Global solutions to a structure acoustic interaction model with nonlinear sources

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Abstract

This article focuses on a structural acoustic interaction system consisting of a semilinear wave equation defined on a smooth bounded domain ΩR3 which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of Ω. In particular, the source terms acting on the wave and plate equations are allowed to have arbitrary growth order. We employ a standard Galerkin approximation scheme to establish a rigorous proof of the existence of local weak solutions. In addition, under some conditions on the parameters in the system, we prove such solutions exist globally in time and depend continuously on the initial data.

Introduction

Let ΩR3 be a bounded, open, connected domain with smooth boundary Ω=Γ0Γ, where Γ0 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:{uttΔu+|u|p1u=0 in Ω×(0,T),wtt+Δ2w+wt+ut|Γ=h(w) in Γ×(0,T),u=0 on Γ0×(0,T),νu=wt on Γ×(0,T),w=νΓw=0 on Γ×(0,T),(u(0),ut(0))=(u0,u1),(w(0),wt(0))=(w0,w1), where the initial data reside in the finite energy space, i.e.,u0HΓ01(Ω)Lp+1(Ω),u1L2(Ω), and (w0,w1)H02(Γ)×L2(Γ). The term |u|p1u represents an internal restoring source acting on the acoustic medium chamber Ω and is allowed to have an arbitrary power p1. The term wt represents a frictional internal damping on the plate, whereas h(w) 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 νΓ denote the outer normal vectors to Γ and ∂Γ; respectively. The part Γ0 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 f(u)=u5 or, more generally f(u)=|u|p1u, 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 (wt+Uwx)3–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 h(w) 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 R3 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 Lp space norms and inner products will be used, respectively:||u||p=||u||Lp(Ω),(u,v)Ω=(u,v)L2(Ω),|u|p=||u||Lp(Γ),(u,v)Γ=(u,v)L2(Γ). We also use the notation γu to denote the trace of u on Γ and we write ddt(γu(t)) as γut or γu. Occasionally, we also use the notation u|Γ to mean γu. We also use at times the notation u to mean ut. As is customary, C shall always denote a positive constant which may change from line to line.

Further, we putHΓ01(Ω)={uH1(Ω):u|Γ0=0}. It is well-known that the standard norm uHΓ01(Ω) is equivalent to u2. Thus, we put:uHΓ01(Ω)=u2. For a similar reason, we put:wH02(Γ)=|Δw|2. Relevant to this work in the entire paper, we define the Banach space X and its norm by:X=HΓ01(Ω)Lp+1(Ω),uX=u2+up+1. For a Banach space Y, we denote the duality pairing between the dual space Y and Y by ,Y,Y. That is,ψ,yY,Y=ψ(y) for yY,ψY. Throughout the paper, the following Sobolev imbeddings will be used often without mention:{H1ϵ(Ω)L61+2ϵ(Ω) for ϵ[0,1],H1ϵ(Ω)γH12ϵ(Γ)L41+2ϵ(Γ) for ϵ[0,12],H1(Γ)Lq(Γ) for all 1q<. 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 R-valued functions satisfying:

  • 1p<,

  • hC1(R) such that |h(u)|C(|u|q1+1) with 1q<.

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{||u|p1u|v|p1v|C(|u|p1+|v|p1)|uv|,|h(u)|C(|u|q+1),|h(u)h(v)|C(|u|q1+|v|q1+1)|uv|.

We begin by introducing the definition of a suitable weak solution for (1.1).

Definition 1.3

A pair of functions (u,w) is said to be a weak solution of (1.1) on the interval [0,T] provided:

  • (i)

    uCw([0,T];X), utCw([0,T];L2(Ω)),

  • (ii)

    wCw([0,T];H02(Γ)), wtCw([0,T];L2(Γ)),

  • (iii)

    (u(0),ut(0))=(u0,u1)HΓ01(Ω)×L2(Ω),

  • (iv)

    (w(0),wt(0))=(w0,w1)H02(Γ)×L2(Γ),

  • (v)

    The functions u and w satisfy the following variational identities for all t[0,T]:(ut(t),ϕ(t))Ω(u1,ϕ(0))Ω0t(ut(τ),ϕt(τ))Ωdτ+0t(u(τ),ϕ(τ))Ωdτ0t(wt(τ),γϕ(τ))Γdτ+0tΩ|u(τ)|p1u(τ)ϕ(τ)dxdτ=0,(wt(t)+γu(t),ψ(t))Γ(w1+γu(0),ψ(0))Γ0t(wt(τ),ψt(τ))Γdτ0t(γu(τ),ψt(τ))Γdτ+0t(Δw(τ),Δψ(τ))Γdτ+0t(wt(τ),ψ(τ))Γdτ=0tΓh(w(τ))ψ(τ)dΓdτ, for all test functions ϕCw([0,T];X) with ϕtL2(0,T;L2(Ω)), and ψCw([0,T];H02(Γ)) with ψtL2(0,T;L2(Γ)).

Remark 1.4

In Definition 1.3 above, Cw([0,T];X) denotes the space of weakly continuous (often called scalarly continuous) functions from [0,T] into a Banach space X. That is, for each uCw([0,T];X) and fX the map tf,u(t)X,X is continuous on [0,T].

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, (u,w), in the sense of Definition 1.3 on a non-degenerate interval [0,T], where T depends upon the initial positive energy E(0) (where E(t) is defined below). Furthermore, if in addition 1p3, then the said solution (u,w) satisfies the following energy identity for all t[0,T]:E(t)+0t|wt(τ)|22dτ=E(0)+0tΓh(w)wtdΓdτ, whereE(t)=12(ut(t)22+u(t)22+|wt(t)|22+|Δw(t)|22)+1p+1u(t)p+1p+1. If p>3, then the solution (u,w) satisfies the energy inequality:E(t)+0t|wt(τ)|22dτE(0)+0tΓh(w)wtdΓdτa.e.[0,T]. Equivalently, (1.6) can also be written asE(t)+0t|wt(τ)|22dτE(0)a.e.[0,T], with E(t)=E(t)ΓH(w(t))dΓ, where H is the primitive of h, i.e., H(w)=0wh(s)ds.

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 q=1. Then any solution (u,w) 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 p3 and U0=(u0,w0,u1,w1)H is an initial data with a corresponding weak solution (u,w) of (1.1), where H=HΓ01(Ω)×H02(Γ)×L2(Ω)×L2(Γ). If U0n=(u0n,w0n,u1n,w1n) is a sequence of initial data such that U0nU0 in H, as n, then the corresponding weak solutions (un,wn) with initial data U0n satisfy:(un,wn,utn,wtn)(u,w,ut,wt)inL(0,T;H),asn, where 0<T< is chosen to be independent of nN.

Corollary 1.8

In addition to Assumptions 1.1, assume p3. 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 {ej}1X=HΓ01(Ω)Lp+1(Ω) with the following properties:{e1,,eN are linearly independent for every NN,andThe set of all finite linear combinations of the form:{j=1Ncjej:cjR,NN} is dense in X. Let B=Δ2:D(B)L2(Γ)L2(Γ) with its domain D(B)=H4(Γ)H02(Γ). 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 {μn:nN} 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 1p3. One is tempted to test (1.2) with ut and (1.3) with wt, and carry out standard calculations to obtain energy identity. However, this procedure is only formal, since ut and wt 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 Dhu and Dhw 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 (u,w) must exist globally in time or else one may find a value of T0 with 0<T0< so thatlim suptT0(E(t)+0t|wt(τ)|22dτ)=, where, E(t)=12(ut(t)22+u(t)22+|wt(t)|22+|Δw(t)|22)+1p+1u(t)p+1p+1.

By demonstrating a bound on the energyE(t)+0t|wt(τ)|22dτ on every interval [0,T] 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 1p3, where the bound (4.2) is crucial in the proof.

Proof

Let U0=(u0,w0,u1,w1)H=HΓ01(Ω)×H02(Γ)×L2(Ω)×L2(Γ). Assume that {U0n=(u0n,w0n,u1n,w1n):nN} is a sequence of initial data that satisfies:U0nU0 in H strongly as n. Let {(un,wn)} and (u,w) be the weak solutions to (1.1) defined on [0,T] in the sense of Definition 1.3, corresponding to the initial data {U0n} and {U0}, respectively. First, we show that the local existence time T

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