Global solutions to a structure acoustic interaction model with nonlinear sources
Introduction
Let be a bounded, open, connected domain with smooth boundary , where and Γ are two disjoint, open, connected sets of positive Lebesgue measure. Moreover, Γ is a flat portion of the boundary of Ω and is referred to as the elastic wall, whose dynamics are described by the Berger plate or beam equation. We refer the reader to [20] and the references quoted therein for more details on the Berger model. The acoustic medium in the chamber Ω is described by a semilinear wave equation influenced by a restoring source. The resulting relationship is represented in the following coupled PDE system: where the initial data reside in the finite energy space, i.e., The term represents an internal restoring source acting on the acoustic medium chamber Ω and is allowed to have an arbitrary power . The term represents a frictional internal damping on the plate, whereas is an internal source on the plate that is allowed to have a bad sign which may cause instability (blow up) in a finite time. In addition, ν and denote the outer normal vectors to Γ and ∂Γ; respectively. The part of the boundary ∂Ω describes a rigid wall, while the coupling takes place on the flexible wall Γ.
Models such as (1.1) arise in the context of modeling gas pressure in an acoustic chamber which is surrounded by a combination of rigid and flexible walls. The pressure in the chamber is described by the solution to a wave equation, while vibrations of the flexible wall are described by the solution to a coupled plate equation. The nonlinear source term or, more generally , in a single wave equation often arises in quantum field theory and additionally has some important mechanical applications. See for example Jörgens [37] and Segal [56].
In the considered system, the linear parts of the system capture plate vibrations influenced by acoustic pressure in the chamber. However, nonlinear source terms allow for refinements on the restoring forces, for example modeling weaker response under smaller deviations from equilibrium. More generally, semilinear component of the system can capture coupling of the model with other types of dynamics. For instance, plate vibrations in hypersonic airflows can be approximately decoupled from the flow by replacing the pressure on the plate with a nonlinear lower-order forcing term of the form –as in piston theory, where U is the speed of the airflow over the surface of the plate. The latter scenario is more complex than we address in this paper, but it prompts us to focus on lower order nonlinearities like in the plate equation.
Structural acoustic interaction models have rich and extensive history. These models are well known in both the physical and mathematical literature and go back to the canonical models considered in [10], [36]. In the context of stabilization and controllability of structural acoustic models there is a very large body of literature. We refer the reader to the monograph by Lasiecka [42] which provides a comprehensive overview and quotes many works on these topics. Other contributions worthy of mention include [2], [3], [4], [5], [17], [29], [30], [41]. For instance, questions of exact controllability or uniform stability are considered in [5] for the interaction of wave/Kirchhoff plates, [17] for the interaction of wave/shell models, and [29] for the interaction of wave/Reissner-Mindlin plates. For the case that corresponds to nonlinear aeroelastic plate problem in a flow of gas, we mention the papers [13], [16], [19] which consider the coupled system of a linear wave equation in the upper-half space in and von Karman equations on the flexible wall.
Other central questions include the existence of global attractors and the analysis of their properties. This particular topic attracted considerable interest in the last three decades or so. In general, structural acoustic models present several technical difficulties in proving existence of attractors, or asserting their regularity and their finite dimensionality in the presence of nonlinear damping. These challenges are an intrinsic character for the hyperbolic-like dynamics involved in studying the long time behavior of structural acoustic models. In the presence of linear damping, there are several interesting results on the existence of global attractors [6], [20], [35], [58]. However, the presence of nonlinear damping has been recognized in the literature as a source of many technical difficulties. Over the years, there has been some novel progress in this area, particularly for wave equations influenced by nonlinear damping [26], [27], [43], [51]. For structural acoustic models and other related models we mention the work of Bucci et al. [15] and the work by Chueshov and Lasiecka and others [21], [22], [23], [24], [25]. In particular, [23] provides a comprehensive account of new abstract results, along with the analysis of relevant PDE examples such as wave and plate equations with nonlinear damping and critical nonlinear source terms.
Nonlinear wave equations under the influence of damping and sources has been attracting considerable attention in the research field of analysis of nonlinear PDEs. We briefly give an overview of some related results in the literature regarding wave equations and systems of wave equations. In [28], Georgiev and Todorova considered a semilinear wave equation with frictional damping and a subcritical source term. The paper [28] provided the local and global solvability of the equation, and also provided a blow up result which ignited considerable interest in the area. Consequent results on wave equations with subcritical sources were established in [1], [18], [50], [53], [60]. We also would like to mention the works [7], [8], [9] on wave equations influenced by degenerate damping and source terms. Well-posedness results for wave equations with supercritical sources include the breakthrough papers by Bociu and Lasiecka [11], [12] and the papers on systems of wave equations [31], [32], [33]. For other related results on wave equations involving supercritical sources we mention [34], [38], [39], [48], [49] and the references therein.
In this manuscript, we follow a similar approach by Lions [46] to establish the existence of local weak solutions. For the case of a critical source acting on the wave equation, we prove such solutions depend continuously on the initial data, and so these solutions are unique in the finite energy space.
Throughout the paper the following notational conventions for space norms and inner products will be used, respectively: We also use the notation γu to denote the trace of u on Γ and we write as or . Occasionally, we also use the notation to mean γu. We also use at times the notation to mean . As is customary, C shall always denote a positive constant which may change from line to line.
Further, we put It is well-known that the standard norm is equivalent to . Thus, we put: For a similar reason, we put: Relevant to this work in the entire paper, we define the Banach space X and its norm by: For a Banach space Y, we denote the duality pairing between the dual space and Y by . That is, Throughout the paper, the following Sobolev imbeddings will be used often without mention: As it occurs so frequently we shall pass to subsequences consistently without re-indexing.
Throughout this paper, we study (1.1) under the following general assumptions: Assumption 1.1 We assume that the sources in (1.1) are -valued functions satisfying: , .
Remark 1.2 As the following bounds will be used often throughout the paper it is worthy of note that the above assumption implies that
We begin by introducing the definition of a suitable weak solution for (1.1). Definition 1.3 A pair of functions is said to be a weak solution of (1.1) on the interval provided: , , , , , , The functions u and w satisfy the following variational identities for all : for all test functions with , and with .
Remark 1.4 In Definition 1.3 above, denotes the space of weakly continuous (often called scalarly continuous) functions from into a Banach space X. That is, for each and the map is continuous on .
Our principal result is the existence of local solutions of problem (1.1) in the following sense. Theorem 1.5 Under the validity of Assumption 1.1, problem (1.1) possesses a local weak solution, , in the sense of Definition 1.3 on a non-degenerate interval , where T depends upon the initial positive energy (where is defined below). Furthermore, if in addition , then the said solution satisfies the following energy identity for all : where If , then the solution satisfies the energy inequality: Equivalently, (1.6) can also be written as with , where H is the primitive of h, i.e., .
Although the source term acting on the plate equation can have a “bad” sign which may cause blow up in finite time, our next result states that solutions established by Theorem 1.5 are indeed global solutions, provided the plate source term is essentially linear. Theorem 1.6 In addition to Assumption 1.1, assume . Then any solution furnished by Theorem 1.5 is a global weak solution and the existence time T may be taken arbitrarily large.
Theorem 1.7 In addition Assumption 1.1, assume and is an initial data with a corresponding weak solution of (1.1), where . If is a sequence of initial data such that in H, as , then the corresponding weak solutions with initial data satisfy: where is chosen to be independent of .
Corollary 1.8 In addition to Assumptions 1.1, assume . Then, weak solutions of (1.1) (in the sense of Definition 1.3) are unique.
The paper is organized as follows. Sections 2 and 3 are devoted to the proof of Theorem 1.5. In Sections 4 and 5 we complete the proofs of Theorem 1.6, Theorem 1.7.
Section snippets
Approximate solutions
We begin by selecting a sequence with the following properties: Let with its domain . It is well known that B is positive, self-adjoint, and B is the inverse of a compact operator. Moreover, B has the infinite sequence of positive eigenvalues and a corresponding sequence of
Energy identity and energy inequality
This section is devoted to derive the energy identity (1.4) in Theorem 1.5 in the case . One is tempted to test (1.2) with and (1.3) with , and carry out standard calculations to obtain energy identity. However, this procedure is only formal, since and are not regular enough and cannot be used as test functions in (1.2) and (1.3). In order to overcome this technicality we shall use the difference quotients and and their well-known properties that appeared in [40] and
Global existence
This section is devoted to prove the existence of global solutions as described in Theorem 1.6. As in [1], [33], [49] and other works, it is the case here that either a given solution must exist globally in time or else one may find a value of with so that where, .
By demonstrating a bound on the energy on every interval which is dependent only upon T and
Continuous dependence on initial data
In this section, we provide the proof to Theorem 1.7 in the case , where the bound (4.2) is crucial in the proof.
Proof Let . Assume that is a sequence of initial data that satisfies: Let and be the weak solutions to (1.1) defined on in the sense of Definition 1.3, corresponding to the initial data and , respectively. First, we show that the local existence time T
References (60)
- et al.
Exact controllability of structural acoustic interactions
J. Math. Pures Appl. (9)
(2003) - et al.
Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping
J. Differ. Equ.
(2010) - et al.
Uniform stability in structural acoustic models with flexible curved walls
J. Differ. Equ.
(2002) - et al.
Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping—source interaction
J. Differ. Equ.
(2007) Global attractors for semilinear damped wave equations with supercritical exponent
J. Differ. Equ.
(1995)- et al.
Existence of a solution of the wave equation with nonlinear damping and source terms
J. Differ. Equ.
(1994) On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam
J. Math. Anal. Appl.
(2006)On a structural acoustic model which incorporates shear and thermal effects in the structural component
J. Math. Anal. Appl.
(2008)- et al.
Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping
J. Differ. Equ.
(2014) - et al.
On wave equations of the p-Laplacian type with supercritical nonlinearities
Nonlinear Anal.
(2019)
Boundary stabilization of a 3-dimensional structural acoustic model
J. Math. Pures Appl. (9)
Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation
J. Math. Anal. Appl.
Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. II. General boundary data
J. Differ. Equ.
Local and global well-posedness of semilinear Reissner–Mindlin–Timoshenko plate equations
Nonlinear Anal.
Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms
Nonlinear Anal.
Global existence for the wave equation with nonlinear boundary damping and source terms
J. Differ. Equ.
Systems of nonlinear wave equations with damping and source terms
Differ. Integral Equ.
Wellposedness of a structural acoustics model with point control
Uniform decay rates for solutions to a structural acoustics model with nonlinear dissipation
Appl. Math. Comput. Sci.
Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls
Adv. Differ. Equ.
Attractors of Evolution Equations
Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms
Control Cybern.
On nonlinear wave equations with degenerate damping and source terms
Trans. Am. Math. Soc.
Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms
Indiana Univ. Math. J.
Spectral properties of an acoustic boundary condition
Indiana Univ. Math. J.
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping
Discrete Contin. Dyn. Syst.
The problem on interaction of von Karman plate with subsonic flow of gas
Math. Methods Appl. Sci.
Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models
Global attractor for a composite system of nonlinear wave and plate equations
Commun. Pure Appl. Anal.
On the oscillations of a von Kármán plate in a potential gas flow
Izv. Akad. Nauk SSSR, Ser. Mat.
Cited by (3)
On the asymptotic behavior of solutions to a structural acoustics model
2023, Journal of Differential Equations