Communicated by X. WangThis work deals with L p -boundedness of mild solutions for an abstract version of the damped wave equation u 00 C˛u 000 Ď u C u 0 C f that models the vibrations of flexible structures possessing internal material damping and external force f. We prove that the set consisting of mild solutions for this problem is compact and connected in the space of continuous functions. This property is known in the literature as the Kneser's property.together with its semilinear version,˛uwhere A is a closed linear operator acting in a Banach space X,˛,ˇ, 2 R C , and f is an X-valued function. Usually, assumptions to the spectrum of operator A allow a systematic treatment of such models. Ideas from theory of resolvent of operators are used to define a so-called .˛,ˇ, /-regularized family. By using this family and Duhamel's principle, we can define mild solutions for this type of equations. Maximal regularity in Hölder spaces for the Equation 1.1 has been recently characterized in [11]. In [12], the authors have characterized L p -maximal regularity of Equation (1.1). To get their characterization, the authors introduced the notion of .˛,ˇ, /regularized families, and they used the technique of operator-valued Fourier multiplier in Unconditional Martingale Difference property spaces ‡ . Note that the operator A D is the generator of an . , c 2 , c 2 /-regularized family in L 2 . /. In Section 2, we review some of the standard properties of .˛,ˇ, /-regularized families. In [14], the authors have studied regularity of mild and strong solutions for Equation (1.1) in a Hilbert space under appropriate initial conditions. The result in [15] for Equation (1.1) assumed that A is a self-adjoint operator defined in a Hilbert space H and rewrite the equation as a first order abstract system on the phase space D.A 1=2 / D.A 1=2 / H. We mention that in [15], the Equation 1.1 (with A D and f D 0) is called Moore-Gibson-Thompson equation, and the nonlinear version is referred as the Jordan-Moore-Gibson-Thompson-Westervelt Equation (see also [16]). In [15], Equation 1.1 (with A D and f D 0) arise as a model in high intensity ultrasound. We observe that real systems usually exhibit internal variations or are submitted to external perturbations. In many situations, we can assume that these variations are approximately periodic in a broad sense. In [17] and [18], the authors have analyzed the asymptotic periodicity of Equation (1.2).In the first part of this work, we examine L p -boundedness properties for the Equation 1.1 and its perturbation (1.2) (see Section 3). In the second one, we establish a Kneser's type property for the set formed by the mild solutions of Equation (1.2) (see Section 4). This kind of subject matter for flexible structures is largely nonexistent at this time, and consequently, it shall be widely investigated. Finally, we show that our abstract results apply to Equations 1.1 and (1.2) in the case of A D , the Laplacian.
Preliminaries and basic resultsIn this section, we introduce notations, definition...