and stability of unique bounded mild solutions of system (2.7). Definition 4.1 (Mild solution). For mild solution z(t) of (2.7) with initial condition z(t0 ) = z0 ∈ Z1 , we understand a function given by t z(t) = T (t − t0 )z0 +
T (t − s)F s, z(s) ds,
t ∈ R.
(4.1)
t0
Remark 4.1. It is easy to prove that any solution of (2.7) is a solution of (4.1). It may be thought that a solution of (4.1) is always a solution of (2.7) but this is not true in general. However, we shall prove in Theorem 5.2 that bounded solutions of (4.1) are solutions of (2.7). We shall consider Zb = Cb (R, Z1 ) the space of bounded and continuous functions defined in R taking values in Z1 . Zb is a Banach space with supremum norm z b = sup z(t)Z : t ∈ R , z ∈ Zb . 1
A ball of radio ρ > 0 and center zero in this space is given by Bρb = z ∈ Zb : z(t) ρ, t ∈ R . The proof of the following lemma is similar to Lemma 3.1 of [15]. Lemma 4.1. Let z be in Zb . Then, z is a mild solution of (2.7) if and only if z is a solution of the following integral equation: t z(t) =
T (t − s)F s, z(s) ds,
t ∈ R.
(4.2)
−∞
The following theorem refers to bounded mild solutions of system (2.7). Theorem 4.1. Suppose that F is locally Lipschitz and there exists ρ > 0 such that 0 < MLF < (µ − MLρ )ρ,
(4.3)
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
203
b where Lρ is the Lipschitz constant of F in the ball B2ρ . Then, Eq. (2.7) has one and only one bounded mild solution zb (t) which belong Bρb . Moreover, this bounded solution is exponentially stable.
Proof. For the existence of such solution, we shall prove that the following operator T : Bρb → Bρb has a unique fixed point in the ball Bρb : t (T z)(t) =
T (t − s)F s, z(s) ds,
t ∈ R.
−∞
In fact, for z ∈ Bρb we have T z(t)
t
−∞
MLρ ρ + MLF . Me−µ(t −s) Lρ z(s) + LF µ
The condition (4.3) implies that Lρ ρ + MLF < µρ
⇔
Lρ ρ + MLF < ρ. µ
Therefore, T z ∈ Bρb for all z ∈ Bρb . Now, we shall see that T is a contraction mapping. In fact, for all z1 , z2 ∈ Bρb we have that t T z1 (t) − T z2 (t) Me−µ(t −s)Lρ z1 (s) − z2 (s) ds −∞
MLρ z1 − z2 b , µ
t ∈ R.
Hence, MLρ T z1 − z2 b , µ The condition (4.3) implies that T z1 − T z2 b
0 < µ − MLρ
⇔
MLρ < µ
z1 , z2 ∈ Bρb .
⇔
MLρ < 1. µ
Therefore, T has a unique fixed point zb in Bρb : t zb (t) = (T zb )(t) =
T (t − s)F s, zb (s) ds ds,
t ∈ R.
−∞
From Lemma 3.1, zb is a bounded solution of Eq. (4.1). Now, we shall prove that zb (·) is exponentially stable. To this end, we consider any other solution z(·) of Eq. (4.1) such that z(0) − zb (0) < ρ/2M. Then, z(0) < 2ρ. As long as z(t) remains less than 2ρ we obtain the following estimate:
204
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
t z(t) − zb (t) T (t) z(0) − zb (0) + T (t − s) F s, z(s) − F s, zb (s) ds 0
Me
−µt
z(0) − zb (0) +
t
Me−µ(t −s) Lρ z(s) − zb (s) ds.
0
Then, e
z(t) − zb (t) M z(0) − zb (0) +
µ(t )
t
Meµs Lρ z(s) − zb (s) ds.
0
Hence, applying the Gronwall’s inequality we obtain z(t) − zb (t) Me(MLρ −µ)t z(0) − zb (0) ,
t ∈ [0, t1).
From (4.3) we get that MLρ − µ < 0. Therefore z(t) − zb (t) ρ/2. Hence, if z(t) < 2ρ on [0, t1 ) with t1 = sup{t > 0: z(t) < 2ρ, t ∈ [0, t1 )}, then either t1 = ∞ or z(t1 ) = 2ρ. But, the second case contradicts the above computation, b then the solution z(t) remains in the ball B2ρ for all t 0. So, z(t) − zb (t) Me(MLρ −µ)t z(0) − zb (0) , t 0. This concludes the proof of the theorem. ✷ Theorem 4.2. Suppose that F globally Lipschitz with a Lipschitz constant L > 0 and µ > ML.
(4.4)
Then, Eq. (2.7) has one and only one bounded mild solution zb (t) on R. Moreover, this bounded solution is the only bounded solution of Eq. (4.1) and is exponentially stable in large. Proof. Condition (4.4) implies that for ρ > 0 big enough we have the following estimate: 0 < MLF < (µ − ML)ρ.
(4.5)
From here, in a similar way we can prove that the following operator is a contraction mapping from Bρb into Bρb : t (T z)(t) =
T (t − s)F s, z(s) ds,
t ∈ R.
−∞
Therefore, T has a unique fixed point zb in Bρb : t zb (t) = (T zb )(t) = −∞
T (t − s)F s, zb (s) ds ds,
t ∈ R.
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
205
From Lemma 3.1, zb is a bounded solution of Eq. (4.1). Since condition (4.5) holds for any ρ > 0 big enough independent of ML < µ, then zb is the unique bounded solution of Eq. (4.1). To prove that zb (t) is exponentially stable in the large, we shall consider any other solution z(t) of (4.1) and the following estimate: t z(t) − zb (t) T (t) z(0) − zb (0) + T (t − s) F s, z(s) − F s, zb (s) ds 0
Me
−µt
z(0) − zb (0) +
t
Me−µ(t −s) Lρ z(s) − zb (s) ds.
0
Then, e
µt
z(t) − zb (t) M z(0) − zb (0) +
t
MLeµs z(s) − zb (s) ds.
0
Hence, applying the Gronwall’s inequality we obtain z(t) − zb (t) Me(ML−µ)t z(0) − zb (0) ,
t 0.
From (4.5) we know that ML − µ < 0 and therefore zb (t) is exponentially stable in the large ✷ Corollary 4.1. If F is periodic in t of period τ (F (t + τ, ξ ) = f (t, ξ )), then the unique bounded solution given by Theorems 4.1 and 4.2 is also periodic of period τ . Proof. Let zb be the unique solution of (4.1) in the ball Bρb . Then, z(t) = zb (t + τ ) is also a solution of Eq. (4.1) lying in the ball Bρb . In fact, consider z0 = zb (0) and t +τ T (t + τ − s)F s.zb (s) ds zb (t + τ ) = T (t + τ )z0 + 0
τ = T (t)T (τ )z0 +
T (t + τ − s)F s.zb (s) ds
0
t +τ + T (t + τ − s)F s.zb (s) ds τ
τ
= T (t) T (τ )z0 + 0
T (τ − s)F s.zb (s) ds
206
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
t +
T (t − s)F s.zb (s + τ ) ds
0
t = T (t)zb (τ ) +
T (t − s)F s.zb (s + τ ) ds.
0
Therefore, t z(t) = T (t)zb (τ ) +
T (t − s)F s.z(s) ds,
0
and by the uniqueness of the fixed point of the contraction mapping T in this ball, we conclude that zb (t) = zb (t + τ ), t ∈ R. ✷ Remark 4.2. Under some condition, the bounded solution given by Theorems 4.1 and 4.2 is almost periodic; for example, we can study the case when the function F has the following form: F (t, z) = g(z) + P (t),
t, ξ ∈ R,
(4.6)
where P ∈ Cb (R, Z1 ) and g : Z1 → Z1 is a locally Lipschitz function. Corollary 4.2. Suppose F has the form (4.6) and g is a globally Lipschitz function with a Lipschitz constant L > 0. Then the bounded solution zb (·, P ) given by Theorem 4.2 depends continuously on P ∈ Cb (R, Z1 ). Proof. Let P1 , P2 ∈ Cb (R, Z1 ) and zb (·, P1 ), zb (·, P2 ) be the bounded functions given by Theorem 4.2. Then t T (t − s) g zb (s, P2 ) − g zb (s, P2 ) ds zb (t, ·, P1 ) − zb (t, ·, P2 ) = −∞
t +
T (t − s) P1 (s) − P2 (s) ds.
−∞
Hence, zb (·, P1 ) − zb (·, P2 ) ML zb (·, P1 ) − zb (·, P2 ) + M P1 − P2 b . b b µ µ Therefore, zb (·, P1 ) − zb (·, P2 ) b
M P1 − P2 b . µ − ML
✷
Lemma 4.2. Suppose F is as in (4.6). Then, if P (t) is almost periodic, then the unique bounded solution of system (4.1) given by Theorems 4.1 and 4.2 is also almost periodic.
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
207
Proof. To prove this lemma, we shall use the following well known fact, due to Bohr (see Hale [5, Appendix]). A function h ∈ C(R; Z1 ) is almost periodic (a.p.) if and only if the hull H (h) of h is compact in the topology of uniform convergence. Here H (h) is the closure of the set of translates of h under the topology of uniform convergence H (h) = {hτ : τ ∈ R},
hτ (t) = h(t + τ ), t ∈ R.
Since the limit of a uniformly convergent sequence of a.p. functions is a.p., then the set Aρ of a.p. functions in the ball Bρb is closed, where ρ is given by Theorem 4.2. Claim. The contraction mapping T given in Theorems 4.1 and 4.2 leaves Aρ invariant. In fact, if z ∈ Aρ , then h(t) = g(z(t)) + P (t) is also an a.p. function. Now, consider the function t t F (t) = (T z)(t) = T (t − s) g z(s) + P (s) ds = T (t − s)h(s) ds, t ∈ R. −∞
−∞
Then, it is enough to establish that H (F ) is compact in the topology of uniform convergence. Let {Fτk } be any sequence in H (F ). Since h is a.p., we can select from {hτk } a Cauchy subsequence {hτkj }, and we have that t +τkj
Fτkj (t) = F (t + τkj ) =
t T (t + τkj − s)h(s) ds =
−∞
T (t − s)h(s + τkj ) ds.
−∞
Hence, Fτ (t) − Fτ (t) kj ki
t
e−µ(t −s) h(s + τkj ) − h(s + τki ) ds
−∞
t hτkj − hτki b −∞
e−µ(t −s) ds =
1 hτ − hτki b . µ kj
Therefore, {Fτkj } is a Cauchy sequence. So, H (F ) is compact in the topology of uniform convergence, F is a.p. and T Aρ ⊂ Aρ . Now, the unique fixed point of T in the ball Bρb lies in Aρ . Hence, the unique bounded solution zb (t) of Eq. (4.1) given in Theorem 4.2 is also almost periodic. ✷ 5. Smoothness of the bounded solution In this section we shall prove that the bounded solution of Eq. (4.1) given by Theorems 4.1 and 4.2 is also solution of the original equation (2.7). That is to say, this bounded solution is a classic solution of Eq. (2.7). To this end, we will use the following theorem from [8].
208
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
Theorem 5.1. Let A on D(A) be a closed operator in the Banach space X and x ∈ C([a, b); X) with b ∞. Suppose that x(t) ∈ D(A), Ax(t) is continuous on [a, b) and that the improper integrals b
b x(s) ds
Ax(s) ds
and
a
a
exist. Then b
b x(s) ds ∈ D(A)
x(s) ds =
and A
a
b
a
Ax(s) ds. a
Theorem 5.2. The bounded Mild solution zb (t) of Eq. (2.7) given by Theorems 4.1 and 4.2 is a classic solution of this equation on R, i.e., zb (t) = Azb (t) + F t, zb (t) , t ∈ R. Proof. Let zb (t) be the only bounded mild solution of (2.7) given by Theorems 4.1 and 4.2. Then, t z(t) =
∞ T (t − s)g(s) ds =
−∞
T (s)g(t − s) ds,
t ∈ R,
0
where g(s) = F (s, zb (s)). Therefore, g ∈ Cb (R, Z1 ) and g(s) g b , s ∈ (−∞, t). Let us put x(s) = T (t − s)g(s), s ∈ (−∞, t). Then x(s) is a continuous function, and since {T (t)}t 0 is analytic, then x(s) ∈ D(A),
for s < t.
Claim. Ax(s) is continuous on (−∞, t) and the improper integral t Ax(s) ds,
t ∈ R,
−∞
exists. From Theorem 3.1, there exists a complete family of orthogonal projections {qi (j )}3i=1 in R3 such that Aj = σ1 (j )q1 (j ) + σ1 (j )q2 (j ) + σ1 (j )q3 (j ), eAj t = e−λj ρ1 t q1 (j ) + e−λj ρ2 t q2 (j ) + e−λj ρ3 t q3 (j ). Hence, Az =
∞ σ1 (j )Pj 1 z + σ2 (j )Pj 2 z + σ3 (j )Pj 3 z j =1
(5.1)
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
209
and T (t)z =
∞ −λ ρ t e j 1 Pj 1 z + e−λj ρ2 t Pj 2 z + e−λj ρ3 t Pj 3 z , j =1
where Pj i = qi (j )Pj is a complete family of orthogonal projections in Z1 . Therefore, Ax(s) =
∞ −λj ρ1 e−λj ρ1 (t −s)Pj 1 g(s) − λj ρ2 e−λj ρ2 (t −s)pj 2 g(s) j =1
So,
− λj ρ3 e−λj ρ3 (t −s)pj 3 g(s) .
Ax(s) max λj |ρi |e−λj Re(ρi )(t −s): i = 1, 2, 3 g b . j 1
Then, using the dominate convergence theorem, we get that Ax(s) is a continuous function on (−∞, t). Now, consider the following improper integrals: t
∞ Ax(s) ds =
−∞
AT (s)g(t − s) ds 0
∞ ∞ = −λj ρ1 e−λj ρ1 s Pj 1 g(t − s) − λj ρ2 e−λj ρ2 s Pj 2 g(t − s) 0 j =1 − λj ρ3 e−λj ρ3 s Pj 3 g(t − s) ds
∞ ∞ = −λj ρ1 e−λj ρ1 s Pj 1 g(t − s) ds j =1
0
∞ −
λj ρ2 e−λj ρ2 s Pj 2 g(t − s) ds
0
∞ −
λj ρ3 e
−λj ρ3 s
Pj 3 g(t − s) ds .
0
On the other hand, we have that ∞ ∞ −λj ρi e−λj ρi s Pj i g(t − s) ds λj |ρi |e−λj Re(ρi )s Pj i g(t − s) ds 0
0
Therefore, the improper integral t Ax(s) ds −∞
|ρi | g b . Re(ρi )
210
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
exists. Now, from Theorem 5.1 we obtain that t
t x(s) ds ∈ D(A)
and A
−∞
t x(s) ds =
−∞
Ax(s) ds,
−∞
i.e., t
t T (t − s)g(s) ds ∈ D(A)
and A
−∞
t T (t − s)g(s) ds =
−∞
AT (t − s)g(s) ds.
−∞
Now, we are ready to prove that zb (t) is a solution of (2.7). In fact, consider zb (t + h) − zb (t) 1 = h h =
t +h t 1 T (t + h − s)g(s) ds − T (t − s)g(s) ds h
−∞
T (h) − I h
−∞
t T (t − s)g(s) ds + −∞
1 h
t +h T (t + h − s)g(s) ds. t
Using the definition of infinitesimal generator of a semigroup and passing to the limit as h → 0+ we get that zb (t) = A
t T (t − s)g(s) ds + T (0)g(t).
−∞
So,
zb (t) = Azb (t) + F t, zb (t) ,
t ∈ R.
✷
References [1] J.M. Alonso, J. Mawhin, R. Ortega, Bounded solutions of second order semilinear evolution equations and applications to the telegraph equation, J. Math. Pures Appl. 78 (1999) 49–63. [2] G. Avalos, I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste Suppl. XXVIII (1997) 1–28. [3] G. Avalos, I. Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions, SIAM J. Control Optim. 38 (2000) 337–383. [4] G. Avalos, I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29 (1998) 155–182. [5] J. Hale, Ordinary Differential Equations, Wiley–Interscience, New York, 1969. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. [7] J.U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (1992) 889. [8] G.E. Ladas, V. Lakshmikantham, Differential Equations in Abstract Spaces, in: Math. in Science and Engineering, Vol. 85, Academic Press, 1972. [9] J. Lagnese, Boundary Stabilization of Thin Plate, in: SIAM Studies in Appl. Math., Vol. 10, 1989. [10] L. de Teresa, E. Zuazua, Controllability of the linear system of thermoelastic plates, Adv. Differential Equations 1 (1996) 369–402.
H. Leiva, Z. Sivoli / J. Math. Anal. Appl. 285 (2003) 191–211
211
[11] L. de Teresa, Approximate controllability of a semilinear heat equation in RN , SIAM J. Control Optim. 36 (1998) 2128–2147. [12] H. Leiva, Stability of a periodic solution for a system of parabolic equations, J. Appl. Anal. 60 (1996) 277–300. [13] H. Leiva, Existence of bounded solutions of a second order system with dissipation, J. Math. Anal. Appl. 237 (1999) 288–302. [14] L. Garcia, H. Leiva, Center manifold and exponentially bounded solutions of a forced Newtonian system with dissipation, Electron. J. Differential Equations Conf. 5 (2000) 69–77. [15] H. Leiva, Existence of bounded solutions of a second order evolution equation and applications, J. Math. Phys. 41 (2000). [16] H. Leiva, A lemma on C0 -semigroups and applications PDEs systems, Report Series of CDSNS 99-353. [17] Y. Shibata, On the exponential decay of the energy of a linear thermoelastic plate, Comp. Appl. Math. 13 (1994) 81–102. [18] J.H. Ortega, E. Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation, SIAM J. Control Optim. 39 (2001) 1585–1614.