#### Abstract

We study abstract equations of the form , which is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of -regularized families, which is a particular case of -regularized families, and characterize maximal regularity in -spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.

#### 1. Introduction

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

We consider here the problem of characterizing -maximal regularity (or 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.

The study of * vibrations* of flexible structures possessing internal material damping was first derived by Bose and Gorain [1]. The consideration of external forces leads to more general equations of the form

where is a closed linear operator acting in a Banach space and is an -valued function. We emphasize that the abstract Cauchy problem associated with (1.1) is in general ill posed; see, for example, [2]. Also it is well known that in order to analyze well-posedness, a direct approach leads to better results than those obtained by a reduction to a first-order equation.

Maximal regularity in Hölder spaces for (1.1) has been recently characterized in [3]. In case , there are more literatures. For example, stability of the solution was studied by Gorain in [4]. In [5], Gorain and Bose studied exact controllability and boundary stabilization. More recently, Batkai and Piazzera [6, page 188] have obtained the exact decay rate. We note that well-posedness in Lebesgue spaces in the case of a damped wave equation has been only recently considered by Chill and Srivastava in [7], and in Hölder spaces by Poblete [8]. We note that the class studied in [8] includes equations with delay. In particular, well-posedness of the homogeneous abstract Cauchy problem has been observed in [9] for under certain assumptions on

This paper is organized as follows. Section 2, collects results essentially contained in [10] and standard literature on -boundedness and maximal regularity (see [11] and [12]). In Section 3 we study, by an operator theoretical method, sufficient conditions for existence of solutions for (1.1). We obtain two results: a description of the solution by means of certain regularized families (Proposition 3.1) and the existence of such families in the particular case of positive self-adjoint operators (Theorem 3.2). In Section 4, we succeed in * characterizing* well-posedness of (1.1) in terms of -boundedness of a resolvent set which involves (Theorem 4.2). This will be achieved in the Lebesgue spaces where is a space (see below the definition). The methods to obtain this goal are those incorporated in [13] where a similar problem in case of the first-order abstract Cauchy problem has been studied. Our main result (Theorem 4.2) is a combination of the well-known (and deep) result due to Weis [14] stated in Theorem 2.8 and a direct calculation involving the parameters , , and .

#### 2. Preliminaries

Let be given. In what follows we denote

In order to give an operator theoretical approach to (1.1) we introduce the following definition.

*Definition 2.1. *Let be a closed and linear operator with domain defined on a Banach space . One calls the generator of an -regularized family if the following conditions are satisfied.(R1) is strongly continuous on and .(R2) and for all .(R3) The following equation holds:
for all In this case, is called the -regularized family generated by .

*Remark 2.2. * It is proved in [10], in the more general context of -regularized families, that an operator is the generator of an -regularized family if and only if there exists and a strongly continuous function such that and

Because of the uniqueness of the Laplace transform, we note that an -regularized family corresponds to an -regularized family studied in [10]. In fact, we have

As in the situation of -semigroups, we have diverse relations of an -regularized family and its generator. The following result is a direct consequence of [10, Proposition and Lemma ].

Proposition 2.3. * Let be an -regularized family on with generator . Then the following hold.*(a)*For all one has *(b)*Let and Then and *

Results on perturbation, approximation, asymptotic behavior, representation, as well as ergodic-type theorems for -regularized families can be also deduced from the more general context of -regularized families (see [10, 15–18]).

We will need the following results on Laplace transform (see [19, Theorem and Corollary ] for a detailed proof).

Lemma 2.4. * Suppose that is holomorphic and satisfies and let Then there exists with such that for all *

Lemma 2.5. * Suppose that is holomorphic and satisfies for all . Then there exists a bounded function such that for all *

We introduce the means

for

*Definition 2.6. *Let , be Banach spaces. A subset of is called -bounded if there exists a constant such that
for all The least such that (2.7) is satisfied is called the -bound of and is denoted as

The notion of -boundedness was implicitly introduced and used by Bourgain [20] and later on also by Zimmermann [21]. Explicitly it is due to Berkson and Gillespie [22] and to Clément et al. [23].

-boundedness clearly implies boundedness. If , the notion of -boundedness is strictly stronger than boundedness unless the underlying space is isomorphic to a Hilbert space [24, Proposition ]. Some useful criteria for -boundedness are provided in [11, 24].

*Remark 2.7. *(a)Let be -bounded sets, then is -bounded.(b)Let and be -bounded sets, then is -bounded and
(c)Also, each subset of the form is -bounded whenever is bounded.

We recall that those Banach spaces for which the Hilbert transform is bounded on for some are called spaces. For more information and details on the Hilbert transform and the Banach spaces we refer to [12]. Examples of spaces include Hilbert spaces, Sobolev spaces (see [25]), Lebesgue spaces , , when is a space, and the Schatten-von Neumann classes , of operators on Hilbert spaces.

After these preliminaries, we state the following operator-valued Fourier multiplier theorem. It is fundamental in our treatment. A proof can be founded in [11].

Theorem 2.8. * Suppose that is a space and let Let be such that the following conditions are satisfied. *(i)*The set is -bounded.*(ii)*The set is -bounded. **Then the operator defined by**
extends to a bounded operator from to *

#### 3. Existence of Solutions

Let Consider the equation

with initial conditions where is the generator of an -regularized family By a solution of (3.1) we understand a function such that and verify (3.1).

Proposition 3.1. * Let be an -regularized family on with generator . If , then given by
**
is a solution of (3.1).*

*Proof. *Given that , we obtain from Proposition 2.3 that and hence , is of class . For all , we have
If , then . Moreover,
Since , from (3.2), we have that and
Hence,
By the other side, for all , we obtain
Since and is closed, from (3.6) we conclude that verify (3.1).

The following remarkable result provides a wide class of generators of -regularized families. In what follows we denote

Theorem 3.2. *Let be a positive self-adjoint operator on a Hilbert space such that
**
Then is the generator of a bounded -regularized family on *

*Proof. *Since is a positive self-adjoint operator in , the spectrum is a subset of the negative real axis and the resolvent operator is defined at least for all negative non real Let such that If , then clearly If , then we claim that In fact, for , with a direct computation we obtain

Note that if and only if or

Since , we have that

Hence, if and only if Since a direct calculation gives
proving the claim. We conclude that for all Hence (see Kato [26, Section ]),
Note that
since has order Define We have by (3.14) and (3.13) that for all
On the other hand,
where
and, by (3.14),
for all We conclude that

By Lemma 2.5 there exists a strongly continuous family such that and for In consequence, for all we have

and, by Remark 2.2, it shows that is a bounded -regularized family generated by .

Since it is a known fact that the Dirichlet-Laplacian operator is a self-adjoint operator on and , we obtain the following corollary.

Corollary 3.3. *Let be a bounded domain in with smooth boundary , and assume that Then the Dirichlet-Laplacian operator with domain is the generator of an -regularized family on *

*Remark 3.4. *In Theorem 3.2 the condition is fundamental to have for all with , which is the key in the proof. Figure 1 is the typical situation, where we have mapped by the lines , , and with , , and .

Note that in case it can happen that For example, taking , , and , we obtain Figure 2 of for and

#### 4. -Well-Posedness

Having presented preliminary material on -boundedness and Fourier multipliers, we will now show how these tools can be used to handle (3.1). Our main result give concrete conditions on the operator under which (3.1) has -maximal regularity.

The definition of -maximal regularity which we investigate in this section is given as follows.

*Definition 4.1. * One says that (3.1) has -maximal regularity (or is -well posed) on if for each there is a unique function such that (3.1) holds a.e.

The following is the main abstract result of this section. It completely characterizes the maximal regularity of solutions for (3.1) in Lebesgue spaces.

Theorem 4.2. * Let be a space, and let be the generator of a bounded -regularized family The following statements are equivalent. *(i)*Equation (3.1) has -maximal regularity on *(ii)* for all and the set*

*Proof. * By (3.1) and Definition 4.1 together with Proposition 3.1, the convolution operator with kernel
is a bounded operator from to Note that the Fourier transform exists for because is bounded and can be analytically extended from to the imaginary axis. Then the symbol of this convolution operator is given by
and the conclusion follows from [11, Proposition ].

Define and

We check that the set is -bounded.

Since , we have that Replacing in (4.4)

Note that

Since the sum of -bounded sets is -bounded, see [11], we obtain that is -bounded.

We now check that the set is -bounded. With a direct computation, we obtain

Hence

Since the set is -bounded and the complex functions appearing in the above equality are bounded, we obtain the claim from the fact that the sum of -bounded sets is again -bounded. We employ now Theorem 2.8 to conclude that the operator defined by
extends to a bounded operator from to

Define

We will prove that the sets and are -bounded.

In fact, note that Hence the set is -bounded. Moreover, we have

obtaining that the set is -bounded. By Theorem 2.8 we conclude that the operator defined by
extends to a bounded operator from to

Finally, define

The set is -bounded from hypothesis and also note that the set is -bounded, since
Again by Theorem 2.8 we conclude that the operator defined by
extends to a bounded operator from to From (4.9), (4.12), and (4.15) and since it is clear that (3.1) has -maximal regularity if the convolution operator with each one of the kernels
is a bounded operator from to (see [11]), we conclude (i) and the proof is complete.

Of course, -boundedness in (4.1) can be replaced by boundedness in case is a Hilbert space.

Corollary 4.3. *The solution of (3.1), under the conditions given by Theorem 4.2, satisfies the following maximal regularity property: and Moreover, there exists a constant independent of such that
*

The proof follows by the closed-graph theorem.

As an example, we consider for the vibration equation subject to the action of an external force. Explicitly, we consider

in a smooth bounded region . Also, we assume that . We have the following application in the Hilbert space setting.

Theorem 4.4. *Let be a bounded domain with smooth boundary in . Suppose that Then the initial value problem (4.18) defined on with Dirichlet boundary conditions has -maximal regularity on *

*Proof. *Let , , and and note that if and only if By Corollary 3.3 we conclude that generates a bounded -regularized family on

Note that we have and there exists a constant such that

for all Here is the first eigenvalue of the Dirichlet-Laplacian. Hence, by Theorem 4.2 the assertion follows.

*Remark 4.5. *In Figure 3, we show in case , , and

#### Acknowledgments

The authors are supported by Laboratorio de Analisis Estocástico, Proyecto Anillo ACT-13. The third author is also partially financed by Proyecto Fondecyt de Iniciación 11075046.