# Well-posedness of a Nonlinear Acoustics – Structure Interaction Model

Barbara Kaltenbacher, Amjad Tuffaha

###### Abstract

We establish local-in-time and global in time well-posedness for small data,
for a coupled system of nonlinear acoustic structure interactions. The model
consists of the nonlinear Westervelt equation on a bounded domain with non
homogeneous boundary conditions, coupled with a 4th order linear equation
defined on a lower dimensional interface occupying part of the boundary of the
domain, with transmission boundary conditions matching acoustic velocities and
acoustic pressures. While the well-posedness of the Westervelt model has been
well studied in the literature, there has been no works on the literature on
the coupled structure acoustic interaction model involving the Westervelt
equation. Another contribution of this work, is a novel variational weak
formulation of the linearized system and a consideration of various boundary
conditions.

Department of Mathematics
University of Klagenfurt
Klagenfurt, Austria
e-mail<EMAIL_ADDRESS>

Department of Mathematics
American University of Sharjah
Sharjah, UAE
e-mail<EMAIL_ADDRESS>

##  1 Introduction

In this paper, we study the interaction of a nonlinear acoustic fluid or gas
with an elastic plate, a situation that is practically relevant, e.g., in the
context of high intensity ultrasound; think, e.g., of a cavity enclosed by a
thin wall [5] or a thin structure immersed in an ultrasound cleaning device
[43].

One of the classical models of nonlinear acoustics (and probably the most
widely used one) is the Westervelt equation [49], a quasilinear second order
damped wave equation. For its analysis in the practically relevant case of a
smooth bounded domain $\Omega$ we refer to, e.g., [28, 25, 21, 38, 47]; the
free space case $\Omega=\mathbb{R}^{3}$ is considered in, e.g., [10] and local
in time results for vanishing damping can be found in, e.g., [13, 11, 29].
Advanced models such as Kuznetsov’s equation [32], the Jordan-Moore-Gibson-
Thompson (JMGT) equation [19, 41, 48, 45] and the Blackstock-Crighton-
Brunnhuber-Jordan (BCBJ) equation [4, 8, 6] have been analyzed in, e.g., [10,
11, 20, 22, 21, 23, 24, 29, 26, 27, 38, 40].

For some recent work on the mathematical analysis of acoustics-structure
interaction problems we refer to, e.g., [1, 2, 3, 34, 35] and the references
therein. Most of these models involve a linear or semi-linear wave equation
coupled with linear and nonlinear plate models. Another important aspect in
the study of the mathematics of acoustics-structure interactions is control
analysis, see for example [16, 17, 37, 50]. However, to the best of the
authors’ knowledge, there are no results on well-posedness of nonlinear
acoustics-structure interaction models involving the Westervelt equation.

The model we consider consists of the Westervelt equation defined on a bounded
domain and coupled with a 4th order linear plate equation defined on a lower
dimensional interface occupying a part of the boundary of the domain. The
coupling is realized through the acoustic pressure which acts as a lifting
force on the structural plate equation, in addition to a velocity matching
condition involving the normal pressure gradient. We also consider the
possibility of mixed type non-homogeneous boundary conditions on the rest of
the boundary such as Dirichlet, Neumann, and absorbing boundary conditions.
Well-posedness of the Westervelt equations under these various mixed boundary
conditions were considered in [12] with vanishing boundary data. Considering
non-zero boundary data, we aim at laying the foundations for practically
relevant control problems in future research.

Our main result in this paper, is firstly the existence of energy level weak
solutions for the linearized model, and secondly the existence of local and
global-in-time smooth solutions for the full nonlinear model. In particular,
we provide a novel variational formulation of the coupled problem, that allows
us to analyze the linearized problem (Proposition 3.5 and Corollary 3.7).
Using this together with a fixed point argument, we establish local and global
in time well-posedness for sufficiently small data (Theorem 4.4). Challenges
in the analysis of the model arise due to the well-known potential degeneracy
of the Westervelt equation (which it shares with basically all other models of
nonlinear acoustics) in addition to the low regularity due to mixed boundary
conditions. In particular, coupling to the plate provides less regularity
than, e.g., prescribed Dirichlet or Neumann boundary values would do. This
requires a different approach as compared to the existing literature on the
analysis of the Westervelt equation, e.g., [28, 39, 40]. We note that in the
mixed Dirichlet/Neumann boundary conditions, additional assumptions on the
domain have to be imposed since classical elliptic regularity does not hold in
general [44, 46]. This assumption allows us to control the $L^{\infty}$ norm
of the pressure and hence avoid degeneracy without having to control the full
Sobolev norm.

The paper is organized as follows. Section 2 introduces the model and Section
3 provides the analysis of its linearization. In Section 4 we prove well-
posedness of the nonlinear acoustics – plate model and in Section 5 we give a
brief outlook on related further research questions.

###  1.1 Notation

We will use the function space $H^{2}_{\Delta}(\Omega):=\\{u\in
L^{2}(\Omega)\,:\,u\in L^{2}(\Omega),\,\Delta u\in L^{2}(\Omega)\\}\supset
H^{2}(\Omega)$

Moreover, we abbreviate by $C^{\Omega}_{\Gamma}$ the constant in the Poincaré-
Friedrichs type inequality

$\|v\|_{H^{1}(\Omega)}\leq C^{\Omega}_{\Gamma}\Bigl{(}\|\nabla
v\|_{L^{2}(\Omega)}^{2}+\|v\|_{L^{2}(\Gamma)}^{2}\Bigr{)}^{1/2}\text{ for all
}v\in H^{1}(\Omega)$ (1.1)

for some regular boundary part $\Gamma\subseteq\partial\Omega$ with
nonvanishing measure.

The norm of some continuous embedding $X(\Omega)\to Y(\Omega)$ for some
Sobolev or Lebesgue spaces $X(\Omega)$, $X(\Omega)$ will be denoted by
$C^{\Omega}_{X,Y}$.

We will sometimes drop the subscript in $\|\cdot\|_{L^{2}(\Omega)}$ (but not
in the boundary norms, since this could cause confusion) and skip the trace
operator whenever it is clear that we are taking some boundary trace of a
function defined in $\Omega$.

The topological dual of a normed space $X$ will be denoted by $X^{\star}$.

##  2 Model

We consider a system of partial differential equations modeling the
interaction between nonlinear acoustic waves and an isotropic, homogeneous
plate. We take a domain $\Omega\subset\mathbb{R}^{3}$ with a $C^{1}$ boundary
$\partial\Omega$ consisting of four different parts $\Gamma_{pl}$,
$\Gamma_{a}$, $\Gamma_{D}$ and $\Gamma_{N}$, the plate coupling, absorbing,
and excitation boundary parts, respectively. Each of these boundary parts is
assumed to be regular in the sense that it is either empty or has positive
measure and the interfaces between them are assumed to be Lipschitz curves. An
exemplary geometric setup is depicted in figure 1.

Figure 1: Domain setup (example)

The model consists of the Westervelt equation defined in the variable $p$
representing acoustic pressure

$\displaystyle(1-2kp)p_{tt}-c^{2}\Delta p-b\Delta
p_{t}=2k(p_{t})^{2}~{}~{}\mbox{in}~{}~{}\Omega\times[0,T]$ (2.1)

and a $4$th order plate equation defined on $\Gamma_{pl}$ in the mid surface
displacement variable $w$, forced by the trace of the pressure variable $p$

$\displaystyle\rho
w_{tt}+\delta\Delta_{pl}^{2}w+\beta(-\Delta_{pl})^{\gamma}w_{t}=\kappa
p+h~{}~{}$ $\displaystyle\mbox{on}~{}~{}\Gamma_{pl}\times[0,T],$ (2.2)

where $\Delta_{pl}$ is the Laplace-Beltrami operator on $\Gamma_{pl}$,
equipped with homogeneous Dirichlet boundary conditions, and $h$ is a given
forcing function. 111See, e.g., [35] for this type of coupling in a velocity
potential formulation (rather than the pressure formulation we are using
here). Here, the constants $\rho>0$, $c>0$, $\delta>0$ represent the mass
density, the speed of sound and the ratio of plate stiffness to plate
thickness respectively, while $b>0$, $\beta>0$, and $\gamma\geq 0$ are damping
parameters. The parameter $\kappa>0$ is a coupling coefficient and $k$ is the
nonlinearity parameter of the acoustic medium. The operator
$(-\Delta_{pl})^{\gamma}$ is the spectrally defined power of the symmetric
positive definite operator $(-\Delta_{pl})$; for its definition, see, e.g.,
[36, Section 2.1.1].

Accordingly, the boundary conditions we assume on the plate are of hinged
type:

$\displaystyle
w=\Delta_{pl}w=0~{}~{}\mbox{on}~{}~{}\partial\Gamma_{pl}\times[0,T].$ (2.3)

We also impose another boundary condition connecting the plate equation and
the Westervelt equation which ensures continuity of inertial forces across the
interface

$\displaystyle\partial_{\nu}{p}=-\rho w_{tt}~{}~{}$
$\displaystyle\mbox{on}~{}~{}\Gamma_{pl}\times[0,T].$ (2.4)

Note that the pressure $p$ is related to the particle velocity $v$ through the
(linearized) momentum balance $\rho_{fl}v_{t}=-\nabla p$ where $\rho_{fl}$ is
the mass density of the fluid.

On the boundary component $\Gamma_{a}$, we impose the absorbing boundary
conditions

$\displaystyle p_{t}+c\partial_{\nu}{p}=a~{}~{}$
$\displaystyle\mbox{on}~{}~{}\Gamma_{a}\times[0,T]$ (2.5)

where the typical value of the boundary function is $a\equiv 0$ in order to
avoid spurious reflections on the boundary of the computational domain
$\Omega$, see, e.g., the review articles [14, 18] and the references therein.

Finally, we allow for mixed Dirichlet and/or Neumann boundary condition on the
boundary components $\Gamma_{D}$ and $\Gamma_{N}$,

$p=g_{D}~{}~{}\mbox{on}~{}~{}\Gamma_{D}\times[0,T]$ (2.6)
$\partial_{\nu}{p}=g_{N}~{}~{}\mbox{on}~{}~{}\Gamma_{N}\times[0,T],$ (2.7)

where $g_{D}$ and $g_{N}$ are given functions. Subsets with $g_{D}=0$ and
$g_{N}=0$ correspond to sound-soft and sound-hard boundary parts,
respectively; nonvanishing values of $g_{D}$ and $g_{N}$ represent an
excitation control action; in case of Neumann conditions, e.g., by some
piezoelectric transducer array in ultrasonics.

######  Remark 2.1 (damping).

While the damping term in the plate equation is not needed for part of the
well-posedness analysis, it is useful to achieve independence of estimates on
the final time $T$ and to extract additional regularity of $p$ via the
interface identity (3.10), see Remark 4.5 below.

On the contrary, strong damping in the Westervelt equation is essential even
for only local in time well-posedness, since due to the lack of spatial
regularity resulting from the mixed boundary conditions, the techniques from,
e.g., [13, 29] would not be applicable here.

##  3 Linearized Model

###  3.1 Reformulation, Linearization, and some Auxiliary Results

We will reformulate the model in terms of $\widetilde{w}$ defined as

$\displaystyle\widetilde{w}=w_{tt}$ (3.8)

so that the boundary conditions on $\Gamma_{pl}$ are written as

$\displaystyle\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}=\kappa
p_{tt}+\widetilde{h}~{}~{}$
$\displaystyle\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$ (3.9)

with $\widetilde{h}=h_{tt}$ and

$\displaystyle\partial_{\nu}{p}=-\rho\widetilde{w}~{}~{}$
$\displaystyle\mbox{in}~{}~{}\Gamma_{pl}\times[0,T].$ (3.10)

We also start our analysis on a linear variable coefficient version of (2.1)

$\displaystyle\alpha\,p_{tt}-c^{2}\Delta p-b\Delta
p_{t}=f~{}~{}\mbox{in}~{}~{}\Omega\times[0,T]$ (3.11)

(having in mind $\alpha=\alpha(t,x)=(1-2kp)$, $f=f(t,x)=2k(p_{t})^{2}$), thus
altogether

$\displaystyle\alpha\,p_{tt}-c^{2}\Delta p-b\Delta p_{t}$
$\displaystyle=f~{}~{}\mbox{in}~{}~{}\Omega\times[0,T]$ (3.12) $\displaystyle
c\partial_{\nu}{p}+p_{t}$
$\displaystyle=a~{}~{}\mbox{in}~{}~{}\Gamma_{a}\times[0,T]$ $\displaystyle p$
$\displaystyle=g_{D}~{}~{}\mbox{on}~{}~{}\Gamma_{D}\times[0,T]$
$\displaystyle\partial_{\nu}{p}$
$\displaystyle=g_{N}~{}~{}\mbox{on}~{}~{}\Gamma_{N}\times[0,T]$
$\displaystyle\partial_{\nu}{p}$
$\displaystyle=-\rho\widetilde{w}~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}$
$\displaystyle=\kappa
p_{tt}+\widetilde{h}~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\widetilde{p}(0)$
$\displaystyle=p_{0},~{}~{}\widetilde{p}_{t}(0)=p_{1}.$

We homogenize the Dirichlet and Neumann conditions on $\Gamma_{D}$,
$\Gamma_{N}$ as well as the absorbing condition on $\Gamma_{a}$ in order to be
able to consider $g_{D}=0$, $g_{N}=0$, $a=0$ in the following. To this end, we
decompose the solution $p$ into $\bar{p}$ and $\widetilde{p}$ where $\bar{p}$
serves as an extension of the inhomogeneous boundary data $g_{D}$, $g_{N}$,
$a$ to the interior by satisfying the initial boundary value problem IBVP

$\displaystyle\alpha\,\bar{p}_{tt}-c^{2}\Delta\bar{p}-b\Delta\bar{p}_{t}$
$\displaystyle=0~{}~{}\mbox{in}~{}~{}\Omega\times[0,T]$ (3.13) $\displaystyle
c\partial_{\nu}{\bar{p}}+\bar{p}_{t}$
$\displaystyle=a~{}~{}\mbox{in}~{}~{}\Gamma_{a}\times[0,T]$
$\displaystyle\bar{p}$
$\displaystyle=g_{D}~{}~{}\mbox{on}~{}~{}\Gamma_{D}\times[0,T]$
$\displaystyle\partial_{\nu}{\bar{p}}$
$\displaystyle=g_{N}~{}~{}\mbox{on}~{}~{}\Gamma_{N}\times[0,T]$
$\displaystyle\partial_{\nu}{\bar{p}}$
$\displaystyle=0~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\bar{p}(0)$ $\displaystyle=0,~{}~{}\bar{p}_{t}(0)=0.$

######  Lemma 3.1.

Under the assumptions $\alpha,\frac{1}{\alpha}\in
L^{\infty}(0,T;L^{\infty}(\Omega))$, $\alpha_{t}\in
L^{1}(0,T;L^{\infty}(\Omega))$ (with small enough norm), $\alpha_{tt}\in
L^{2}(0,T;L^{3}(\Omega))$, $\nabla\alpha\in L^{2}(0,T;L^{3}(\Omega))$,
$a_{ttt}\in L^{2}(0,t;L^{2}(\Gamma_{a}))$, $g_{D\,ttt}\in
L^{\infty}(0,T;H^{1/2}(\Gamma_{D}))$, $g_{D\,tttt}\in
L^{1}(0,T;H^{1/2}(\Gamma_{D}))$, and $g_{N\,ttt}\in
W^{1,1}(0,T;H^{-1/2}(\Gamma_{N}))$, there exists a unique solution $\bar{p}\in
W^{3,\infty}(0,T;H^{1}(\Omega))\cap H^{3}(0,T;H^{2}_{\Delta}(\Omega))$ with
$\textup{tr}_{\Gamma_{a}}\bar{p}_{tttt}\in L^{2}(0,T;L^{2}(\Gamma_{a}))$ to
the IBVP (3.13).

In addition, for any $t\in(0,T)$, $\bar{p}$ satisfies the energy estimate

$\mathcal{E}^{p}_{1}[\bar{p}](t)+\int_{0}^{T}\mathcal{E}^{p}_{1}[\bar{p}](s)\,ds\leq
C\Bigl{(}\mathcal{E}^{p}_{1}[\bar{p}](0)+G[a,g_{D},g_{N}]+G[a_{t},g_{D\,t},g_{N\,t}]\Bigr{)},$
(3.14)

where
$\mathcal{E}^{p}_{1}[\bar{p}]=\mathcal{E}^{p}[\bar{p}](t)+\mathcal{E}^{p}[\bar{p}_{t}](t)$,
cf. (3.24), (3.25), and

$\displaystyle
G[a,g_{D},g_{N}]:=\|a_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}+\|g_{D\,tt}\|_{L^{2}(0,T;H^{1/2}(\Gamma_{D}))}^{2}+\|g_{N\,t}\|_{W^{1,1}(0,t;H^{-1/2}(\Gamma_{N}))}^{2}$
(3.15)

The constant $C$ depends on the coefficients $b$, $c$, $\rho$, $\delta$,
$\kappa$, $\|\frac{1}{\alpha}\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}$,
$\|\alpha_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$, but not directly on $T$.

The proof of the above lemma goes analogously to the proof of e.g., [30,
Theorem 1]; its most crucial steps are given in the appendix.

The remaining part $\widetilde{p}$ together with $\widetilde{w}$ then
satisfies the equation

$\displaystyle\alpha\,\widetilde{p}_{tt}-c^{2}\Delta\widetilde{p}-b\Delta\widetilde{p}_{t}$
$\displaystyle=f~{}~{}\mbox{in}~{}~{}\Omega\times[0,T]$ (3.16) $\displaystyle
c\partial_{\nu}{\widetilde{p}}+\widetilde{p}_{t}$
$\displaystyle=0~{}~{}\mbox{in}~{}~{}\Gamma_{a}\times[0,T]$
$\displaystyle\widetilde{p}$
$\displaystyle=0~{}~{}\mbox{on}~{}~{}\Gamma_{D}\times[0,T]$
$\displaystyle\partial_{\nu}{\widetilde{p}}$
$\displaystyle=0~{}~{}\mbox{on}~{}~{}\Gamma_{N}\times[0,T]$
$\displaystyle\partial_{\nu}{\widetilde{p}}$
$\displaystyle=-\rho\widetilde{w}~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}$
$\displaystyle=\kappa\widetilde{p}_{tt}+\kappa\bar{p}_{tt}+\widetilde{h}~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\widetilde{p}(0)$
$\displaystyle=p_{0},~{}~{}\widetilde{p}_{t}(0)=p_{1}.$

We thus continue by considering (3.16) with $\widetilde{h}+\kappa\bar{p}_{tt}$
replaced by $\widetilde{h}\in W^{1,1}(0,T;L^{2}(\Gamma_{pl}))$ while
supressing the tilde notation for the $p$ variable, that is, we study (3.12)
with $g_{D}=0$, $g_{N}=0$, $a=0$. In fact, the high temporal regularity
$\bar{p}\in W^{3,\infty}(0,T;L^{2}(\Omega))$ in Lemma 3.1 is motivated by the
need for $\widetilde{h}+\kappa\bar{p}_{tt}\in
W^{1,1}(0,T;L^{2}(\Gamma_{pl}))$.

In the energy estimates below, we will see that the plate equation sometimes
does not yield sufficient regularity of the trace $p|_{\Gamma_{pl}}$ and we
thus need to bound the values of $p$ on $\Gamma_{pl}$ by relying on its
interior regularity in $\Omega$ and a trace estimate. While our energy
estimates will allow us to bound $\|\Delta p\|_{L^{2}(\Omega)}+\|\nabla
p\|_{L^{2}(\Omega)}$ we cannot expect full $H^{2}(\Omega)$ regularity from
elliptic estimates – again due to the fact that boundary conditions are too
weak. Thus, the following trace estimate (with $\vec{\varphi}=\nabla p$ in
mind) will be useful.

######  Lemma 3.2.

For a function $\vec{\varphi}\in L^{2}(\Omega)^{d}$ with $\mathop{\rm
div}\nolimits{\vec{\varphi}}\in L^{2}(\Omega)$, the trace
$\vec{\varphi}\cdot\nu$ is a well defined element in
$H^{-1/2}(\partial\Omega)$, the following inequality holds

$\displaystyle\|\vec{\varphi}\cdot\nu\|_{H^{-1/2}(\partial\Omega)}\leq
C^{tr}_{-1/2}\Bigl{(}\|\vec{\varphi}\|_{L^{2}(\Omega)^{d}}+\|\mathop{\rm
div}\nolimits\vec{\varphi}\|_{L^{2}(\Omega)}\Bigr{)}$ (3.17)

where $C^{tr}_{-1/2}$ is the norm of the inverse trace operator
$H^{1/2}(\partial\Omega)\to H^{1}(\Omega)$.

###### Proof.

The estimate follows by a duality argument, using the identity
$\int_{\partial\Omega}\vec{\varphi}\cdot\nu\,v\,dS=\int_{\Omega}\mathop{\rm
div}\nolimits(\vec{\varphi}\bar{v})\,dx$ for sufficiently smooth
$\vec{\varphi}$, where $\bar{v}\in H^{1}(\Omega)$ is an extension of $v\in
H^{1/2}(\partial\Omega)$, such that $\text{tr}_{\partial\Omega}\bar{v}=v$, and
employing density. $\Box$

For the same reason of $p(t)$ only being contained in the space
$H^{2}_{\Delta}(\Omega)$ but not necessarily in $H^{2}(\Omega)$, when taking
limits in the Galerkin approximation below, we will rely on the following
auxiliary result.

######  Lemma 3.3.

The operator $\Delta:H^{2}_{\Delta}(\Omega)\to L^{2}(\Omega)$ is weakly
closed.

###### Proof.

For any sequence $(p^{n})_{n\in\mathbb{N}}$ such that (a)
$p_{m}\rightharpoonup p$, (b) $\Delta p_{m}\rightharpoonup P$ in
$L^{2}(\Omega)$, we have, for any $\phi\in C_{0}^{\infty}(\Omega)$

$\displaystyle\int_{\Omega}\Delta
p^{n}\,\phi\,dx=\int_{\Omega}p^{n}\Delta\phi\,dx\to\int_{\Omega}p\Delta\phi\,dx\text{
as }n\to\infty\ \text{ due to (a)}$ $\displaystyle\int_{\Omega}\Delta
p^{n}\,\phi\,dx\to\int_{\Omega}P\phi\,dx\text{ as }n\to\infty\ \text{ due to
(b)}$

and therefore

$\text{for all }\phi\in C_{0}^{\infty}(\Omega)\,:\
\int_{\Omega}p\Delta\phi\,dx=\int_{\Omega}P\phi\,dx.$

Thus $\Delta p$ exists as a weak derivative and by uniqueness coincides with
$P$ a.e., which also implies that $\Delta p\in L^{2}(\Omega)$. $\Box$

###  3.2 Weak Formulation of the Linearized System

With the test and Ansatz space

$V=\\{(\phi,\psi)\in H^{2}(\Omega)\times
H^{2}(\Gamma_{pl})\,:\,\partial_{\nu}{\phi}=-\rho\psi\text{ on }\Gamma_{pl},\
\partial_{\nu}{\phi}=0\mbox{ on }\Gamma_{N}\\}$ (3.18)

a variational formulation of (3.12) can be given by

$\displaystyle(p(t),\widetilde{w}(t))\in V\text{ for a.e. }t\in(0,T)\text{ and
}\text{for all }(\phi,\psi)\in V\,:$ (3.19)
$\displaystyle\int_{\Omega}\Bigl{(}\nabla
p_{tt}\cdot\nabla\phi+\tfrac{c^{2}}{\alpha}\Delta
p\,\Delta\phi+\tfrac{b}{\alpha}\Delta
p_{t}\,\Delta\phi\Bigr{)}\,dx+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\Bigl{(}\rho\widetilde{w}_{tt}\psi+\delta\Delta_{pl}\widetilde{w}\,\Delta_{pl}\psi+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}\psi\Bigr{)}\,dS$
$\displaystyle+\int_{\Gamma_{a}}c\partial_{\nu}p_{t}\,\partial_{\nu}\phi\,dS\
=\int_{\Omega}\tfrac{f}{\alpha}\,(-\Delta\phi)\,dx+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\widetilde{h}\,\psi\,dS$
$\displaystyle(p,p_{t})(0)=(p_{0},p_{1}),\quad(\widetilde{w},\widetilde{w}_{t})(0)=(\widetilde{w}_{0},\widetilde{w}_{1})$

Indeed, we have the following equivalence

######  Lemma 3.4.

If $(p,\widetilde{w})\in\bigl{(}H^{2}(0,T;H^{1}(\Omega))\cap
H^{1}(0,T;H^{2}_{\Delta}(\Omega))\bigr{)}$
$\times\bigl{(}H^{2}(0,T;H^{2}(\Gamma_{pl})^{*})\cap
H^{1}(0,T;L^{2}(\Gamma_{pl}))\cap L^{\infty}(0,T;H^{2}(\Gamma_{pl}))\bigr{)}$
222Recall the notation $H^{2}_{\Delta}(\Omega):=\\{u\in
L^{2}(\Omega)\,:\,u,\,\Delta u\in L^{2}(\Omega)\\}\supset H^{2}(\Omega)$
solves (3.19) and the following compatibility conditions are satisfied:

$p_{1}+c\partial_{\nu}p_{0}=0\text{ on }\Gamma_{a},\quad p_{0}=0\text{ and
}p_{1}=0\text{ on }\Gamma_{D}$ (3.20)

Then $(p,\widetilde{w})$ solves (3.12) with $g_{D}=0$, $g_{N}=0$, $a=0$ (that
is, (3.16)) in an $L^{2}(0,T;L^{2}(\Omega))\times
L^{2}(0,T;H^{2}(\Gamma_{pl})^{*})$ sense.

If additionally the following compatibility conditions hold on $\Gamma_{pl}$

$\displaystyle\widetilde{w}_{0}=\kappa
p_{0}-\delta\Delta_{pl}^{2}w_{0}-\beta(-\Delta_{pl})^{\gamma}w_{1}+\widetilde{h}(0),\quad\widetilde{w}_{1}=\kappa
p_{1}-\delta\Delta_{pl}^{2}w_{1}-\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{0}+\widetilde{h}_{t}(0),$
(3.21)
$\displaystyle\partial_{\nu}p_{0}=-\rho\widetilde{w}_{0},\quad\partial_{\nu}p_{1}=-\rho\widetilde{w}_{1}$

then with
$w(t):=w_{0}+w_{1}t+\int_{0}^{t}\int_{0}^{s}\widetilde{w}(r)\,dr\,ds$, the
pair $(p,w)$ solves (3.11) and (2.2) in an $L^{2}(0,T;L^{2}(\Omega))\times
L^{2}(0,T;L^{2}(\Gamma_{pl}))$ sense with boundary conditions (2.4), (2.5),
and (2.6).

###### Proof.

Integrating by parts and then using $\partial_{\nu}\phi=-\rho\psi$ on
$\Gamma_{pl}$ we obtain from (3.19)

$\displaystyle 0=$
$\displaystyle\int_{\Omega}\Bigl{(}p_{tt}-\tfrac{c^{2}}{\alpha}\Delta
p-\tfrac{b}{\alpha}\Delta
p_{t}-\tfrac{f}{\alpha}\Bigr{)}(-\Delta\phi)\,dx+\int_{\partial\Omega}p_{tt}\partial_{\nu}\phi\,dS$
$\displaystyle+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\Bigl{(}\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}-\widetilde{h}\Bigr{)}\psi\,dS+\int_{\Gamma_{a}}c\partial_{\nu}p_{t}\,\partial_{\nu}\phi\,dS$
$\displaystyle=$
$\displaystyle\int_{\Omega}\Bigl{(}p_{tt}-\tfrac{c^{2}}{\alpha}\Delta
p-\tfrac{b}{\alpha}\Delta
p_{t}-\tfrac{f}{\alpha}\Bigr{)}(-\Delta\phi)\,dx+\int_{\Gamma_{a}}(c\partial_{\nu}p_{t}+p_{tt})\,\partial_{\nu}\phi\,dS+\int_{\Gamma_{D}}p_{tt}\partial_{\nu}\phi\,dS$
$\displaystyle+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\Bigl{(}\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}-\kappa
p_{tt}-\widetilde{h}\Bigr{)}\psi\,dS$

Now, if we choose test functions $(\phi,\psi)\in V$ such that $\psi=0$ and
$\partial_{\nu}{\phi}=0$ on $\partial\Omega$, we obtain

$\displaystyle 0=$
$\displaystyle\int_{\Omega}\Bigl{(}p_{tt}-\tfrac{c^{2}}{\alpha}\Delta
p-\tfrac{b}{\alpha}\Delta p_{t}-\tfrac{f}{\alpha}\Bigr{)}(-\Delta\phi)\,dx.$

Note that given any function $q\in L^{2}(\Omega)$, one can find a function
$\phi\in H^{2}(\Omega)$ such that $-\Delta\phi=q$ in $\Omega$ and
$\partial_{\nu}{\phi}=0$ on the boundary. Hence, we may conclude that

$\displaystyle 0=$
$\displaystyle\int_{\Omega}\Bigl{(}p_{tt}-\tfrac{c^{2}}{\alpha}\Delta
p-\tfrac{b}{\alpha}\Delta p_{t}-\tfrac{f}{\alpha}\Bigr{)}\,q\,dx$

holds for all $q\in L^{2}(\Omega)$, and thus (3.11) is satisfied in the
$L^{2}$ sense. On the other hand, if we choose the test functions
$(\phi,\psi)\in V$ such that $\Delta\phi=0$ in $\Omega$, with
$\partial_{\nu}{\phi}=\psi=0$ on $\Gamma_{pl}$, and $\partial_{\nu}{\phi}=0$
on $\Gamma_{a}$, $\partial_{\nu}{\phi}=u$ on $\Gamma_{D}$ or vice versa
$\partial_{\nu}{\phi}=0$ on $\Gamma_{D}$, $\partial_{\nu}{\phi}=u$ on
$\Gamma_{a}$ respectively for some arbitrary function $u\in
H^{1/2}(\Gamma_{D})$ or $u\in H^{1/2}(\Gamma_{a})$, we obtain

$\displaystyle\int_{\Gamma_{D}}p_{tt}\,u\,dS=0\mbox{ for all }u\in
H^{1/2}(\Gamma_{D})$

and

$\displaystyle\int_{\Gamma_{a}}(c\partial_{\nu}p_{t}+p_{tt})\,u\,dS=0\mbox{
for all }u\in H^{1/2}(\Gamma_{a}).$

Therefore, from the compatibility conditions (3.20), we conclude that (2.6)
and (2.5) hold in $H^{-1/2}$ on $\Gamma_{D}$ and $\Gamma_{a}$ with $g_{D}=0$,
$a=0$, respectively, .

To recover the plate equation (3.9), we select test functions $(\phi,\psi)\in
V$ with $\Delta\phi=0$ and $\partial_{\nu}{\phi}=0$ on $\Gamma_{a}$ and
$\Gamma_{D}$, so that we obtain

$\displaystyle\int_{\Gamma_{pl}}\Bigl{(}\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}-\kappa
p_{tt}-\widetilde{h}\Bigr{)}\psi\,dS=0$

for all $\psi\in H^{2}(\Gamma_{pl})$, and hence the pair
$(\widetilde{w},\widetilde{w}_{t})$ satisfies equation (3.9) in
$H^{2}(\Gamma_{pl})^{*}$.

If we integrate (3.9) and (3.10) twice in time, and utilize the additional
compatibility conditions (3.21), we obtain that $(w,w_{t})$ satisfy (2.2) and
(2.4).

$\Box$

###  3.3 Well-Posedness of the Linearized System

To prove well-posedness, as an intermediate result we will show existence of a
solution to the following weaker variational formulation

$\displaystyle(p,\widetilde{w})\in U\text{ and }\text{for all
}(q,\widetilde{v})\in C^{\infty}([0,T],V),\,(q(T),\widetilde{v}(T))=(0,0)\,:$
(3.22) $\displaystyle\int_{0}^{T}\Bigl{\\{}\int_{\Omega}\Bigl{(}-\nabla
p_{t}\cdot\nabla q_{t}+\tfrac{c^{2}}{\alpha}\Delta p\,\Delta
q+\tfrac{b}{\alpha}\Delta p_{t}\,\Delta q\Bigr{)}\,dx$
$\displaystyle\qquad+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\Bigl{(}-\rho\widetilde{w}_{t}\widetilde{v}_{t}+\delta\Delta_{pl}\widetilde{w}\,\Delta_{pl}\widetilde{v}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}\widetilde{v}\Bigr{)}\,dS+\int_{\Gamma_{a}}c\partial_{\nu}p_{t}\,\partial_{\nu}q\,dS\Bigr{\\}}\,dt$
$\displaystyle=\int_{0}^{T}\Bigl{\\{}\int_{\Omega}\tfrac{f}{\alpha}\,(-\Delta
q)\,dx+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\widetilde{h}\,\widetilde{v}\,dS+\int_{\Omega}\nabla
p_{1}\cdot\nabla
q(0)\,dx+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\rho\widetilde{w}_{1}\widetilde{v}(0)\,dS\Bigr{\\}}$
$\displaystyle p(0)=p_{0},\quad\widetilde{w}(0)=\widetilde{w}_{0}$

in the solution space

$U=(W^{1,\infty}(0,T;H^{1}(\Omega))\cap
H^{1}(0,T;H^{2}_{\Delta}(\Omega)))\times(W^{1,\infty}(0,T;L^{2}(\Gamma_{pl}))\cap
L^{\infty}(0,T;H^{2}(\Gamma_{pl})))$ (3.23)

induced by the energy

$\displaystyle\mathcal{E}[p,\widetilde{w}](t):=$
$\displaystyle\mathcal{E}^{p}[p](t)+\mathcal{E}^{w}[\widetilde{w}](t)\text{
where }$ (3.24) $\displaystyle\mathcal{E}^{p}[p](t):=$
$\displaystyle\|p_{t}(t)\|_{H^{1}(\Omega)}^{2}+\|\Delta
p(t)\|_{L^{2}(\Omega)}^{2}+b\|\Delta
p_{t}\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\|\partial_{\nu}p_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}$
$\displaystyle\mathcal{E}^{w}[\widetilde{w}](t):=$
$\displaystyle\|\widetilde{w}_{t}(t)\|_{L^{2}(\Gamma_{pl})}^{2}+\|\Delta_{pl}\widetilde{w}(t)\|_{L^{2}(\Gamma_{pl})}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{pl}))}^{2}$

where we track dependency on the damping parameters $b$, $\beta$. Later on,
also the higher energy

$\mathcal{E}_{1}[p,\widetilde{w}]:=\mathcal{E}[p,\widetilde{w}]+\mathcal{E}[p_{t},\widetilde{w}_{t}]$
(3.25)

will be used.

Accordingly, we define the energy of the initial data

$\displaystyle\mathcal{E}[p,\widetilde{w}](0):={\|p_{1}\|_{H^{1}(\Omega)}^{2}}+\|\Delta
p_{0}\|_{L^{2}(\Omega)}^{2}+\|\widetilde{w}_{1}\|_{L^{2}(\Gamma_{pl})}^{2}+\|\Delta_{pl}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}$
(3.26)

and the norms of the boundary/interior sources

$\displaystyle
F[f,\widetilde{h}]:=\|f\|_{L^{1}(0,T;L^{2}(\Omega))}^{2}+\|\widetilde{h}\|_{L^{1}(0,T;L^{2}(\Gamma_{pl}))}^{2}$
(3.27)

######  Proposition 3.5.

  1. (i)

Suppose for some arbitrary $T\in(0,\infty]$, that $\alpha$, $\frac{1}{\alpha}$
$\in L^{\infty}(0,T;L^{\infty}(\Omega))$, $\alpha_{t}~{}\in
L^{1}(0,T;L^{\infty}(\Omega))$ with
$\bar{\alpha}:=\|\frac{\alpha_{t}}{\alpha}\|_{L^{1}(0,T;L^{\infty}(\Omega))}<1/4$,
$f\in L^{1}(0,T;L^{2}(\Omega))$, $\widetilde{h}\in
L^{1}(0,T;L^{2}(\Gamma_{pl}))$, $\nabla p_{1},\Delta p_{0}\in L^{2}(\Omega)$,
and $\widetilde{w}_{1},\Delta_{pl}\widetilde{w}_{0}\in L^{2}(\Gamma_{pl})$. If
$\Gamma_{D}=\emptyset$ we additionally assume $\nabla\alpha\in
L^{\infty}(0,T;L^{3}(\Omega))$ with
$\widetilde{\alpha}_{2}=\|\alpha^{-3/2}\nabla\alpha\|_{L^{2}(0,t;L^{3}(\Omega))}$
and
$\widetilde{\alpha}_{\infty}=\|\alpha^{-3/2}\nabla\alpha\|_{L^{\infty}(0,t;L^{3}(\Omega))}$
small enough.

Then there exists a solution $(p,\widetilde{w})$ to (3.22) in $U$ defined in
(3.23), which for any $t\in(0,T)$ satisfies the energy estimate

$\mathcal{E}[p,\widetilde{w}](t)\leq
C\Bigl{(}\mathcal{E}[p,\widetilde{w}](0)+F[f,\widetilde{h}]\Bigr{)}$ (3.28)

  2. (ii)

Moreover,

$\displaystyle\mathcal{E}[p,\widetilde{w}](t)+\int_{0}^{t}\mathcal{E}[p,\widetilde{w}](s)\,ds$
(3.29) $\displaystyle\leq C\Bigl{(}\mathcal{E}[p,\widetilde{w}](0)+b\|\Delta
p_{0}\|_{L^{2}(\Omega)}^{2}+\|\partial_{\nu}{p_{0}}\|_{L^{2}(\Gamma_{a})}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}+F[f,\widetilde{h}]\Bigr{)}$

  3. (iii)

If, in addition to (i), $f_{t}\in L^{1}(0,T;L^{2}(\Omega))$,
$\widetilde{h}_{t}\in L^{1}(0,T;L^{2}(\Gamma_{pl}))$, $\nabla p_{2}$, $\Delta
p_{1}$ $\in L^{2}(\Omega)$, and
$\widetilde{w}_{2},\Delta_{pl}\widetilde{w}_{1}$ $\in L^{2}(\Gamma_{pl})$,
$\partial_{\nu}{p_{2}}=-\rho\widetilde{w}_{2}$ on $\Gamma_{pl}$, where
$p_{2}:=\tfrac{1}{\alpha(0)}\Bigl{(}c^{2}\Delta p_{0}+b\Delta
p_{1}+f(0)\Bigr{)}$,
$\widetilde{w}_{2}\>=\frac{1}{\rho}\Bigl{(}-\delta\Delta_{pl}^{2}\widetilde{w}_{0}-\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{1}+\kappa
p_{2}+\widetilde{h}(0)\Bigr{)}$ then $(p,\widetilde{w})$ satisfies the energy
estimate

$\mathcal{E}_{1}[p,\widetilde{w}](t)\leq
C\Bigl{(}\mathcal{E}_{1}[p,\widetilde{w}](0)+F[f,\widetilde{h}]+F[f_{t},\widetilde{h}_{t}]\Bigr{)}$
(3.30)

for
$\mathcal{E}_{1}[p,\widetilde{w}](t):=\mathcal{E}[p,\widetilde{w}](t)+\mathcal{E}[p_{t},\widetilde{w}_{t}](t)$.
The solution $(p,\widetilde{w})$ to (3.16) exists and is unique in

$U^{\prime}:=\\{(p,\widetilde{w})\in U\,:\,(p_{t},\widetilde{w}_{t})\in
U\\}\text{ with $U$ defined in \eqref{U}.}$ (3.31)

  4. (iv)

Moreover, under the conditions (i), (ii), (iii),

$\displaystyle\mathcal{E}_{1}[p,\widetilde{w}](t)+\int_{0}^{t}\mathcal{E}_{1}[p,\widetilde{w}](s)\,ds$
(3.32) $\displaystyle\leq
C\Bigl{(}\mathcal{E}_{1}[p,\widetilde{w}](0)+b\|\Delta
p_{0}\|_{L^{2}(\Omega)}^{2}+b\|\Delta
p_{1}\|_{L^{2}(\Omega)}^{2}+\|\partial_{\nu}{p_{0}}\|_{L^{2}(\Gamma_{a})}^{2}+\|\partial_{\nu}{p_{1}}\|_{L^{2}(\Gamma_{a})}^{2}$
$\displaystyle\qquad\qquad+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{1}\|_{L^{2}(\Gamma_{pl})}^{2}+F[f,\widetilde{h}]+F[f_{t},\widetilde{h}_{t}]\Bigr{)}$

The constants $C$ in (i), (ii), (iii) depend on the coefficients $b$, $c$,
$\rho$, $\delta$, $\kappa$,
$\|\frac{1}{\alpha}\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}$,
$\|\alpha_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$, but not directly on $T$.

###### Proof.

Step 1 Galerkin approximation: (note that this is still done on (3.19) rather
than (3.22))
We use an $L^{2}$ orthonormal eigensystem
$(\phi_{j},\lambda_{j})_{j\in\mathbb{N}}$ of the negative Laplacian on
$\Omega$ with homogeneous Neumann values on $\Gamma_{pl}\cup\Gamma_{N}$ and
homogeneous Dirichlet values on $\Gamma_{D}$

$-\Delta\phi_{j}=\lambda_{j}\phi_{j}\text{ in
}\Omega,\qquad\partial_{\nu}{\phi}_{j}=0\text{ on
}\Gamma_{pl}\cup\Gamma_{N}\,,\qquad\phi=0\text{ on }~{}\Gamma_{D}$

and an $L^{2}_{\rho}$ orthonormal eigensystem
$(\psi_{i},\mu_{i})_{i\in\mathbb{N}}$ of the negative Laplace Beltrami
operator with Dirichlet boundary conditions on $\Gamma_{pl}$, which we
harmonically extend to functions $\widetilde{\phi}_{i}$ on $\Omega$ such that

$-\Delta\widetilde{\phi}_{i}=0\text{ in
}\Omega,\quad\partial_{\nu}{\widetilde{\phi}}_{i}=-\rho\psi_{i}\text{ on
}\Gamma_{pl}\,,\quad\partial_{\nu}{\widetilde{\phi}}_{i}=0\text{ on
}\Gamma_{a}\cup\Gamma_{N}\,,\quad\widetilde{\phi}_{i}=0\text{ on
}~{}\Gamma_{D}.$

Our Galerkin spaces are then defined by
$V^{n}:=\mbox{span}((\phi_{1},0),\ldots,(\phi_{n},0),\,(\widetilde{\phi}_{1},\psi_{1}),\ldots,(\widetilde{\phi}_{n},\psi_{n}))$.
For any fixed $n\in\mathbb{N}$ we make the ansatz

$p^{n}:=\sum_{j=1}^{n}p^{n}_{j}(t)\phi_{j}+\sum_{i=1}^{n}\widetilde{w}_{i}^{n}(t)\widetilde{\phi}_{i},\qquad\widetilde{w}^{n}:=\sum_{i=1}^{n}\widetilde{w}_{i}^{n}(t)\psi_{i}$

so that in particular $\partial_{\nu}{p}^{n}=-\rho\widetilde{w}^{n}$ holds on
$\Gamma_{pl}$ and due to smoothness of the involved eigenfunctions,
$(p^{n},\widetilde{w}^{n})\,\in V$. Inserting into (3.19) and testing with
$(\phi_{\ell},0)$, $(\widetilde{\phi}_{k},\psi_{k})$, we end up with the
following ODE system for the coefficient vector functions $\underline{p}^{n}$,
$\underline{\widetilde{w}}^{n}$:

$\displaystyle\text{diag}(\lambda_{1},\ldots,\lambda_{n})(\underline{p}^{n})^{\prime\prime}+bM^{n}(t)(\underline{p}^{n})^{\prime}+c^{2}M^{n}(t)(\underline{p}^{n})+D^{n}(\underline{\widetilde{w}}^{n})^{\prime\prime}+C^{n}(\underline{\widetilde{w}}^{n})^{\prime}(t)=f^{p,n}$
(3.33)
$\displaystyle\rho(\underline{\widetilde{w}}^{n})^{\prime\prime}+\beta\text{diag}(\mu_{1}^{\gamma},\ldots,\mu_{n}^{\gamma})(\underline{\widetilde{w}}^{n})^{\prime}+\delta\text{diag}(\mu_{1}^{2},\ldots,\mu_{n}^{2})(\underline{\widetilde{w}}^{n})=f^{\widetilde{w},n}$
(3.34)

where

$\displaystyle
M^{n}_{j,\ell}(t)=\lambda_{j}\lambda_{\ell}\int_{\Omega}\tfrac{1}{\alpha}\phi_{j}\,\phi_{\ell}\,dx+c\int_{\Gamma_{a}}\partial_{\nu}{\phi}_{j}\,\partial_{\nu}{\phi}_{\ell}\,dS\,,\quad$
$\displaystyle
C^{n}_{i,\ell}=c\int_{\Gamma_{a}}\partial_{\nu}{\widetilde{\phi}}_{i}\,\partial_{\nu}{\phi}_{\ell}\,dS\,,\quad
D^{n}_{i,\ell}=\rho\int_{\Gamma_{pl}}\psi_{i}\,\phi_{\ell}\,dS\,,$
$\displaystyle
f^{p,n}_{\ell}(t)=\lambda_{\ell}\int_{\Omega}\tfrac{f}{\alpha}\phi_{\ell}\,dx,\qquad
f^{\widetilde{w},n}_{k}(t)=\int_{\Gamma_{pl}}\rho\,\widetilde{h}\,\psi_{k}\,dS,$

due to the fact that

$\int_{\Omega}\nabla\widetilde{\phi}_{i}\,\nabla\phi_{\ell}\,dx=\int_{\Gamma_{pl}}\partial_{\nu}{\widetilde{\phi}_{i}}\phi_{\ell}\,dS=-\rho\int_{\Gamma_{pl}}\psi_{i}\,\phi_{\ell}\,dS,\quad\int_{\Omega}\tfrac{b}{\alpha}\Delta\widetilde{\phi}_{i}\,\Delta\phi_{\ell}\,dx=0,\quad\int_{\Omega}\tfrac{c^{2}}{\alpha}\Delta\widetilde{\phi}_{i}\,\Delta\phi_{\ell}\,dx=0.$

Note that $\alpha$ depends on space and time so $M^{n}$ is in general not a
diagonal matrix but clearly positive definite. Existence of a solution (3.33),
(3.34) on $(0,T)$ follows from standard ODE results and the fact that this ODE
system is only triangularly coupled so that (3.34) can be solved for
$\underline{\widetilde{w}}^{n}$ independently of (3.33) and then
$\underline{\widetilde{w}}^{n}$ inserted into (3.33).

Step 2 energy estimates:
(We here aim for $T$ independence of estimates in view of our goal of proving
global in time well-posednenss.)
Replacing $\phi$ with $p^{n}_{t}$ and $\psi$ with
$\frac{\kappa}{\rho}\widetilde{w}^{n}_{t}$ in (3.19), abbreviating
$\|\cdot\|=\|\cdot\|_{L^{2}(\Omega)}$ and using the identity

$\frac{1}{2}\frac{d}{dt}\int_{\Omega}\frac{1}{\alpha}v^{2}\,dx=\int_{\Omega}\Bigl{(}\frac{1}{\alpha}v\,v_{t}-\frac{1}{2}\frac{\alpha_{t}}{\alpha^{2}}v^{2}\Bigr{)}\,dx$
(3.35)

we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla
p^{n}_{t}\|^{2}+\frac{1}{2}c^{2}\frac{d}{dt}\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|^{2}+b\,\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}_{t}\right\|^{2}+c\,\|\partial_{\nu}p^{n}_{t}\|_{L^{2}(\Gamma_{a})}^{2}$
(3.36)
$\displaystyle+\frac{1}{2}\rho\frac{d}{dt}\|\widetilde{w}^{n}_{t}\|_{L^{2}(\Gamma_{pl})}^{2}+\frac{1}{2}\delta\frac{d}{dt}\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{2}(\Gamma_{pl})}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}^{n}\|_{L^{2}(\Gamma_{pl})}^{2}$
$\displaystyle=\int_{\Omega}\Bigl{(}-\frac{c^{2}}{2}\frac{\alpha_{t}}{\alpha^{2}}(\Delta
p^{n})^{2}-\frac{f}{\alpha}\Delta
p^{n}_{t}\Bigr{)}\,dx+\int_{\Gamma_{pl}}\widetilde{h}\,\widetilde{w}^{n}_{t}\,dS$
$\displaystyle\leq\frac{c^{2}}{2}\left\|\frac{\alpha_{t}}{\alpha}\right\|_{L^{\infty}(\Omega)}\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|^{2}+\frac{1}{2b}\left\|\frac{1}{\sqrt{\alpha}}f\right\|^{2}+\frac{b}{2}\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}_{t}\right\|^{2}$
$\displaystyle\qquad+\|\widetilde{h}\|_{L^{2}(\Gamma_{pl})}\,\|\widetilde{w}^{n}_{t}\|_{L^{2}(\Gamma_{pl})}$

After integrating with respect to time and taking the supremum with respect to
$t$ on both sides of (3.36) we obtain

$\displaystyle\|\nabla
p^{n}_{t}\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+c^{2}(1-\bar{\alpha})\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+b\,\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}_{t}\right\|_{L^{2}(0,T;L^{2}(\Omega))}^{2}+2c\,\|\partial_{\nu}p^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Gamma_{a}))}^{2}$
(3.37)
$\displaystyle+\frac{\rho}{2}\|\widetilde{w}^{n}_{t}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}^{2}+\delta\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{t}\|_{L^{2}(0,T;L^{2}(\Gamma_{pl}))}^{2}$
$\displaystyle\leq\|\nabla
p_{1}\|^{2}+c^{2}\left\|\frac{1}{\sqrt{\alpha(0)}}\Delta
p_{0}\right\|^{2}+\rho\|\widetilde{w}_{1}\|_{L^{2}(\Gamma_{pl})}^{2}+\delta\|\Delta_{pl}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}$
$\displaystyle\qquad+\frac{1}{b}\left\|\frac{1}{\sqrt{\alpha}}f\right\|_{L^{1}(0,T;L^{2}(\Omega))}^{2}+\frac{2}{\rho}\|\widetilde{h}\|_{L^{1}(0,T;L^{2}(\Gamma_{pl}))}^{2}$

where we have used Young’s inequality in
$\int_{0}^{t}\|\widetilde{h}\|_{L^{2}(\Gamma_{pl})}\,\|\widetilde{w}^{n}_{t}\|_{L^{2}(\Gamma_{pl})}\,ds\leq\|\widetilde{h}\|_{L^{1}(0,T;L^{2}(\Gamma_{pl}))}\|\widetilde{w}^{n}_{t}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}\leq\frac{1}{\rho}\|\widetilde{h}\|_{L^{1}(0,T;L^{2}(\Gamma_{pl}))}^{2}+\frac{\rho}{4}\|\widetilde{w}^{n}_{t}\|^{2}_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}$.

In case of $\Gamma_{D}\not=\emptyset$, this together with the Poincaré-
Friedrichs inequality (1.1) already yields the energy estimate (3.28) for
$(p^{n},\widetilde{w}^{n})$ in place of $(p,\widetilde{w})$. Otherwise we need
to bound the $L^{2}(\Omega)$ norm of $p^{n}_{t}(t)$ separately in order to
arrive at its full $H^{1}(\Omega)$ norm.

Step 2.b control of $L^{2}(\Omega)$ norm in case $\Gamma_{D}=\emptyset$:

Taking $(\phi,\psi)\in V$ in (3.19) such that $-\Delta\phi=p^{n}_{t}$,
$\partial_{\nu}{\phi}=\begin{cases}0&\text{if}~{}~{}B_{a}p^{n}_{t}=0\\\
-\frac{B_{a}p^{n}_{t}}{|B_{a}p^{n}_{t}|^{2}}\frac{1}{\alpha}(c^{2}B_{a}p^{n}+bB_{a}p^{n}_{t})&\text{
otherwise }\end{cases}$ with $B_{a}v:=c\partial_{\nu}{v}+v_{t}$,
$\partial_{\nu}{\phi}=0$ on $\Gamma_{N}\cup\Gamma_{pl}$ and accordingly
$\psi=0$, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|p^{n}_{t}\|^{2}+\frac{1}{2}c^{2}\Bigl{(}\frac{d}{dt}\left\|\frac{1}{\sqrt{\alpha}}\nabla
p^{n}\right\|^{2}+\int_{\Omega}\frac{\alpha_{t}}{\alpha^{2}}|\nabla
p^{n}|^{2}\,dx\Bigr{)}+b\,\left\|\frac{1}{\sqrt{\alpha}}\nabla
p^{n}_{t}\right\|^{2}$
$\displaystyle=\int_{\Omega}\Bigl{(}\frac{f}{\alpha}p^{n}_{t}+\frac{1}{\alpha^{2}}p^{n}_{t}\nabla\alpha\cdot(c^{2}\nabla
p^{n}+b\nabla
p^{n}_{t})\Bigr{)}\,dx-\int_{\partial\Omega}\Bigl{(}B_{a}p^{n}_{t}\,\partial_{\nu}{\phi}-(c^{2}\partial_{\nu}{p^{n}}+b\partial_{\nu}{p^{n}_{t}})\frac{1}{\alpha}p^{n}_{t}\Bigr{)}\,dS,$

where

$\int_{\Gamma_{a}}\Bigl{(}B_{a}p^{n}_{t}\,\partial_{\nu}{\phi}-(c^{2}\partial_{\nu}{p^{n}}+b\partial_{\nu}{p^{n}_{t}})\frac{1}{\alpha}p^{n}_{t}\Bigr{)}\,dS=c\int_{\Gamma_{a}}\frac{1}{\alpha}p_{t}^{2}\,dS+\frac{b}{c}\int_{\Gamma_{a}}\frac{1}{\alpha}\frac{d}{dt}[p_{t}^{2}]\,dS.$

Integrating in time yields

$\displaystyle\frac{1}{2}\|p^{n}_{t}(t)\|^{2}+\frac{c^{2}}{2}\Bigl{(}\left\|\frac{1}{\sqrt{\alpha}}\nabla
p^{n}(t)\right\|^{2}+\int_{0}^{t}\int_{\Omega}\frac{\alpha_{t}}{\alpha^{2}}|\nabla
p^{n}|^{2}\,dx\,ds\Bigr{)}+\int_{0}^{t}b\left\|\frac{1}{\sqrt{\alpha}}\nabla
p^{n}_{t}\right\|^{2}\,dx\,ds$ (3.38)
$\displaystyle+c\int_{0}^{t}\left\|\frac{1}{\sqrt{\alpha}}p^{n}_{t}\right\|^{2}_{L^{2}(\Gamma_{a})}\,ds+\frac{b}{c}\Bigl{(}\left\|\frac{1}{\sqrt{\alpha}}p^{n}_{t}(t)\right\|^{2}_{L^{2}(\Gamma_{a})}+\int_{0}^{t}\int_{\Gamma_{a}}\frac{\alpha_{t}}{\alpha^{2}}|p^{n}|^{2}\,ds\Bigr{)}$
$\displaystyle=\frac{1}{2}\|p^{n}_{1}\|^{2}+\frac{c^{2}}{2}\left\|\frac{1}{\sqrt{\alpha}}\nabla
p^{n}_{0}\right\|+\frac{b}{c}\left\|\frac{1}{\sqrt{\alpha}}p^{n}_{1}\right\|^{2}_{L^{2}(\Gamma_{a})}+\underbrace{\int_{0}^{t}\int_{\Omega}\Bigl{(}\frac{f}{\alpha}p^{n}_{t}+\frac{1}{\alpha^{2}}p^{n}_{t}\nabla\alpha\cdot(c^{2}\nabla
p^{n}+b\nabla p^{n}_{t})\Bigr{)}\,dx\,ds}_{(0)}$
$\displaystyle\quad-c^{2}\rho\underbrace{{\int_{0}^{t}}\int_{\Gamma_{pl}}\frac{1}{\alpha}{\widetilde{w}^{n}}p^{n}_{t}\,dS\,ds}_{(I)}-b\rho\underbrace{{\int_{0}^{t}}\int_{\Gamma_{pl}}\frac{1}{\alpha}{\widetilde{w}^{n}_{t}}p^{n}_{t}\,dS\,ds}_{(II)}.$

We estimate

$\displaystyle(O)\leq$
$\displaystyle\|\tfrac{f}{\alpha}\|_{L^{1}(0,t;L^{2}(\Omega))}\|p^{n}_{t}\|_{L^{\infty}(0,t;L^{2}(\Omega))}+c^{2}\|\tfrac{1}{\sqrt{\alpha}}\nabla
p^{n}\|_{L^{\infty}(0,t;L^{2}(\Omega))}\|p^{n}_{t}\|_{L^{2}(0,t;L^{6}(\Omega))}\|\alpha^{-3/2}\nabla\alpha\|_{L^{2}(0,t;L^{3}(\Omega))}$
$\displaystyle+b\|\tfrac{1}{\sqrt{\alpha}}\nabla
p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Omega))}\|p^{n}_{t}\|_{L^{2}(0,t;L^{6}(\Omega))}\|\alpha^{-3/2}\nabla\alpha\|_{L^{\infty}(0,t;L^{3}(\Omega))}$

where due to (1.1)

$\|p^{n}_{t}\|_{L^{2}(0,t;L^{6}(\Omega))}\leq
C_{H^{1},L^{6}}^{\Omega}\|p^{n}_{t}\|_{L^{2}(0,t;H^{1}(\Omega))}\leq
C_{H^{1},L^{6}}^{\Omega}C^{\Omega}_{\Gamma_{a}}\Bigl{(}\|\nabla
p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\|p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}\Bigr{)}^{1/2}\,.$

The terms (I) and (II) can be estimated by

$\displaystyle(I)\leq$
$\displaystyle\frac{c^{2}\rho}{\underline{\alpha}}\|\widetilde{w}^{n}\|_{L^{2}(0,t;H^{1/2}(\Gamma_{pl}))}\|p^{n}_{t}\|_{L^{2}(0,t;H^{-1/2}(\Gamma_{pl}))}$
$\displaystyle(II)\leq$
$\displaystyle\frac{b\rho}{\underline{\alpha}}\|\widetilde{w}^{n}_{t}\|_{L^{2}(0,t;H^{1/2}(\Gamma_{pl}))}\|p^{n}_{t}\|_{L^{2}(0,t;H^{-1/2}(\Gamma_{pl}))}$

where the terms containing ${\widetilde{w}^{n}}$ can be estimated by the 8th
and 9th terms in (3.44) and

$\displaystyle\|p^{n}_{t}\|_{L^{2}(0,t;H^{-1/2}(\Gamma_{pl}))}$
$\displaystyle\leq
C^{\Gamma_{pl}}_{H^{1/2},H^{-1/2}}C^{tr}_{1/2}\|p^{n}_{t}\|_{L^{2}(0,t;H^{1}(\Omega){)}}$
$\displaystyle\leq
C^{\Gamma_{pl}}_{H^{1/2},H^{-1/2}}C^{tr}_{1/2}C^{\Omega}_{\Gamma_{a}}\Bigl{(}\|\nabla
p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\|p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}\Bigr{)}^{1/2}\,.$

After using Young’s inequality, this provides us with the lower order energy
estimate

$\displaystyle\frac{1}{2}\|p^{n}_{t}\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+c^{2}(1-\bar{\alpha}-\widetilde{\alpha}_{2})\left\|\frac{1}{\sqrt{\alpha}}\nabla
p^{n}\right\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}$ (3.39) $\displaystyle\
+\Bigl{(}\frac{\min\\{b(1-\widetilde{\alpha}_{\infty}),c\\}}{\|\alpha\|_{L^{\infty}(\Omega)}(C^{\Omega}_{\Gamma_{a}})^{2}}-(c^{2}\widetilde{\alpha}_{2}+b\widetilde{\alpha}_{\infty})(C^{\Omega}_{H^{1},L^{6}})^{2}\Bigr{)}\|p^{n}_{t}\|_{L^{2}(0,T;H^{1}(\Omega))}^{2}+\tfrac{b}{c}(1-\bar{\alpha})\left\|\frac{1}{\sqrt{\alpha}}p^{n}_{t}\right\|_{L^{2}(0,T;L^{2}(\Gamma_{a}))}^{2}$
$\displaystyle\leq\|p_{1}\|^{2}+c^{2}\left\|\frac{1}{\sqrt{\alpha(0)}}\nabla
p_{0}\right\|^{2}+\frac{1}{2}\left\|\frac{1}{\sqrt{\alpha}}f\right\|_{L^{1}(0,T;L^{2}(\Omega))}^{2}+\widetilde{C}(\mathcal{E}[p,\widetilde{w}](0)+F[f,\widetilde{h}])$

where
$\widetilde{\alpha}_{2}=\|\alpha^{-3/2}\nabla\alpha\|_{L^{2}(0,t;L^{3}(\Omega))}$,
$\widetilde{\alpha}_{\infty}=\|\alpha^{-3/2}\nabla\alpha\|_{L^{\infty}(0,t;L^{3}(\Omega))}$.

Step 2.c equipartition of energy: To show (ii), we substitute $\phi=p^{n}$ and
$\psi=\frac{\kappa}{\rho}\widetilde{w}^{n}$ in (3.19) and use (3.35), which
yields the equipartition of energy identity

$\displaystyle-\|\nabla p^{n}_{t}\|^{2}+\frac{d}{dt}\int_{\Omega}\nabla
p^{n}_{t}\cdot\nabla p^{n}\,dx+c^{2}\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|^{2}+\frac{1}{2}b\,\frac{d}{dt}\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|^{2}+\frac{1}{2}c\,\frac{d}{dt}\|\partial_{\nu}p^{n}\|_{L^{2}(\Gamma_{a})}^{2}$
$\displaystyle-\rho\|\widetilde{w}^{n}_{t}\|_{L^{2}(\Gamma_{pl})}^{2}+\rho\frac{d}{dt}\int_{\Gamma_{pl}}\widetilde{w}^{n}_{t}\widetilde{w}^{n}\,dS+\delta\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{2}(\Gamma_{pl})}^{2}+\frac{1}{2}\beta\frac{d}{dt}\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}\|_{L^{2}(\Gamma_{pl})}^{2}$
$\displaystyle=\int_{\Omega}\Bigl{(}-b\frac{\alpha_{t}}{{2}\alpha^{2}}(\Delta
p^{n})^{2}-\frac{f}{\alpha}\Delta
p^{n}\Bigr{)}\,dx+\int_{\Gamma_{pl}}\,\widetilde{h}\,\widetilde{w}^{n}\,dS$

and, similarly to above, using
$\int_{\Gamma_{pl}}\,\widetilde{h}\,\widetilde{w}^{n}\,dS$
$\leq\|\widetilde{h}\|_{H^{2}(\Gamma_{pl})^{*}}\|\widetilde{w}^{n}\|_{H^{2}(\Gamma_{pl})}$
$\leq\|\widetilde{h}\|_{H^{2}(\Gamma_{pl})^{*}}\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{2}(\Gamma_{pl})}\|(-\Delta_{pl})^{-1}\|_{L^{2}(\Omega)\to
H^{2}(\Omega)}$, the estimate

$\displaystyle c^{2}\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|_{L^{2}(0,T;L^{2}(\Omega))}^{2}+\frac{b}{4}(1-2\bar{\alpha})\,\left\|\frac{1}{\sqrt{\alpha}}\Delta
p^{n}\right\|_{L^{\infty}(0,T;L^{2}(\Omega))}^{2}+\frac{c}{2}\,\|\partial_{\nu}p^{n}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{a}))}^{2}$
(3.40)
$\displaystyle+\frac{\delta}{2}\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{2}(0,T;L^{2}(\Gamma_{pl}))}^{2}+\frac{1}{2}\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}^{n}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}^{2}$
$\displaystyle\leq\underbrace{\|\nabla
p^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Omega))}^{2}}_{{I}}+\int_{\Omega}\nabla
p^{n}_{1}\cdot\nabla p^{n}_{0}\,dx+\sup_{t\in(0,T)}\int_{\Omega}-\nabla
p^{n}_{t}\cdot\nabla p^{n}\,dx$
$\displaystyle+\underbrace{\rho\|\widetilde{w}^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Gamma_{pl}))}^{2}}_{{II}}+\rho\int_{\Gamma_{pl}}\widetilde{w}^{n}_{1}\,\widetilde{w}^{n}_{0}\,dS+\rho\sup_{t\in(0,T)}\int_{\Gamma_{pl}}-\widetilde{w}^{n}_{t}\,\widetilde{w}^{n}\,dS$
$\displaystyle+\frac{b}{2}\,\left\|\frac{1}{\sqrt{\alpha(0)}}\Delta
p_{0}\right\|^{2}+\frac{c}{2}\,\|\partial_{\nu}p_{0}\|_{L^{2}(\Gamma_{a})}^{2}+\frac{1}{2}\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}$
$\displaystyle\qquad+\frac{1}{b}\left\|\frac{1}{\sqrt{\alpha}}f\right\|_{L^{1}(0,T;L^{2}(\Omega))}^{2}+\frac{1}{2\delta\rho^{2}}\|(-\Delta_{pl})^{-1}\|_{L^{2}(\Omega)\to
H^{2}(\Omega)}\||\widetilde{h}\||_{L^{2}(0,T;H^{2}(\Gamma_{pl})^{*})}^{2}.$

In case $\Gamma_{D}\neq\emptyset$, we may estimate the underlined term $I$ by
the following

$\displaystyle\|\nabla p^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Omega)}\leq
C_{\Delta}(\|\Delta
p^{n}_{t}\|_{L^{2}(0,T;H^{-1}(\Omega))}+\|\partial_{\nu}p^{n}_{t}\|_{L^{2}(0,T;H^{-1/2}(\Gamma_{a}\cup\Gamma_{N}\cup\Gamma_{pl})})$
(3.41) $\displaystyle\leq C_{\Delta}(C^{\Omega}_{L^{2},H^{-1}}\|\Delta
p^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Omega)}+\|\partial_{\nu}p^{n}_{t}\|_{L^{2}(0,T;H^{-1/2}(\Gamma_{a})}+\rho\|\widetilde{w}^{n}_{t}\|_{L^{2}(0,T;H^{-1/2}(\Gamma_{a})}),$

which together with (3.10) allows us to bound I by the damping terms on the
left hand side of (3.37), that is, the third, fourth, and seventh terms.
Estmiate (3.41) follows from elliptic estimates applied to the Laplace
equation with mixed Neumann Dirichlet boundary conditions

$\displaystyle{-}\Delta\phi$ $\displaystyle=f\text{ in }\Omega,$
$\displaystyle\partial_{\nu}\phi$ $\displaystyle=g\text{ on
}\Gamma_{a}\cup\Gamma_{N}\cup\Gamma_{pl},$ $\displaystyle\phi$
$\displaystyle=0\text{ on }\Gamma_{D},$

whose unique solution by the Lax-Milgram Theorem satisfies the inequality

$\|\phi\|_{H^{1}}\leq{C_{\Delta}}(\|f\|_{H^{-1}}+\|g\|_{H^{-1/2}(\Gamma_{N}\cup\Gamma_{pl}\cup\Gamma_{a})})$
(3.42)

see, e.g., [31, p. 125].

Note that in case $\Gamma_{D}=\emptyset$, considering instead

$\displaystyle-\Delta\phi+\phi$ $\displaystyle=f+\phi\text{ in }\Omega,$
$\displaystyle\partial_{\nu}\phi$ $\displaystyle=g\text{ on
}\Gamma_{a}\cup\Gamma_{N}\cup\Gamma_{pl},$

we only have

$\displaystyle\|\nabla p^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Omega){)}}$ (3.43)
$\displaystyle\leq C_{\Delta}(C^{\Omega}_{L^{2},H^{-1}}\|\Delta
p^{n}_{t}\|_{L^{2}(0,T;L^{2}(\Omega)}+\|\partial_{\nu}p^{n}_{t}\|_{L^{2}(0,T;H^{-1/2}(\Gamma_{a}\cup\Gamma_{N}\cup\Gamma_{pl})}+\|p_{t}\|_{L^{2}(0,T;H^{-1}(\Omega))})$

which requires an additional estimate to control the $L^{2}$ norm of $p_{t}$.
For this purpose we use the third term in (3.39).

The term II in (3.40) is bounded by the plate damping term with coefficient
$\beta$ in (3.37).

We recall that $\Delta_{pl}$ is the Laplace-Beltrami operator equipped with
homogeneous Dirichlet boundary conditions, cf. (2.3), which by elliptic
regularity allows us to recover the full $H^{2}$ norm of $\widetilde{w}^{n}$.

Combining (3.40) with (3.37) and the control of the $L^{2}$ norm by either
Poincaré-Friedrichs or Step 2.b yields the energy estimate

$\displaystyle\|p^{n}_{t}\|_{L^{\infty}(0,t;H^{1}(\Omega))}^{2}+(1+b)\|\Delta
p^{n}\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+b\|\Delta
p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\|\Delta
p^{n}\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}$ (3.44)
$\displaystyle+\|\partial_{\nu}p^{n}\|_{L^{\infty}(0,t;L^{2}(\Gamma_{a}))}^{2}+\|\partial_{\nu}p^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}$
$\displaystyle+\|\widetilde{w}^{n}_{t}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}^{2}+\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{\infty}(0,T;L^{2}(\Gamma_{pl}))}^{2}+{\|\Delta_{pl}\widetilde{w}^{n}\|_{L^{2}(0,T;L^{2}(\Gamma_{pl}))}^{2}}$
$\displaystyle{}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}^{n}_{t}\|_{L^{2}(0,t;L^{2}(\Gamma_{pl}))}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}^{n}\|_{L^{\infty}(0,t;L^{2}(\Gamma_{pl}))}^{2}$
$\displaystyle\leq C\Bigl{(}\|p_{1}\|_{H^{1}(\Omega)}^{2}+(1+b)\|\Delta
p_{0}\|_{L^{2}(\Omega)}^{2}+\|\partial_{\nu}p_{0}\|_{L^{2}(\Gamma_{a})}^{2}+\|\widetilde{w}_{1}\|_{L^{2}(\Gamma_{pl})}^{2}+\|\Delta_{pl}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}$
$\displaystyle\qquad\qquad+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}+F[f,\widetilde{h}]+\|\widetilde{h}\|_{L^{1}(0,t;H^{2}(\Gamma_{pl})^{*})}^{2}\Bigr{)}$

whose weak limit can be written as (3.29).

Step 3 weak limits:
According to Step 1, for each $n\in\mathbb{N}$, $\nabla p^{n}_{tt}$ and
$\widetilde{w}^{n}_{tt}$ exist and lie in $L^{2}(0,T;L^{2}(\Omega))$ and
$L^{2}(0,T;L^{2}(\Gamma_{pl}))$, respectively (even though we cannot estimate
these norms uniformly in $n$). Therefore, we can take the $L^{2}(0,T)$ inner
product of (3.33), (3.34) (that is, the Galerkin projection of (3.22)) with
arbitrary $C_{c}^{\infty}(0,T)$ functions $v$ and integrate by parts with
respect to time to arrive at

$\displaystyle\text{for all }(\phi^{n},\psi^{n})\in V^{n},\ v\in
C^{\infty}(0,T),\,v(T)=0\,:$ (3.45)
$\displaystyle\int_{0}^{T}\Bigl{\\{}\int_{\Omega}\Bigl{(}-\nabla
p^{n}_{t}\cdot\nabla\phi^{n}v^{\prime}+\tfrac{c^{2}}{\alpha}\Delta
p^{n}\,\Delta\phi^{n}v+\tfrac{b}{\alpha}\Delta
p^{n}_{t}\,\Delta\phi^{n}v\Bigr{)}\,dx$
$\displaystyle\qquad+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\Bigl{(}-\rho\widetilde{w}^{n}_{t}\psi^{n}v^{\prime}+\delta\Delta_{pl}\widetilde{w}^{n}\,\Delta_{pl}\psi^{n}v+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}^{n}_{t}\,\psi^{n}v\Bigr{)}\,dS+\int_{\Gamma_{a}}c\partial_{\nu}p^{n}_{t}\,\partial_{\nu}\psi^{n}v\,dS\Bigr{\\}}\,dt$
$\displaystyle=\int_{0}^{T}\Bigl{\\{}\int_{\Omega}\tfrac{f}{\alpha}\,(-\Delta\phi^{n})\,dx+\tfrac{\rho}{\kappa}\int_{\Gamma_{pl}}\widetilde{h}\,\psi^{n}\,dS\Bigr{\\}}\,v\,dt+\Bigl{(}\int_{\Omega}\nabla
p^{n}_{1}\cdot\nabla\phi^{n}\,dx+\int_{\Gamma_{pl}}{\tfrac{\rho^{2}}{\kappa}}\Bigl{(}-\widetilde{w}^{n}_{1}\psi^{n}\,dS\Bigr{)}v(0)\,.$

The energy estimates from Step 2 allow us to extract a subsequence of the
Galerkin approximations that weakly * converges in $U$ (which is the dual of a
separable space) to some $(p,\widetilde{w})\in U$. Taking limits along this
subsequence in (3.45) relying on Lemma 3.3 yields (3.22) and by weak* lower
semicontinuity of the energy functional on the left hand side of (3.28), the
limit $(p,\widetilde{w})$ also satisfies the energy estimate (3.28).

Step 4 attainment of the initial conditions:
By uniform boundedness of $(p^{n},\widetilde{w}^{n})$ in
$L^{\infty}(0,T;{H_{\Delta}^{2}(\Omega)}\times H^{2}(\Gamma_{pl}))\cap
C([0,T],H^{1}(\Omega)\times L^{2}(\Gamma_{pl}))$, respectively, using [51,
Lemma 3.1.7] we can conclude that along the subsequence from Step 3 we have
weak convergence of $(p^{n}(0),\widetilde{w}^{n}(0))$ in
${H_{\Delta}^{2}(\Omega)}\times H^{2}(\Gamma_{pl})$ to
$(p(0),\widetilde{w}(0))$ and since at the same time
$(p^{n}(0),\widetilde{w}^{n}(0))$ (the Galerkin projected initial data)
converges to $(p_{0},\widetilde{w}_{0})$ even strongly in
${H_{\Delta}^{2}(\Omega)}\times H^{2}(\Gamma_{pl})$, the initial conditions
are satisfied by $(p,\widetilde{w})$ in ${H_{\Delta}^{2}(\Omega)}\times
H^{2}(\Gamma_{pl})$.

Step 5 higher regularity in time; proof of (iii):
To prove (iii) we use the fact that by time differentiation,
$(p^{\prime},\widetilde{w}^{\prime}):=(p^{(n)}_{t},\widetilde{w}^{n}_{t})$
solves (3.19) with $(f,\widetilde{h})$ replaced by
$(f^{\prime},\widetilde{h}^{\prime})$, where
$f^{\prime}=f_{t}-\frac{\alpha_{t}}{\alpha}(f+c^{2}\Delta p^{n}+b\Delta
p^{n}_{t})$, $\widetilde{h}^{\prime}=\widetilde{h}_{t}$. Applying (i) with

$\displaystyle\|f^{\prime}\|_{L^{1}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq\|f_{t}\|_{L^{1}(0,T;L^{2}(\Omega))}+\|\tfrac{\alpha_{t}}{\alpha}\|_{L^{1}(0,T;L^{\infty}(\Omega))}\|f+c^{2}\Delta
p^{n}+b\Delta p^{n}_{t}\|_{L^{\infty}(0,T;L^{2}(\Omega))}$
$\displaystyle\leq\|f_{t}\|_{L^{1}(0,T;L^{2}(\Omega))}+\bar{\alpha}\Bigl{(}\|f\|_{L^{\infty}(0,T;L^{2}(\Omega))}+C\Bigl{(}\mathcal{E}[p,\widetilde{w}](0)+F[f,\widetilde{h}]\Bigr{)}\Bigr{)}\Bigr{)}$

and taking weak limits implies the assertion.

Step 6 uniqueness:
In view of linearity it only remains to prove that the solution to the
homogeneous equation vanishes. To this end, note that for
$(p,\widetilde{w})\in U$ we can perform the testing from Step 2 of the proof
of Proposition 3.5 to the infinite dimensional version of (3.19). In the
homogeneous case $f=0$, $\widetilde{h}=0$, $(p_{0},\widetilde{w}_{0})=(0,0)$,
$(p_{1},\widetilde{w}_{1})=(0,0)$, this results in
$\mathcal{E}[p,\widetilde{w}]\equiv 0$, thus $(p,\widetilde{w})\equiv 0$.

Step 7 proof of (iv):
The proof of (iv) goes analogously to the one of (ii), by testing the time
differentiated equation (cf. Step 5) with $(p^{n}_{t},\widetilde{w}^{n}_{t})$.
$\Box$

######  Remark 3.6 (exponential decay).

In particular, if $f=0$, $\widetilde{h}=0$, the energy estimate (3.29) in
Proposition 3.5 implies exponential decay of $\|\nabla
p_{t}\|_{L^{2}(\Omega)}^{2}+\|\Delta p\|_{L^{2}(\Omega)}^{2}$ and — via (3.10)
as well as $\|\partial_{\nu}{p}\|_{H^{-1/2}(\Gamma_{pl})}\leq
C^{tr}_{1/2}\Bigl{(}\|\nabla p\|_{L^{2}(\Omega)}+\|\Delta
p\|_{L^{2}(\Omega)}\Bigr{)}$, cf. Lemma 3.2 — also of
$\|\widetilde{w}\|_{H^{-1/2}(\Gamma_{pl})}$.

To extend the result to inhomogeneous Dirichlet, Neumann, and impedance
conditions, we make use of the decomposition $p=\bar{p}+\widetilde{p}$,
estimating $\bar{p}$ by Lemma 3.1 and $\widetilde{p}$ by Proposition 3.5. To
obtain energy estimates on $p=\bar{p}+\widetilde{p}$, we use the fact that
$\sqrt{\mathcal{E}^{p}}$ is a seminorm, in particular it satisfies the
triangle inequality. For the definitions of $\mathcal{E}$, $\mathcal{E}^{p}$,
$F[f,\widetilde{h}]$ and $G[a,g_{D},g_{N}]$, we refer to (3.24), (3.27),
(3.15).

######  Corollary 3.7.

Let, the regularity assumptions on the data of Lemma 3.1 and of Proposition
3.5 (i), (iii) as well as the compatibility conditions

$\displaystyle p_{1}+c\partial_{\nu}{p_{0}}=a(0),\
p_{2}+c\partial_{\nu}{p_{1}}=a_{t}(0)$ $\displaystyle\text{ on }\Gamma_{a}$
(3.46) $\displaystyle p_{0}=g_{D}(0),\ p_{1}=g_{D\,t}(0),\ p_{2}=g_{D\,tt}(0)$
$\displaystyle\text{ on }\Gamma_{D}$
$\displaystyle\partial_{\nu}{p_{0}}=g_{N}(0),\
\partial_{\nu}{p_{1}}=g_{N\,t}(0)\ \partial_{\nu}{p_{2}}=g_{N\,tt}(0)$
$\displaystyle\text{ on }\Gamma_{N}$
$\displaystyle\partial_{\nu}p_{0}=-\rho\widetilde{w}_{0},\
\partial_{\nu}p_{1}=-\rho\widetilde{w}_{1},\
\partial_{\nu}p_{2}=-\rho\widetilde{w}_{2}$ $\displaystyle\text{ on
}\Gamma_{pl}$

where $p_{2}:=\tfrac{1}{\alpha(0)}\Bigl{(}c^{2}\Delta p_{0}+b\Delta
p_{1}+f(0)\Bigr{)}$,
$\widetilde{w}_{2}\>=\frac{1}{\rho}\Bigl{(}-\delta\Delta_{pl}^{2}\widetilde{w}_{0}-\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{1}+\kappa
p_{2}+\widetilde{h}(0)\Bigr{)}$

be satisfied.

Then there exists a solution $(p,\widetilde{w})$ to (3.12) in $U^{\prime}$ and
the estimate

$\displaystyle\|\mathcal{E}_{1}[p,\widetilde{w}]\|_{L^{\infty}(0,T)}\leq
C\Bigl{(}\mathcal{E}_{1}[p,\widetilde{w}](0)+F[f,\widetilde{h}]+F[f_{t},\widetilde{h}_{t}]+G[a,g_{D},g_{N}]+G[a_{t},g_{D\,t},g_{N\,t}]\Bigr{)}$
(3.47)

holds. The solution is unique in $U^{\prime}$.

If additionally the assumptions of Proposition 3.5 (ii), (iv) hold, then

$\displaystyle\|\mathcal{E}_{1}[p,\widetilde{w}]\|_{L^{\infty}(0,T)}+\|\mathcal{E}_{1}[p,\widetilde{w}]\|_{L^{1}(0,T)}$
(3.48) $\displaystyle\leq
C\Bigl{(}\mathcal{E}_{1}[p,\widetilde{w}](0)+b\|\Delta
p_{0}\|_{L^{2}(\Omega)}^{2}+b\|\Delta
p_{1}\|_{L^{2}(\Omega)}^{2}+\|\partial_{\nu}{p_{0}}\|_{L^{2}(\Gamma_{a})}^{2}+\|\partial_{\nu}{p_{1}}\|_{L^{2}(\Gamma_{a})}^{2}$
$\displaystyle\qquad\qquad+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{1}\|_{L^{2}(\Gamma_{pl})}^{2}$
$\displaystyle\qquad\qquad+F[f,\widetilde{h}]+F[f_{t},\widetilde{h}_{t}]+G[a,g_{D},g_{N}]+G[a_{t},g_{D\,t},g_{N\,t}]\Bigr{)}$

The constant $C$ depends on the coefficients $b$, $c$, $\rho$, $\delta$,
$\kappa$, $\|\frac{1}{\alpha}\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}$,
$\|\alpha_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$, but not directly on $T$.

##  4 Local and Global in Time Well-posedness of the Nonlinear Model

To first of all prove local in time well-posedness of the quasilinear coupled
system

$\displaystyle(1-2kp)\,p_{tt}-c^{2}\Delta p-b\Delta\widetilde{p}_{t}$
$\displaystyle=2k(p_{t})^{2}~{}~{}\mbox{in}~{}~{}\Omega\times[0,T]$ (4.49)
$\displaystyle c\partial_{\nu}{p}+p_{t}$
$\displaystyle=a~{}~{}\mbox{in}~{}~{}\Gamma_{a}\times[0,T]$ $\displaystyle p$
$\displaystyle=g_{D}~{}~{}\mbox{on}~{}~{}\Gamma_{D}\times[0,T]$
$\displaystyle\partial_{\nu}{p}$
$\displaystyle=g_{N}~{}~{}\mbox{on}~{}~{}\Gamma_{N}\times[0,T]$
$\displaystyle\partial_{\nu}{p}$
$\displaystyle=-\rho\widetilde{w}~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\rho\widetilde{w}_{tt}+\delta\Delta_{pl}^{2}\widetilde{w}+\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{t}$
$\displaystyle=\kappa
p_{tt}+\widetilde{h}~{}~{}\mbox{on}~{}~{}\Gamma_{pl}\times[0,T]$
$\displaystyle\widetilde{p}(0)$
$\displaystyle=p_{0},~{}~{}\widetilde{p}_{t}(0)=p_{1}.$

we construct $(p,\widetilde{w})$ as a fixed point of the map

$\Lambda:W\mapsto U^{\prime},\quad(q,v)\to(p,\widetilde{w})\text{ solving
\eqref{pwlin} with $\alpha=1-2kq$, $f=2k(q_{t})^{2}$}$

on the following subset $W$ of the solution space $U^{\prime}$ (cf. (3.31),
(3.23))

$\displaystyle W=\\{(q,v)\in U^{\prime}:$
$\displaystyle\|q\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq M_{1},\
{\|\mathcal{E}[q,v]\|_{L^{\infty}(0,T)}}\leq M_{2},\
{\|\mathcal{E}[q_{t},v_{t}]\|_{L^{\infty}(0,T)}}\leq M_{3},$
$\displaystyle(q(0),q_{t}(0))=(p_{0},p_{1}),\
(v(0),v_{t}(0))=(\widetilde{w}_{0},\widetilde{w}_{1})$ $\displaystyle
c\partial_{\nu}{q}+q=a\text{ on }\Gamma_{a},\ q=g_{D}\text{ on }\Gamma_{D},\
\partial_{\nu}{q}=g_{N}\text{ on }\Gamma_{N},\ \partial_{\nu}{q}=-\rho v\text{
on }\Gamma_{pl}\\},$

where $0<M_{1}<\frac{1}{2k}$ and $0<M_{2}$, $0<M_{3}$ are to be chosen in a
suitable way, where the first of these requirements is supposed to avoid
degeneracy of the coefficient $\alpha=1-2kq$.

For the latter purpose, a crucial step in the proof invariance of the set $W$
under the map $\Lambda$ is to control the $L^{\infty}([0,T]\times\Omega)$ norm
of $q$ in terms of its energy. This is here impeded by the fact that the
reduced boundary regularity limits applicability of elliptic regularity in the
sense that we cannot conclude $H^{2}(\Omega)\subseteq H^{2}_{\Delta}(\Omega)$
(and then use continuity of the embedding $H^{2}(\Omega)\to
L^{\infty}(\Omega)$). We thus proceed with some results allowing us to obtain
an $L^{\infty}(\Omega)$ bound on $p(t)$ from reasonable assumptions on the
boundary data.

To this end, in case of nonempty Dirichlet boundary, we make use of the
following result,

######  Lemma 4.1.

([44, Theorem 5]) Suppose that $\Omega$, $\Gamma_{D}$ and
$\partial\Omega\setminus\Gamma_{D}$ are regular in the sense that for any $r$
in a dense subset of $L^{2}(\Omega)$, the elliptic problem $-\Delta u+u=r$ has
an $H^{2}(\Omega)$ solution such that $u|_{\Gamma_{D}}$,
$\partial_{\nu}{u}|_{\Gamma_{D}}$ are essentially bounded and $z\in
H^{2}(\Omega)$ exist such that

$\text{essinf}_{x\in\Omega}z(x)>0,\quad\text{esssup}_{x\in\Omega}\Delta
z(x)<\infty,\quad z\geq 1\text{ a.e. on }\Gamma_{D},\quad\partial_{\nu}{z}\geq
1\text{ a.e. on }\partial\Omega\setminus\Gamma_{D}$

Then there exist $K>0$ such that any solution $u$ to

$\displaystyle-\Delta u+u$ $\displaystyle=\mathfrak{f}\textup{ in }\Omega$
(4.50) $\displaystyle u$ $\displaystyle=\mathfrak{g}_{D}\textup{ on
}\Gamma_{D}$ $\displaystyle\partial_{\nu}{u}$
$\displaystyle=\mathfrak{g}_{N}\textup{ on }\partial\Omega\setminus\Gamma_{D}$

satisfies the estimate

$\|u\|_{L^{\infty}(\Omega)}\leq
K(\|\mathfrak{f}\|_{L^{2}(\Omega)}+\|\mathfrak{g}_{D}\|_{L^{\infty}(\Gamma_{D})}+\|\mathfrak{g}_{N}\|_{L^{\infty}(\Gamma_{D})})$

We mention in passing that [44, Theorem 5] also gives an explicit expression
of $K$ in terms of $z$, as well as examples of domains satisfying these
conditions referring also to [33, Chapter 3, Section 9]. We also point to [9,
12, 46].

######  Lemma 4.2.

Let $s\in(0,\frac{1}{2})$, $\mathfrak{f}\in L^{2}(\Omega)$,
$\mathfrak{g}_{D}\in L^{\infty}(\Gamma_{D})$, ${\mathfrak{g}_{N}}\in
X_{\mathfrak{g}}$, and either

  * (a)

$\Gamma_{D}\not=\emptyset$ and the regularity conditions of Lemma 4.1 on
$\partial\Omega$, $\Gamma_{D}$ hold and
$X_{\mathfrak{g}}=H^{s}(\partial\Omega\setminus\Gamma_{D})\cap
L^{\infty}(\partial\Omega\setminus\Gamma_{D})$ or

  * (b)

$\Gamma_{D}=\emptyset$, $X_{\mathfrak{g}}=H^{s}(\partial\Omega)$.

Then there exist a constant $K$ independent of ${\mathfrak{g}_{D}}$,
${\mathfrak{g}_{N}}$, $\mathfrak{f}$, such that for any solution $u$ to
(4.50), the estimate

$\|u\|_{L^{\infty}(\Omega)}\leq
K(\|\mathfrak{f}\|_{L^{2}(\Omega)}{+\|\mathfrak{g}_{D}\|_{L^{\infty}(\Gamma_{D})}+\|\mathfrak{g}_{N}\|_{X_{\mathfrak{g}}}})$

holds. In case (b) we additionally have

$\|u\|_{H^{3/2+s}(\Omega)}\leq
K(\|\mathfrak{f}\|_{L^{2}(\Omega)}+\|\mathfrak{g}_{N}\|_{X_{\mathfrak{g}}}).$

###### Proof.

In case (a) we apply Lemma 4.1. In case (b) the assertion follows from
standard results for the Laplace Neumann problem together with continuity of
the embedding $H^{3/2+s}(\Omega)\to L^{\infty}(\Omega)$ $\Box$

We use Lemma 4.2 in order to bound $\|\alpha\|_{L^{\infty}}$,
$\|\frac{1}{\alpha}\|_{L^{\infty}}$,
$\|\alpha_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$ and in case of
$\Gamma_{D}=\emptyset$ also $\|\nabla\alpha\|_{L^{2}(0,T;L^{3}(\Omega))}$,
$\|\nabla\alpha\|_{L^{\infty}(0,T;L^{3}(\Omega))}$, as required by Proposition
3.5.

######  Lemma 4.3.

Let $M_{1}\leq\frac{1}{4k}$ and $M_{2}(1+T^{1/2})$, $M_{3}(1+T)$ as well as
$\|g_{D}\|_{L^{\infty}(0,T;H^{1+s}(\Gamma_{D}))}$,
$\|g_{N}\|_{L^{\infty}(0,T;H^{s}(\Gamma_{N}))}$,
$\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))}$ be small enough for some
$s\in(0,1/2)$, where
in case (a) $\Gamma_{D}\not=\emptyset$ we assume that $\Gamma_{a}=\emptyset$
and the regularity conditions from Lemma 4.1 on $\partial\Omega$, $\Gamma_{D}$
on $\partial\Omega$, $\Gamma_{D}$ are satisfied

Then for any $(q,v)\in W$, the functions $\alpha=1-2kq$, $f=2kq_{t}^{2}$
satisfy the following.

  1. 1.

For all $(t,x)\in\Omega\times[0,T]$, we have
$\frac{1}{2}\leq\alpha(t,x)\leq\frac{3}{2}$.

  2. 2.

$\alpha_{t}\in L^{1}(0,T;L^{\infty}(\Omega))$ with
$\bar{\alpha}=\|\frac{\alpha_{t}}{\alpha}\|_{L^{1}(0,T;L^{\infty}(\Omega))}<1/2$

  3. 3.

In case $\Gamma_{D}=\emptyset$, $\nabla\alpha\in L^{2}(0,T;L^{3}(\Omega))$
with
$\widetilde{\alpha}_{2}=\|\alpha^{-3/2}\nabla\alpha\|_{L^{2}(0,T;L^{3}(\Omega))}<1/4$,
$\widetilde{\alpha}_{\infty}=\|\alpha^{-3/2}\nabla\alpha\|_{L^{\infty}(0,T;L^{3}(\Omega))}<1/4$

  4. 4.

$f\in W^{1,1}(0,T;L^{2}(\Omega))$.

###### Proof.

The estimate 1. immediately follows from the definition of $W$ with
$M_{1}\leq\frac{1}{4k}$ and

$1-\|\alpha-1\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq\alpha(t,x)\leq
1+\|\alpha-1\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}$

with

$\|\alpha-1\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}=2k\|q\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq
2kM_{1}\leq\frac{1}{2}$

To prove 2. we invoke the two cases of Lemma 4.2 separately:

In case (b) $\Gamma_{D}=\emptyset$, relying on the control of the
$L^{2}(\Omega)$ norm of $q_{t}$ and $\Delta q_{t}$ as established in the proof
of Proposition 3.5), we can simply set

${\mathfrak{g}_{N}}=\begin{cases}-\tfrac{1}{c}q_{tt}(t)&\textup{ on
}\Gamma_{a}\\\ -\rho v_{t}(t)&\textup{ on }\Gamma_{pl}\\\ g_{N\,t}&\textup{ on
}\Gamma_{N}\end{cases}\qquad\mathfrak{f}=-\Delta q_{t}(t)+q_{t}(t).$ (4.51)

to obtain

$\displaystyle\|\alpha_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}=2k\|q_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$
(4.52) $\displaystyle\leq 2kK\Bigl{(}\|-\Delta
q_{t}+q_{t}\|_{L^{1}(0,T;L^{2}(\Omega))}+C^{tr}_{H^{s}}\tfrac{1}{c}\|q_{tt}\|_{L^{1}(0,T;H^{s+1/2}(\Omega))}+\rho\|v_{t}\|_{L^{1}(0,T;H^{s}(\Gamma_{pl}))}+\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))}\Bigr{)}$
$\displaystyle\leq
2kK(\widetilde{C}\sqrt{\sup_{t\in(0,T)}\mathcal{E}[q_{t},v_{t}]+\int_{0}^{T}\mathcal{E}[q_{t},v_{t}]\,ds}+\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))})$
$\displaystyle\leq
2kK(\widetilde{C}\sqrt{(1+T)\sup_{t\in(0,T)}\mathcal{E}[q_{t},v_{t}]}+\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))}).$

Here we have used the fact that for $s\in[0,1/2)$ the concatenation of $H^{s}$
boundary functions is again an $H^{s}$ boundary function, see, e.g., [15,
Corollary 1.4.4.5.] combined with the arguments in [30, Appendix]. Thus by
choosing $s\geq 0$ $(1+T)M_{3}$ and
$\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))}$ small enough, so that
$2kK(\widetilde{C}M_{3}+\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))})\leq 1/2$,
we obtain 2.

Likewise

$\displaystyle\|\nabla\alpha\|_{L^{\infty}(0,T;L^{3}(\Omega))}$
$\displaystyle\leq
C^{\Omega}_{H^{s+1/2},L^{3}}\|\alpha\|_{L^{\infty}(0,T;H^{s+3/2}(\Omega))}=2kC^{\Omega}_{H^{s+1/2},L^{3}}\|q\|_{L^{\infty}(0,T;H^{s+3/2}(\Omega))}$
$\displaystyle\leq
2kKC^{\Omega}_{H^{s+1/2},L^{3}}(\widetilde{C}(1+T)\sup_{t\in(0,T)}\mathcal{E}[q,v](t)+\|g_{N}\|_{L^{\infty}(0,T;H^{s}(\Gamma_{N})}).$

and similarly for $\|\nabla\alpha\|_{L^{2}(0,T;L^{3}(\Omega))}$ in
$\widetilde{\alpha}_{2}$ thus implying 3. provided $(1+T)M_{2}$, $(1+T)M_{3}$,
$\|g_{N}\|_{L^{\infty}(0,T;H^{s}(\Gamma_{N}))}$ are small enough.

In case (a) $\Gamma_{D}\not=\emptyset$, the energy controls
$\|q_{t}(t)\|_{L^{2}(\Omega)}$ due to Poincaré’s inequality and we set
${\mathfrak{g}_{N}}$, $\mathfrak{f}$ according to (4.51). From $v_{t}(t)\in
H^{r}(\Gamma_{pl})\subseteq L^{\infty}(\Gamma_{pl})$ and $q_{tt}\in
H^{r}(\Gamma_{a})\subseteq L^{\infty}(\Gamma_{a})$ with $r>1$ we would obtain
${\mathfrak{g}_{N}}\in H^{s}(\Gamma_{N})\cap L^{\infty}(\Gamma_{N})$ for any
$s\in(0,\frac{1}{2})$ and could use Lemma 4.2 (a) to conclude 2.

However, the available energy estimates do not provide $q_{tt}(t)\in
H^{r}(\Gamma_{a})$ for any $r>1$. To achieve this, we would have to derive
higher order energy estimates, which would lead to stronger assumptions on
smoothness of the data. In order to avoid this problem we assume
$\Gamma_{a}=\emptyset$ in case (a) $\Gamma_{D}\not=\emptyset$. Therewith it
suffices to bound

$\|v_{t}(t)\|_{H^{r}(\Gamma_{pl})}\leq
C^{\Gamma_{pl}}_{H^{2},H^{r}}\mathcal{E}[q_{t},v_{t}](t),$

and the assertion follows analogously to case (b).

For $f$ and $f_{t}$ we obtain

$\|f\|_{L^{1}(0,T;L^{2}(\Omega))}=2k\|q_{t}\|_{L^{1}(0,T;L^{4}(\Omega))}^{2}\leq
2k(C^{\Omega}_{H^{1},H^{4}})^{2}\|q_{t}\|_{L^{1}(0,T;H^{1}(\Omega))}^{2},$
$\|f_{t}\|_{L^{1}(0,T;L^{2}(\Omega))}=4k\|q_{tt}q_{t}\|_{L^{1}(0,T;L^{2}(\Omega))}\leq
4k\|q_{tt}\|_{L^{\infty}(0,T;L^{2}(\Omega))}\|q_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$

where $\|q_{t}\|_{L^{1}(0,T;H^{1}(\Omega))}$ and
$\|q_{tt}\|_{L^{\infty}(0,T;L^{2}(\Omega))}$ can be bounded by some constant
multiple of $\int_{0}^{T}\mathcal{E}[q,v](t)\,dt$ and
$\sup_{t\in(0,T)}\mathcal{E}[q_{t},v_{t}](t)$, respectively, and
$\|q_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$ has already been bounded in the
proof of 2. $\Box$

######  Theorem 4.4.

In case $\Gamma_{D}\not=\emptyset$ assume that $\Gamma_{a}=\emptyset$ and the
regularity conditions from Lemma 4.1 on $\partial\Omega$, $\Gamma_{D}$ are
satisfied.

Let $T>0$ be arbitrary, then there exists constants $m_{1}>0$, $m_{2}>0$ and
$m_{3}>0$ possibly depending on $T$ such that for initial conditions
$(p_{0},p_{1},\widetilde{w}_{0},\widetilde{w}_{1})$ and data
$(a,g_{D},g_{N},\widetilde{h})$ satisfying the regularity assumptions of
Corollary 3.7 as well as the compatibility conditions (3.46) with
$p_{2}:=\tfrac{1}{1-2kp_{0}}\Bigl{(}c^{2}\Delta p_{0}+b\Delta
p_{1}+2kp_{1}^{2}\Bigr{)}$,
$\widetilde{w}_{2}\>=\frac{1}{\rho}\Bigl{(}-\delta\Delta_{pl}^{2}\widetilde{w}_{0}-\beta(-\Delta_{pl})^{\gamma}\widetilde{w}_{1}+\kappa
p_{2}+\widetilde{h}(0)\Bigr{)}$ and

$\displaystyle\mathcal{E}_{1}[p,\widetilde{w}](0)\leq m_{1},$ $\displaystyle
F[{2k\bar{p}_{t}^{2},\widetilde{h}+\bar{p}_{tt}}]+F[{4k\bar{p}_{t}\bar{p}_{tt},\widetilde{h}_{t}+\bar{p}_{ttt}}]+G[a,g_{D},g_{N}]{+G[a,g_{D\,t},g_{N\,t}]}\leq
m_{2},$
$\displaystyle\|g_{D}\|_{L^{\infty}(0,T;H^{1+s}(\Gamma_{D}))}+\|g_{N}\|_{L^{\infty}(0,T;H^{s}(\Gamma_{N}))}{+\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))}}\leq
m_{3}$

for some $s\in(0,1/2)$, there exists a unique solution $(p,\widetilde{w})$ in
the set $W$ to (4.49).

This solution satisfies the energy estimate

$\displaystyle\mathcal{E}_{1}[p,\widetilde{w}](t)+\int_{0}^{t}\mathcal{E}_{1}[p,\widetilde{w}](s)\,ds\leq
C\Bigl{(}\int_{0}^{t}\mathcal{E}_{1}[p,\widetilde{w}](s)\,ds\Bigr{)}^{2}$
(4.53) $\displaystyle+C\Bigl{(}\mathcal{E}_{1}[p,\widetilde{w}](0)+b\|\Delta
p_{0}\|_{L^{2}(\Omega)}^{2}+b\|\Delta
p_{1}\|_{L^{2}(\Omega)}^{2}+\|\partial_{\nu}{p_{0}}\|_{L^{2}(\Gamma_{a})}^{2}+\|\partial_{\nu}{p_{1}}\|_{L^{2}(\Gamma_{a})}^{2}$
$\displaystyle\qquad\qquad+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{0}\|_{L^{2}(\Gamma_{pl})}^{2}+\beta\|(-\Delta_{pl})^{\gamma/2}\widetilde{w}_{1}\|_{L^{2}(\Gamma_{pl})}^{2}+F[0,\widetilde{h}]+F[0,\widetilde{h}_{t}]\Bigr{)}$

If additionally $\widetilde{h}+\bar{p}_{tt}\in
W^{1,1}(0,\infty;H^{2}(\Gamma_{pl})^{*})$, $g_{N\,t}\in
W^{2,1}(0,{\infty};H^{-1/2}(\Gamma_{N}))$, then $m_{1},m_{2},m_{3}$ can be
chosen independently of $T$ and the solution exists globally in time, that is,
the assertion above is valid with $T=\infty$.

If additionally the forcing data $a,g_{D},g_{N},\widetilde{h}$ vanish, then
$\mathcal{E}_{1}[p,\widetilde{w}](t)$ decays exponentially

###### Proof.

Step 1 Invariance of solution map:
We let $(q,v)\in W$ and consider equation (3.12) with $\alpha=1-2kq$ and
$f=2k(q_{t})^{2}$. By Lemma 4.3, $\alpha$ and $f$ satisfy the assumptions of
Proposition 3.5 (i), (iii) and thus, due to our assumptions on the data there
exists a solution $\Lambda(q,v)=(p,\widetilde{w})\in U^{\prime}$. To show that
$\Lambda$ maps $W$ into itself for suitably chosen $m_{1},m_{2}>0$, we employ
the energy estimate (3.47) in Corollary 3.7, that together with our
assumptions on smallness of the data implies

$\mathcal{E}[p,\widetilde{w}](t)+\mathcal{E}[p_{t},\widetilde{w}_{t}](t)\leq
C\Bigl{(}m_{1}+m_{2}).$

Choosing $m_{1}$, $m_{2}$ small enough thus allows us to conclude
$\mathcal{E}[p,\widetilde{w}](t)\leq M_{2}$,
$\mathcal{E}[p_{t},\widetilde{w}_{t}](t)\leq M_{3}$ Additionally, analogously
to Lemma 4.3 2., we estimate the $L^{\infty}(0,T;L^{\infty}(\Omega))$ norm of
$p$ using Lemma 4.2 to obtain

$\|p\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq C(m_{1}+m_{2}+m_{3})$

thus, by choosing $m_{1},m_{2},m_{3}$ small enough,
$\|p\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq M_{1}$.

Step 2 Contraction estimates:
We next show that the solution map $\Lambda:(q,v)\to(p,\widetilde{w})$ is a
contraction on $W$ in the topology of the space $U$. In particular, let
$(p_{1},\widetilde{w}_{1})$ and $(p_{2},\widetilde{w}_{2})$ be the solutions
of (3.12)corresponding to $(q_{1},v_{1})$ and $(q_{2},v_{2})$ in $W$. We
denote the differences of solutions by $(\hat{p},\hat{w})$ and
$(\hat{q},\hat{v})$; that is $\hat{p}=p_{1}-p_{2}$,
$\hat{w}=\widetilde{w}_{1}-\widetilde{w}_{2}$, $\hat{q}=q_{1}-q_{2}$,
$\hat{v}=v_{1}-v_{2}$. We also denote by $\alpha_{1}=1-2kq_{1}$,
$\alpha_{2}=1-2kq_{2}$ while $f_{1}=2kq_{1\,t}^{2}$ and
$f_{2}=2kq_{2\,t}^{2}$. Then $(\hat{p},\hat{w})$ satisfies (3.16) with
$(f,\widetilde{h})$ replaced by $(\hat{f},0)$ where
$\hat{f}=f_{1}-f_{2}-\frac{\alpha_{1}-\alpha_{2}}{\alpha_{1}\alpha_{2}}(c^{2}\Delta
p_{2}+b\Delta p_{2\,t})$ and homogeneous initial conditions. We apply
Proposition 3.5 (i) with the estimate

$\displaystyle\|\hat{f}\|_{L^{1}(0,T;L^{2}(\Omega))}\leq$ $\displaystyle
2k\|q_{1\,t}+q_{2\,t}\|_{L^{2}(0,T;L^{4}(\Omega))}\|\hat{q}_{t}\|_{L^{2}(0,T;L^{4}(\Omega))}$
$\displaystyle+8k\|\hat{q}\|_{L^{2}(0,T;L^{\infty}(\Omega))}(c^{2}{\|\Delta
p_{2}\|_{L^{2}(0,T;L^{2}(\Omega))}}+b\|\Delta
p_{2\,t}\|_{L^{2}(0,T;L^{2}(\Omega))}),$

where we have used

$f_{1}-f_{2}=2k(q_{1\,t}+q_{2\,t})(q_{1\,t}-q_{2\,t}),\qquad\left|\frac{\alpha_{1}-\alpha_{2}}{\alpha_{1}\alpha_{2}}\right|\leq
4|\alpha_{1}-\alpha_{2}|=8k|q_{1}-q_{2}|,$

Together with continuity of the embedding $H^{1}(\Omega)\to L^{4}(\Omega)$ and
Lemma 4.2 this yields

$\displaystyle\|(\hat{p},\hat{w})\|_{U}=$
$\displaystyle\sqrt{\|\mathcal{E}[\hat{p},\hat{w}]\|_{L^{\infty}(0,T)}}\leq\sqrt{CF[\hat{f},0]}=\sqrt{C}\|\hat{f}\|_{L^{1}(0,T;L^{2}(\Omega))}$
(4.54) $\displaystyle\leq$ $\displaystyle
2k\sqrt{C}\|(\hat{q},0)\|_{U}\sqrt{\mathcal{E}[q_{1},0]+\mathcal{E}[q_{2},0]+\|\mathcal{E}[p_{2},0]\|_{L^{\infty}(0,T)}+\|\mathcal{E}[p_{2},0]\|_{L^{2}(0,T)}}$
$\displaystyle\leq$ $\displaystyle
2k\sqrt{C}\|(\hat{q},\hat{v})\|_{U}(1+\sqrt{T})\sqrt{3M_{2}}.$

We then choose $M_{2}$ sufficiently small so that

$2k\sqrt{C}(1+\sqrt{T})\sqrt{3M_{2}}<1$

and we get a contraction estimate.

Local in time well-posedness:
The Picard sequence $(p_{n},\widetilde{w}_{n})$ therefore converges to a fixed
point of $\Lambda$ (that is, a solution $(p,\widetilde{w})$ to (4.49)) in the
norm topology of the weaker space $U$. Moreover, since $W$ is a ball in
$U^{\prime}$ and thus weakly-* compact in $U^{\prime}$ due to the Banach-
Alaoglu Theorem, we can extract a weakly* convergent subsequence
$(p_{n_{k}},\widetilde{w}_{n_{k}})$ with limit
$(\bar{p},\bar{\widetilde{w}})\in W$. Due to uniqueness of limits we have
$(p,\widetilde{w})=(\bar{p},\bar{\widetilde{w}})\in W$.

The energy estimate (4.53) follows from (4.53) together with
$F(f,\widetilde{h}]=F[f,0]+F[0,\widetilde{h}]$ cf. (3.27) and the estimate

$\displaystyle
F[f,0]+F[f_{t},0]=\|f\|_{L^{1}(0,t;L^{2}(\Omega))}^{2}+\|f_{t}\|_{L^{1}(0,t;L^{2}(\Omega))}^{2}$
$\displaystyle=4k^{2}\Bigl{(}\|p_{t}\|_{L^{2}(0,t;L^{4}(\Omega))}^{4}+2\|p_{t}\,p_{tt}\|_{L^{1}(0,t;L^{2}(\Omega))}^{2}\Bigr{)}$
$\displaystyle\leq
12k^{2}(C^{\Omega}_{H^{1},L^{4}})^{2}\Bigl{(}\int_{0}^{t}\mathcal{E}_{1}(s)\,ds\Bigr{)}^{2}$

Global in time well-posedness:
If additionally $\widetilde{h}\in W^{1,1}(0,\infty;H^{2}(\Gamma_{pl})^{*})$,
$g_{N\,t}\in W^{2,1}(0,\infty;H^{-1/2}(\Gamma_{N}))$, we can replace the
definition of $W$ by

$\displaystyle W=\\{(q,v)\in U^{\prime}:$
$\displaystyle\|q\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq M_{1},\
{\|\mathcal{E}[q,v]\|_{L^{\infty}(0,T)}+\|\mathcal{E}[q,v]\|_{L^{1}(0,T)}}\leq
M_{2},$
$\displaystyle{\|\mathcal{E}[q_{t},v_{t}]\|_{L^{\infty}(0,T)}+\|\mathcal{E}[q_{t},v_{t}]\|_{L^{1}(0,T)}}\leq
M_{3},$ $\displaystyle(q(0),q_{t}(0))=(p_{0},p_{1}),\
(v(0),v_{t}(0))=(\widetilde{w}_{0},\widetilde{w}_{1})$ $\displaystyle
c\partial_{\nu}{q}+q=a\text{ on }\Gamma_{a},\ q=g_{D}\text{ on }\Gamma_{D},\
\partial_{\nu}{q}=g_{N}\text{ on }\Gamma_{D},\ \partial_{\nu}{q}=-\rho v\text{
on }\Gamma_{pl}\\},$

and use the fact that Lemma 4.3 then holds with $M_{2}$, $M_{3}$ as well as
$\|g_{D}\|_{L^{\infty}(0,T;H^{1+s}(\Gamma_{D}))}$,
$\|g_{N}\|_{L^{\infty}(0,T;H^{s}(\Gamma_{N}))}$,
$\|g_{N\,t}\|_{L^{1}(0,T;H^{s}(\Gamma_{N}))}$ small enough, cf., e.g., the
next to last estimates in (4.52), (4.54).

Exponential decay follows from the energy estimate (4.53) by barrier’s method,
cf., e.g., [28]. $\Box$

######  Remark 4.5.

When dropping the possibility of having Dirichlet and absorbing boundary
conditions, that is, consider the case
$\partial\Omega=\Gamma_{N}\cup\Gamma_{pl}$,
$\partial\Gamma_{N}=\partial\Gamma_{pl}$, we can actually achieve more spatial
regularity. Indeed, replacing part of the hinged boundary condition by the
trace of the Neumann data

$\rho\widetilde{w}=-g_{N}|_{\partial\Gamma_{N}}\,\quad\Delta\widetilde{w}=0\quad\text{
on }\partial\Gamma_{pl},$

and assuming $g_{N}(t)\in H^{1/2}(\Gamma_{N})$, via (3.10) we achieve
$\partial_{\nu}{p}\in H^{1/2}(\partial\Omega)$ and thus, together with
$p(t)\in H^{2}_{\Delta}(\Omega)$, recover $p(t)$ in $H^{2}(\Omega)$ by means
of elliptic regularity. The same holds true for the first and second time
derivative of $p$ so that we end up with the solution spaces

$\displaystyle U=$ $\displaystyle(W^{1,\infty}(0,T;H^{1}(\Omega))\cap
H^{1}(0,T;H^{2}(\Omega)))\times(W^{1,\infty}(0,T;L^{2}(\Gamma_{pl}))\cap
L^{\infty}(0,T;H^{2}(\Gamma_{pl})))$ $\displaystyle U^{\prime}=$
$\displaystyle\\{(p,\widetilde{w})\in U\,:\,(p_{t},\widetilde{w}_{t})\in U\\}$

in place of (3.23), (3.31).

##  5 Outlook

We have kept track of the damping coefficients $b$ and $\beta$ in the
estimates in order to open up the possibility of considering vanishing damping
(see, e.g., [29] for Westervelt alone). While this appears hopeless for $b$,
it might be possible to recover some of estimates in a $\beta$ uniform way. A
similar question arises for boundary damping if we replace (2.5) by
$p_{t}+\sigma\partial_{\nu}{p}=a$ considering $\sigma\to 0$.
As already mentioned above, $g_{N}$, and $h$ could be used as controls and an
analysis of related optimization problems along the lines of, e.g., [7] could
be of interest.
Likewise, optimizing the shape of $\Gamma_{pl}$ could be of practical
relevance.

## Appendix

###### Proof.

(Lemma 3.1)
Since the Galerkin discretiztion step can be done very similarly to, e.g.,
[42], we focus on energy estimates here.

Differentiating (3.13) twice with respect to time and testing with
$-\frac{1}{\alpha}\Delta\bar{p}_{ttt}$, we obtain

$\displaystyle 0=$
$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla\bar{p}_{ttt}\|^{2}-\int_{\partial\Omega}\bar{p}_{tttt}\partial_{\nu}{\bar{p}_{ttt}}\,dS-2\int_{\Omega}\frac{\alpha_{t}}{\alpha}\bar{p}_{ttt}\Delta\bar{p}_{ttt}\,dx-\int_{\Omega}\frac{\alpha_{tt}}{\alpha}\bar{p}_{tt}\Delta\bar{p}_{ttt}\,dx$
$\displaystyle+\frac{1}{2}c^{2}\frac{d}{dt}\left\|\frac{1}{\sqrt{\alpha}}\Delta\bar{p}_{tt}\right\|^{2}+\frac{1}{2}c^{2}\int_{\Omega}\frac{\alpha_{t}}{\alpha^{2}}|\Delta\bar{p}_{tt}|^{2}\,dx+b\left\|\frac{1}{\sqrt{\alpha}}\Delta\bar{p}_{ttt}\right\|^{2}$

where

$\displaystyle-\int_{\partial\Omega}\bar{p}_{tttt}\partial_{\nu}{\bar{p}_{ttt}}\,dS=$
$\displaystyle\frac{1}{c}\Bigl{(}\|\bar{p}_{tttt}\|_{L^{2}(\Gamma_{a})}^{2}-\int_{\Gamma_{a}}\bar{p}_{tttt}a_{ttt}\,dS\Bigr{)}-\int_{\Gamma_{D}}g_{D\,tttt}\partial_{\nu}{\bar{p}_{ttt}}\,dS$
$\displaystyle-\frac{d}{dt}\int_{\Gamma_{N}}\bar{p}_{ttt}g_{N\,ttt}\,dS+\int_{\Gamma_{N}}\bar{p}_{ttt}g_{N\,tttt}\,dS$

Thus, integrating with respect to time we obtain

$\displaystyle\frac{1}{2}\|\nabla\bar{p}_{ttt}\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+\frac{1}{2}c^{2}\left\|\frac{1}{\sqrt{\alpha}}\Delta\bar{p}_{tt}\right\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+b\left\|\frac{1}{\sqrt{\alpha}}\Delta\bar{p}_{ttt}\right\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\frac{1}{c}\|\bar{p}_{tttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}$
(5.55)
$\displaystyle\leq\frac{1}{c}\|\bar{p}_{tttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}\|a_{ttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}+\|\partial_{\nu}{\bar{p}_{ttt}}\|_{{L^{\infty}}(0,t;H^{-1/2}(\Gamma_{D}))}\|g_{D\,tttt}\|_{{L^{1}}(0,t;H^{1/2}(\Gamma_{D}))}$
$\displaystyle\quad+\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;H^{1/2}(\Gamma_{N}))}\Bigl{(}\|g_{N\,ttt}\|_{L^{\infty}(0,t;H^{-1/2}(\Gamma_{N}))}+\|g_{N\,tttt}\|_{L^{1}(0,t;H^{-1/2}(\Gamma_{N}))}\Bigr{)}$
$\displaystyle\quad+c^{2}\tfrac{1}{\underline{\alpha}}\|\tfrac{\alpha_{t}}{\alpha}\|_{L^{1}(0,T;L^{\infty}(\Omega))}\|\Delta\bar{p}_{tt}\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+2\|\tfrac{\alpha_{t}}{\alpha}\|_{L^{2}(0,T;L^{3}(\Omega))}\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;L^{6}(\Omega))}\|\Delta\bar{p}_{ttt}\|_{L^{2}(0,t;L^{2}(\Omega))}$
$\displaystyle\quad+\|\tfrac{\alpha_{tt}}{\alpha}\|_{L^{2}(0,T;L^{3}(\Omega))}\|\bar{p}_{tt}\|_{L^{\infty}(0,t;L^{6}(\Omega))}^{2}\|\Delta\bar{p}_{ttt}\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}$

with
$\tfrac{1}{\underline{\alpha}}=\left\|\frac{1}{\alpha}\right\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}$.

We also test with $\frac{1}{\alpha}\bar{p}_{ttt}$, which yields

$\displaystyle 0=$
$\displaystyle\frac{1}{2}\frac{d}{dt}\|\bar{p}_{ttt}\|^{2}-2\int_{\Omega}\frac{\alpha_{t}}{\alpha}\bar{p}_{ttt}^{2}\,dx-\int_{\Omega}\frac{\alpha_{tt}}{\alpha}\bar{p}_{tt}\bar{p}_{ttt}\,dx$
$\displaystyle\quad+\frac{1}{2}c^{2}\frac{d}{dt}\left\|\frac{1}{\sqrt{\alpha}}\nabla\bar{p}_{tt}\right\|^{2}+\frac{1}{2}c^{2}\int_{\Omega}\frac{\alpha_{t}}{\alpha^{2}}|\nabla\bar{p}_{tt}|^{2}\,dx+b\left\|\frac{1}{\sqrt{\alpha}}\nabla\bar{p}_{ttt}\right\|^{2}$
$\displaystyle-\int_{\Omega}(c^{2}\nabla{\bar{p}_{tt}}+b\nabla{\bar{p}_{ttt}})\cdot\nabla\alpha\,\tfrac{1}{\alpha^{2}}p_{ttt}-\int_{\partial\Omega}(c^{2}\partial_{\nu}{\bar{p}_{tt}}+b\partial_{\nu}{\bar{p}_{ttt}})\tfrac{1}{\alpha}p_{ttt}\,dS,$

where

$\displaystyle\int_{\partial\Omega}(c^{2}\partial_{\nu}{\bar{p}_{tt}}+b\partial_{\nu}{\bar{p}_{ttt}})\tfrac{1}{\alpha}p_{ttt}\,dS=c\Bigl{(}\|\tfrac{1}{\sqrt{\alpha}}\bar{p}_{ttt}\|_{L^{2}(\Gamma_{a})}^{2}-\int_{\Gamma_{a}}\tfrac{1}{\alpha}\bar{p}_{ttt}a_{tt}\,dS\Bigr{)}$
$\displaystyle\quad+\frac{b}{c}\Bigl{(}\frac{1}{2}\frac{d}{dt}\|\tfrac{1}{\sqrt{\alpha}}\bar{p}_{ttt}\|_{L^{2}(\Gamma_{a})}^{2}+\int_{\Gamma_{a}}\bigl{(}\tfrac{\alpha_{t}}{\alpha^{2}}|\bar{p}_{ttt}|^{2}-\tfrac{1}{\alpha}\bar{p}_{ttt}a_{ttt}\bigr{)}\,dS\Bigr{)}$
$\displaystyle-\int_{\Gamma_{D}}(c^{2}\partial_{\nu}{\bar{p}_{tt}}+b\partial_{\nu}{\bar{p}_{ttt}})\tfrac{1}{\alpha}g_{D\,ttt}\,dS-\int_{\Gamma_{N}}(c^{2}g_{N\,tt}+bg_{N\,ttt})\tfrac{1}{\alpha}\bar{p}_{ttt}\,dS,$

thus, after integration with respect to time,

$\displaystyle\frac{1}{2}(1-2\bar{\alpha})\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+\frac{1}{2}c^{2}(1-\bar{\alpha})\left\|\frac{1}{\sqrt{\alpha}}\nabla\bar{p}_{tt}\right\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+b\left\|\frac{1}{\sqrt{\alpha}}\nabla\bar{p}_{ttt}\right\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}$
(5.56) $\displaystyle\hskip
142.26378pt+c\|\frac{1}{\sqrt{\alpha}}\bar{p}_{ttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}+\frac{b}{c}(1-\bar{\alpha})\|\frac{1}{\sqrt{\alpha}}\bar{p}_{ttt}\|_{L^{\infty}(0,t;L^{2}(\Gamma_{a}))}^{2}$
$\displaystyle\leq\|\tfrac{\alpha_{tt}}{\alpha}\|_{L^{2}(0,T;L^{2}(\Omega))}\|\bar{p}_{tt}\|_{L^{\infty}(0,t;L^{4}(\Omega))}\|\bar{p}_{ttt}\|_{L^{2}(0,t;L^{4}(\Omega))}$
$\displaystyle\quad+\tfrac{1}{\underline{\alpha}}\|\tfrac{\nabla\alpha}{\alpha}\|_{L^{2}(0,t;L^{3}(\Omega)}\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;L^{6}(\Omega))}\Bigl{(}c^{2}\|\nabla\bar{p}_{tt}\|_{L^{2}(0,t;L^{2}(\Omega))}+b\|\nabla\bar{p}_{ttt}\|_{L^{2}(0,t;L^{2}(\Omega))}\Bigr{)}$
$\displaystyle\quad+c\|\tfrac{1}{\sqrt{\alpha}}\bar{p}_{ttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}\|\tfrac{1}{\sqrt{\alpha}}a_{tt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}+\tfrac{b}{c}\|\tfrac{1}{\sqrt{\alpha}}\bar{p}_{ttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}\|\tfrac{1}{\sqrt{\alpha}}a_{ttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}$
$\displaystyle\quad+\tfrac{1}{\underline{\alpha}}\Bigl{(}{c^{2}\|\partial_{\nu}{\bar{p}_{tt}}\|_{L^{\infty}(0,t;H^{-1/2}(\Gamma_{D}))}\|g_{D\,ttt}\|_{L^{1}(0,t;H^{1/2}(\Gamma_{D}))}+b\|\partial_{\nu}{\bar{p}_{ttt}}\|_{L^{\infty}(0,t;H^{-1/2}(\Gamma_{D}))}\|g_{D\,ttt}\|_{L^{1}(0,t;H^{1/2}(\Gamma_{D}))}}\Bigr{)}$
$\displaystyle\quad+\tfrac{1}{\underline{\alpha}}\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;H^{1/2}(\Gamma_{N}))}\Bigl{(}c^{2}\|g_{N\,tt}\|_{L^{1}(0,t;H^{-1/2}(\Gamma_{N}))}+b\|g_{N\,ttt}\|_{L^{1}(0,t;H^{-1/2}(\Gamma_{N}))}\Bigr{)}$

In case $\Gamma_{D}\not=\emptyset$ we can estimate

$\|\partial_{\nu}{\bar{p}_{ttt}}\|_{L^{\infty}(0,t;H^{-1/2}(\Gamma_{D}))}\leq
C^{tr}_{-1/2}\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;H^{1})}\leq
C^{tr}_{-1/2}C^{\Omega}_{\Gamma_{D}}\Bigl{(}\|\nabla\bar{p}_{ttt}\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+\|g_{D\,ttt}\|_{L^{\infty}(0,t;L^{2}(\Gamma_{D}))}^{2}\Bigr{)}^{1/2}$

using (1.1) and employing the higher order energy estimate (5.55).
If $\Gamma_{D}=\emptyset$ an estimate of
$\|\partial_{\nu}{\bar{p}_{ttt}}\|_{L^{\infty}(0,t;H^{-1/2}(\Gamma_{D}))}$ is
anyway not needed and actually the lower order energy estimate (5.56) suffices
to obtain $p_{ttt}\in L^{1}(0,T;L^{2}(\Omega))$, which suffices for our
purposes.

We combine (5.55), (5.56), assume
$\bar{\alpha}=\|\tfrac{\alpha_{t}}{\alpha}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$
to be small enough and apply Young’s inequality as well as continuity of the
embedding $H^{1}(\Omega)\to L^{6}(\Omega)$, to arrive at the energy estimate

$\displaystyle\|\bar{p}_{ttt}\|_{L^{\infty}(0,t;H^{1}(\Omega))}^{2}+\left\|\frac{1}{\sqrt{\alpha}}\Delta\bar{p}_{tt}\right\|_{L^{\infty}(0,t;L^{2}(\Omega))}^{2}+b\left\|\frac{1}{\sqrt{\alpha}}\Delta\bar{p}_{ttt}\right\|_{L^{2}(0,t;L^{2}(\Omega))}^{2}+\|\bar{p}_{tttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}$
$\displaystyle\leq
C\Bigl{(}\|a_{ttt}\|_{L^{2}(0,t;L^{2}(\Gamma_{a}))}^{2}{+\|g_{D\,ttt}\|_{L^{\infty}(0,T;H^{1/2}(\Gamma_{D}))}^{2}+\|g_{D\,tttt}\|_{L^{1}(0,T;H^{1/2}(\Gamma_{D}))}^{2}}+\|g_{N\,ttt}\|_{W^{1,1}(0,t;H^{-1/2}(\Gamma_{N}))}^{2}\Bigr{)}$

Analogous estimates obtained by testing the original equation and its time
differentiated version with $-\frac{1}{\alpha}\Delta\bar{p}_{t}$ and
$-\frac{1}{\alpha}\Delta\bar{p}_{tt}$, respectively, yield

$\mathcal{E}^{p}_{1}[\bar{p}](t)\leq
C\Bigl{(}\mathcal{E}^{p}_{1}[\bar{p}](0)+G[a,g_{D},g_{N}]+G[a_{t},g_{D\,t},g_{N\,t}]\Bigr{)}.$
(5.57)

Likewise, with two equipartition of energy estimates obtained by testing the
original equation with $-\frac{1}{\alpha}\Delta\bar{p}$ and its time
differentiated version by $-\frac{1}{\alpha}\Delta\bar{p}_{t}$, we obtain
(3.14).

$\Box$

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