We characterize well-posedness in Hölder spaces for an abstract version of
the equation (∗) u′′+λu′′′=c2(Δu+μΔu′)+f which model the vibrations of flexible structures possessing internal material damping and external force f. As a consequence, we show that in case of the Laplacian with Dirichlet boundary conditions, equation (∗) is always well-posed provided 0<λ<μ.

1. Introduction

During the last few decades, the use of flexible structural systems has steadily increased importance. The study of a flexible aerospace structure is a problem of dynamical system theory governed by partial differential equations.

We consider here the problem of characterize well posedness, for a mathematical model of a flexible space structure like a thin uniform rectangular panel. For example, a solar cell array or a spacecraft with flexible attachments. This problem is motivated by both engineering and mathematical considerations.

Such mechanical system was mathematically introduced in [1] and consists of a short rigid hub, connected to a flexible panel of length l. Control torque Q(t) is applied to the hub. The panel is made of viscoelastic material with internal Voigt-type damping with coefficient μ, that is, an ideal dashpot damping which is directly proportional to the first derivative of the longitudinal displacement, and opposing the direction of motion. The equation of motion of the panel is given by
u′′=c2(Δu+μΔu′),
where c is the velocity of longitudinal wave propagation, c2=Dp/ρJp, and Dp,ρ,Jp are, respectively, torsional rigidity, density and radius of gyration about the central axis of the panel. Initial position and deflection angle are known. In [1] exact controllability and boundary stabilization for the solution of (1.1) was analyzed and in [2, p. 188], the exact decay rate was obtained.

More generally, the study of vibrations of flexible structures possessing internal material damping is modeled by a equation of the form
u′′+λu”'=c2(Δu+μΔu′),0<λ<μ,
in a bounded domain Ω in ℝn with smooth boundary Γ, see [3, 4].

In [4] the explicit exponential energy decay rate was obtained for the solution of (1.2) subject to mixed boundary conditions. However, consideration of external forces interacting with the system, which lead us naturally with the well posedness for the nonhomogeneous version of (1.2), appears as an open problem.

In the first part of this paper we study well posedness of the following abstract version of (1.2):
u′′(t)+λu′′′(t)=c2Au(t)+c2μAu′(t)+f(t),0<λ<μ,

where A is a closed linear operator acting in a Banach space X and f is a X-valued function. We emphasize that when A=Δ in general one cannot expect that (1.3) is well posed due to the presence of the term u′′′. In fact, it is well known that the abstract Cauchy problem associated with (1.3) is in general ill posed, see for example [5].

We are able to characterize well posedness, that is, temporal maximal regularity, of solutions of (1.3) solely in terms of boundedness of the resolvent set of A. This will be achieved in the Hölder spaces Cα(ℝ,X), where 0<α<1. The methods to obtain this goal are those incorporated in [6] where a similar problem in case of the first order abstract Cauchy problem has been studied.

2. Preliminaries

Let X,Y be Banach spaces, we write ℬ(X,Y) for the space of bounded linear operators from X to Y and let 0<α<1. We denote by Ċα(ℝ,X) the spaces
Ċα(ℝ,X)={f:ℝ→X:f(0)=0,∥f∥α<∞}
normed by
∥f∥α=supt≠s∥f(t)-f(s)∥∥t-s∥α.
Let Ω⊂ℝ be an open set. By Cc∞(Ω) we denote the space of all C∞-functions in Ω⊆ℝ having compact support in Ω.

We denote by ℱf or f̃ the Fourier transform, that is,
(ℱf)(s):=f̃(s):=∫ℝe-istf(t)dt(s∈ℝ,f∈L1(ℝ;X)).

Definition 2.1.

Let M:ℝ∖{0}→ℬ(X,Y) be continuous. We say that M is a Ċα-multiplier in ℬ(X,Y) if there exists a mapping L:Ċα(ℝ,X)→Ċα(ℝ,Y) such that
∫ℝ(Lf)(s)(ℱϕ)(s)ds=∫ℝ(ℱ(ϕ·M))(s)f(s)ds
for all f∈Cα(ℝ,X) and all ϕ∈Cc∞(ℝ∖{0}).

Here (ℱ(ϕ·M))(s)=∫ℝe-istϕ(t)M(t)dt∈ℬ(X,Y). Note that L is well defined, linear and continuous (cf. [6, Definition 5.2]).

Define the space Cα(ℝ,X) as the set
Cα(ℝ,X)={f:ℝ→X:∥f∥Cα<∞}
with the norm
∥f∥Cα=∥f∥α+∥f(0)∥.
Let Cα+k(ℝ,X) (where k is a positive integer) be the Banach space of all u∈Ck(ℝ,X) such that u(k)∈Cα(ℝ,X), equipped with the norm
∥u∥Cα+k=∥u(k)∥Cα+∥u(0)∥.
Observe from Definition 2.1 and the relation
∫ℝ(ℱ(ϕM))(s)ds=2π(ϕM)(0)=0,
that for f∈Cα(ℝ,X) we have Lf∈Cα(ℝ,X). Moreover, if f∈Cα(ℝ,X) is bounded then Lf is bounded as well (see [6, Remark 6.3]). The following multiplier theorem is due to Arendt, Batty and Bu [6, Theorem 5.3].

Theorem 2.2.

Let M∈C2(ℝ∖{0},ℬ(X,Y)) be such that
supt≠0∥M(t)∥+supt≠0∥tM′(t)∥+supt≠0∥t2M′′(t)∥<∞.
Then M is a Ċα-multiplier.

Remark 2.3.

If X is B-convex, in particular if X is a UMD space, Theorem 2.2 remains valid if condition (2.2) is replaced by the following weaker condition:
supt≠0∥M(t)∥+supt≠0∥tM′(t)∥<∞,
where M∈C1(ℝ∖{0},ℬ(X,Y)) (cf. [6, Remark 5.5]).

3. A Characterization of Well Posedness in Hölder Spaces

In this section we characterize Cα-well posedness. Given f∈Cα(ℝ,X), we consider in this section the linear problem
u′′(t)+au′′′(t)=bAu(t)+cAu′(t)+f(t),t∈ℝ,

where A is a closed linear operator in X and a,b,c>0. Note that the solution of (3.1) does not have to satisfy any initial condition. In the case a=0, solutions of (3.1) with periodic boundary conditions has been recently studied in [7]. On the other hand, well posedness of the homogeneous abstract Cauchy problem has been observed recently in [8] for a=0 and all b∈ℂ under certain assumptions on A. See also [9] for related maximal regularity results in the case of a damped wave equation.

We denote by [D(A)] the domain of A considered as a Banach space with the graph norm.

Definition 3.1.

We say that (3.1) is Cα-well posed if for each f∈Cα(ℝ,X) there is a unique function u∈Cα+3(ℝ,X)∩Cα+1(ℝ,[D(A)])∩Cα(ℝ,[D(A)]) such that (3.1) is satisfied.

In the next proposition, as usual we denote by ρ(T),R(λ,T) the resolvent set and resolvent of the operator T, respectively.

Proposition 3.2.

Assume that (3.1) is Cα-well-posed. Then

l(η):=-η2((1+iaη)/(b+icη))∈ρ(A) for all η∈ℝ and,

supη∈ℝ||(η3/(b+icη))R(l(η),A)||<∞.

Proof.

Denote by L:Cα(ℝ,X)→Cα+3(ℝ,X) the bounded operator which associates to each f∈Cα(ℝ,X) the unique solution u of (3.1). Let η∈ℝ. Let x∈D(A) be such that Ax-l(η)x=0. Define u(t)=eiηtx. Then it is not difficult to see that u is a solution of (3.1) with f≡0. Hence, by uniqueness, x=0.

Let y∈X and define f(t)=eiηty. Let u=Lf. For fixed s∈ℝ we define
v1(t)=u(t+s),v2(t)=eiηsu(t).
Then is easy to check that v1 and v2 are both solutions of (3.1) with f replaced by eiηsf. By uniqueness, u(t+s)=eiηsu(t) for all t,s∈ℝ. In particular, it follows that u(s)=eiηsu(0) for all s∈ℝ. Let x=u(0)∈D(A). Replacing u(t)=eiηtx in (3.1) we obtain
(-η2-iaη3)u(t)=(b+icη)Au(t)+eiηty.
Taking t=0 we conclude that (l(η)-A) is bijective and
u(t)=1b+icηR(l(η),A)eiηty.
Define eη(t)=eiηt and (eη⊗y)(t)=eη(t)y. We have the identity ||eη⊗x||α=Kα|η|α||x|| where Kα=2supt>0t-αsin(t/2) (see [6, section 3]). Hence
Kα|η|α∥η3b+icηR(l(η),A)y∥=∥eη⊗η3b+icηR(l(η),A)y∥α=∥u′′′∥α≤∥u∥α+3=∥Lf∥α+3≤∥L∥∥f∥α≤∥L∥(∥f∥α+∥f(0)∥)=∥L∥(∥eη⊗y∥α+∥y∥)≤∥L∥(Kα|η|α+1)∥y∥.
Therefore, for ε>0 we have
sup|η|>ε∥η3b+icηR(l(η),A)y∥≤∥L∥sup|η|>ε(1+1Kα|η|α)∥y∥<∞.
On the other hand, since {1/(b+icη)}η∈ℝ is bounded and η→η3R(l(η),A) is continuous at η=0, we obtain (ii) and the proof is complete.

In what follows, we denote by idk the function: s→(is)k for all s∈ℝ, and k∈ℕ. As before, we also use the notation
l(s):=-s21+iasb+ics,M(s):=1b+icsR(l(s),A),∀s∈ℝ.

Lemma 3.3.

Assume that
sups∈ℝ∥s3M(s)∥<∞,
then id2·M and id3·M are Ċα- multipliers in ℬ(X). Moreover M and id·M are Ċα-multipliers in ℬ(X,D(A)).

Proof.

Define κ(s):=1/(b+ias). We first observe that the functions θ(s):=κ′(s)/κ(s) and ϑ(s):=l′(s)/l(s) have the property that sθ(s),sϑ(s),s2θ′(s) and s2ϑ′(s) are bounded on ℝ. We next claim that M is a Ċα-multiplier. In fact, note that by hypothesis sup|s|>ε∥M(s)∥<∞ for each ε>0, and the function s→M(s) is continuous at t=0 since b>0. Hence M(s) is bounded. Moreover, defining ξ(s):=l(s)/κ(s)=-s2-ias3 we have
M′(s)=θ(s)M(s)-ϑ(s)ξ(s)[M(s)]2,
where sξ(s) is of order s4 and then Q(s):=ξ(s)[M(s)]2 is bounded by (3.8). It follows that sM′(s) is bounded. Next, we have the identity
s2M′′(s)=s2θ′(s)M(s)+sθ(s)sM′(s)-s2ϑ′(s)Q(s)-s2ϑ(s)Q′(s).
where the first three terms on the right hand side are bounded. For the last term, we have
s2ϑ(s)Q′(s)=[sϑ(s)]2Q(s)-sϑ(s)sθ(s)Q(s)+2sϑ(s)ξ(s)M(s)sM′(s).
It is clear that the first two terms on the right hand side are bounded. We observe that the last term also is bounded. In fact, note that by hypothesis sup|s|>ε∥ξ(s)M(s)∥<∞ for each ε>0 and the function s→ξ(s)M(s) is continuous at s=0. Hence ξ(s)M(s) is bounded. This completes the proof of the Lemma.

Lemma 3.4.

Let 0<α<1,k∈ℕ and u,v∈Cα(ℝ,X). The following assertions are equivalent:

u∈Cα+k(ℝ,X) and u(k)-v is constant.

∫ℝv(s)(ℱφ)(s)ds=∫ℝu(s)(ℱidk·φ)(s)ds for all φ∈𝒟(ℝ∖{0}).

Proof.

(i)⇒(ii). Let Φ∈𝒟(ℝ{0}). Then ∫ℝv(s)(ℱφ)(s)ds=∫ℝu(k)(s)(ℱφ)(s)ds=(-1)k∫ℝu(s)(ℱφ)(k)(s)ds=∫ℝu(s)ℱ(idk·φ)(s)ds.

(ii)⇒(i). Let Φ∈𝒟(ℝ∖{0}) and ψ(s)=φ(s)/sk. Then ψ∈𝒟(ℝ∖{0}) and ℱφ=(ℱψ)(k). Let w(t)=∫0t(t-s)k-1v(s)ds. Then integration by parts and assumption give ∫ℝw(s)(ℱφ)(s)ds=∫ℝu(s)(ℱφ)(s)ds. It follows from [10, Theorems 4.8.2 and 4.8.1] that w-u is a polynomial. Since ∥w(t)∥≤c(1+|t|α+k) it follows that u(t)=w(t)+y0+ty1+t2y2+⋯+tkyk=∫0t(t-s)k-1v(s)ds+y0+ty1+t2y2+⋯+tkyk for some vectors y0,y1,…,yk∈X. Thus u(k)=v+x for some vector x∈X.

The following theorem, which is one of the main results in this paper, shows that the converse of Proposition 3.2 is valid.

Theorem 3.5.

Let A be a closed linear operator defined on a Banach space X. Then the following assertions are equivalent:

Equation (3.1) is Cα-well posed;

l(η):=-η2((1+iaη)/(b+icη))∈ρ(A) for all η∈ℝ and
supη∈ℝ∥η3b+icηR(l(η),A)∥<∞.

Proof.

The implication (i)⇒(ii) follows by Proposition 3.2. We now prove the converse implication.

Let f∈Cα(ℝ,X). By Lemma 3.3 there exists u1,u2∈Cα(ℝ,[D(A)]) and u3,u4∈Cα(ℝ,X) such that
∫ℝu1(s)(ℱϕ1)(s)ds=∫ℝℱ(ϕ1·M)(s)f(s)ds,∫ℝu2(s)(ℱϕ2)(s)ds=∫ℝℱ(ϕ2·id·M)(s)f(s)ds,∫ℝu3(s)(ℱϕ3)(s)ds=∫ℝℱ(ϕ3·id2·M)(s)f(s)ds,∫ℝu4(s)(ℱϕ4)(s)ds=∫ℝℱ(ϕ4·id3·M)(s)f(s)ds
for all Φi∈C0∞(ℝ∖{0})(i=1,2,3,4). Choosing Φ1=id·Φ2 in (3.13), it follows from Lemma 3.4 that u1∈C1+α(ℝ,X) and
u1′=u2+y1,
for some y1∈X. Now we can choose ϕ2=id·ϕ3 in (3.14), it follows that u1∈Cα+2(ℝ,X) and
u1”=u3+y2,
for some y2∈X. In a similar way, we can see that u1∈Cα+3(ℝ,X) and
u1”'=u4+y3,
for some y3∈X. From the definition of M(s) we obtain (b+ics)(l(s)-A)M(s)=I. Taking into account the definition of l(s) we get [-s2(1+ics)-(b+ics)A]M(s)=I. Then we deduce the identity
(is)2M(s)+a(is)3M(s)=bAM(s)+icsAM(s)+I.
We multiply the above identity by ϕ, take Fourier transforms and then integrate over ℝ after taking the values at f(s), we obtain
∫ℝℱ(ϕ·id2·M)(s)f(s)ds+a∫ℝℱ(ϕ·id3·M)(s)f(s)ds=b∫ℝAℱ(ϕ·M)(s)f(s)ds+c∫ℝAℱ(ϕ·id·M)(s)f(s)ds+∫ℝℱ(ϕ)(s)f(s)ds
for all ϕ∈C0∞(ℝ∖{0}). Using (3.17), (3.18) and (3.19) in the above identity we conclude that
∫ℝu1′′(s)(ℱϕ)(s)ds+a∫ℝu1′′′(s)(ℱϕ)(s)ds=b∫ℝAu1(s)(ℱϕ)(s)ds+c∫ℝAu1′(s)(ℱϕ)(s)ds+∫ℝℱ(ϕ)(s)f(s)ds
for all ϕ∈C0∞(ℝ∖{0}). By Lemma 3.4 there exists z∈X such that
u1′′(t)+au1′′′(t)=bAu1(t)+cAu1′(t)+f(t)+z,t∈ℝ.
We define
u(t)=u1(t)+1bA-1(z).
Then, we can show that u solves (3.1) and that u∈Cα+3(ℝ,X)∩Cα+1(ℝ,[D(A)])∩Cα(ℝ,[D(A)]).

In order to prove uniqueness, suppose that
u′′(t)+au′′′(t)=bAu(t)+cAu′(t),t∈ℝ.
where u∈Cα+3(ℝ,X)∩Cα+1(ℝ,[D(A)])∩Cα(ℝ,[D(A)]). Let σ>0. We define Lσu by
(Lσu)(ρ)=û(σ+iρ)-û(-σ+iρ),ρ∈ℝ
where the hat indicates the Carleman transform (see e.g. [11]). By [12, Proposition A.2(i)], we have that
∫ℝu(ρ)(ℱϕ)(ρ)dρ=limσ↓0∫ℝ(Lσu)(ρ)ϕ(ρ)dρ
for all ϕ∈𝒮(ℝ), the Schwartz space of smooth rapidly decreasing functions on ℝ. We will prove that the right term in (3.27) is zero, from which u≡0 proving the theorem. In fact, by [12, Proposition A.2(iii)] we have
(b+c(σ+iρ))(lσ(ρ)-A)(Lσu)(ρ)=2σcAû(-σ+iρ)-[2σ(σ+iρ)+2σa(σ+iρ)2]û(-σ+iρ)-[2σ+2σa(σ+iρ)]û′(-σ+iρ)-2σaû′′′(-σ+iρ):=Ha,c(σ,ρ),
where
lσ(ρ)=(σ+iρ)21+a(σ+iρ)b+c(σ+iρ).
Observe that l0(ρ)=l(ρ)∈ρ(A) for all ρ∈ℝ. Therefore we have
(lσ(ρ)-l(ρ))(l(ρ)-A)-1(Lσu)(ρ)+(Lσu)(ρ)=1b+c(σ+iρ)(l(ρ)-A)-1Ha,c(σ,ρ).
Let ϕ∈C0∞(ℝ). Multiplying by ϕ and integrating over ℝ the above identity we obtain
∫ℝ(Lσu)(ρ)ϕ(ρ)dρ=∫ℝNσ(ρ)Ha,c(σ,ρ)dρ-∫ℝMσ(ρ)(Lσu)(ρ)dρ
where
Nσ(ρ)=1b+c(σ+iρ)ϕ(ρ)(l(ρ)-A)-1Mσ(ρ)=ϕ(ρ)(lσ(ρ)-l(ρ))(l(ρ)-A)-1.
We note that in [12, Lemma A.4],
limσ↓0∫ℝMσ(ρ)Lσ(u)(ρ)dρ=0.
It remains to prove that
limσ↓0∫ℝNσ(ρ)Ha,c(σ,ρ)dρ=0.
In fact, since (Lσu)(ρ)=∫ℝe-σ|t|e-iρtu(t)dt, we have
∫ℝMσ(ρ)Lσ(u)(ρ)dρ=∫ℝ∫ℝe-iρtMσ(ρ)dρe-σ|t|u(t)dt=∫ℝ(ℱMσ)(t)e-σ|t|u(t)dt.
Then
∥∫ℝMσ(ρ)Lσ(u)(ρ)dρ∥≤∫ℝ∥(ℱMσ)(t)∥∥u(t)∥dt.≤2C(∥Mσ∥L1+∥Mσ′′∥L1).
It is easy to check that ∥Mσ∥L1+∥Mσ′′∥L1→0 as σ→0, proving (3.34).

We write
Ha,c(σ,ρ)=I1(σ,ρ)+I2(σ,ρ)+I3(σ,ρ)+I4(σ,ρ).
We first prove that
limσ↓0∫ℝNσ(ρ)I1(σ,ρ)dρ=0.
In fact, we apply Fubini's theorem to obtain
∫ℝNσ(ρ)I1(σ,ρ)dρ=2σc∫ℝNσ(ρ)Aû(-σ+iρ)dρ=-2σc∫-∞0[∫ℝe-iρtNσ(ρ)dρ]eσtAu(t)dt=-2σc∫-∞0(ℱNσ)(t)eσtAu(t)dt
It follows from [[12], Lemma A.3] that
∥∫-∞0(ℱNσ)(t)eσtAu(t)dt∥≤2C(∥Nσ∥L1+∥Nσ′′∥L1),
where C is a positive constant. Taking into account (3.39) and (3.40) we deduce (3.38).

We next prove that
limσ↓0∫ℝNσ(ρ)I2(σ,ρ)dρ=0.
In fact, define Nσa(ρ)=[1+a(σ+iρ)](σ+iρ)Nσ(ρ). Then
∫ℝNσ(ρ)I2(σ,ρ)dρ=-2σ∫ℝ∫-∞0e(σ-iρ)sNσa(ρ)u(s)dsdρ=-2σ∫-∞0(ℱNσa)(s)easu(s)ds.
By [12, Lemma A.3], we have for 0≤σ≤ε,∥∫-∞0(ℱNσa)(s)easu(s)ds∥≤2Csup0≤σ≤ε(∥Nσa∥L1+∥(Nσa)”∥L1),
where C>0. Therefore, we deduce (3.41). Proceeding in the same way we obtain
limσ↓0∫ℝNσ(ρ)Ij(σ,ρ)dρ=0,j=3,4.
This completes the proof of the assertion (3.34).

Corollary 3.6.

The solution u of problem (3.1) given by Theorem 3.5 satisfies the following maximal regularity property: u,u′∈Cα(ℝ;[D(A)]) and Au,Au′,u′′,u′′′”∈Cα(ℝ;X). Moreover, there exists a constant C>0 independent of f∈Cα(ℝ;X) such that
∥u∥α+∥u′∥α+∥u′′∥α+∥u′′′∥α+∥Au∥α+∥Au′∥α≤C∥f∥α.

The following consequence of Theorem 3.5 is remarkable in the study of Cα well posedness for flexible structural systems. We recall that l(η):=-η2((1+iaη)/(b+icη)).

Corollary 3.7.

If A is the generator of a bounded analytic semigroup, then (3.1) is Cα-well posed.

Proof.

Since A generates a bounded analytic semigroup, we have that {τ:Reτ>0}⊆ρ(A) and there is a constant M>0 such that ∥τ(τ-A)-1∥≤M for Reτ>0. Note that
η3b+icηR(l(η);A)=-η1+iαηl(η)R(l(η);A)
and that
Re(l(η))=b+acη2b2+c2η2>0for eachη∈ℝ.
We conclude that l(η)∈ρ(A) and
supη∈ℝ∥η3b+icηR(l(η),A)∥<∞.
The conclusion follows by Theorem 3.5.

For example, if A is a normal operator on a Hilbert space H satisfying
σ(A)⊂{z∈ℂ:arg(-z)<δ}
for some δ∈[0,π/2), then A generates a bounded analytic semigroup. In particular, the semigroup generated by a self-adjoint operator that is bounded above is analytic of angle π/2. Another important class of generators of analytic semigroups is provided by squares of group generators.

Example 3.8.

Since the Laplacian Δ is the generator a bounded analytic semigroup (the diffusion semigroup) in X=Lp(ℝN)(1≤p<∞), we obtain that for each f∈Cα(ℝ,Lp(ℝN)) the problem
utt(t,x)+λuttt(t,x)=c2(Δu(t,x)+μΔut(t,x))+f(t,x)
has a unique solution u∈Cα+3(ℝ,Lp(ℝN))∩Cα+1(ℝ,W2,p(ℝN))∩Cα(ℝ,W2,p(ℝN)).

Since it is also well known that the Dirichlet Laplacian Δ generates a bounded analytic semigroup on L2(Ω), where Ω is a bounded domain with smooth boundary ∂Ω in ℝ3, we obtain the following consequence for our initial problem.

Corollary 3.9.

If Ω is a bounded domain with boundary of class C2 in ℝ3 then for each f∈Cα(ℝ,L2(Ω)), the problem (3.50) is Cα-well posed, that is, has a unique solution u∈Cα+3(ℝ,L2(Ω))∩Cα+1(ℝ,H2(Ω)∩H01(Ω))∩Cα(ℝ,H2(Ω)∩H01(Ω)).

We note that the same assertion remain true for all p∈[1,∞).

Acknowledgment

The first author is partially supported by CNPQ/Brazil. The second author is partially financed by Laboratorio de Análisis Estocástico, Proyecto Anillo ACT-13.

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