# Damped wave equations with dynamic boundary conditions

Delio Mugnolo  Institut für Angewandte Analysis, Universität Ulm,
Helmholtzstraße 18, D-89081 Ulm, Germany<EMAIL_ADDRESS>
###### Abstract.

We discuss several classes of linear second order initial-boundary value
problems, where damping terms appear in the main wave equation as well as in
the dynamic boundary condition. We investigate their well-posedness and
describe some qualitative properties of their solutions, including
boundedness, stability, or almost periodicity. In particular, we are able to
characterize the analyticity of certain $C_{0}$-semigroups associated to such
problems. Applications to several problems on domains and networks are shown,
mostly borrowed from [10, 39].

###### Key words and phrases:

Second order damped initial-boundary value problems; Operator matrices;
Dynamical or Wentzell boundary conditions; Semigroups of operators

###### 1991 Mathematics Subject Classification:

47D06,35L20

##  1\. Introduction

In recent years, wave and beam equations with dynamic boundary conditions have
been studied by many authors, see e.g. [27, 21, 3, 18, 19, 39, 40, 6, 10, 22,
31] and references therein. Wave equations with such boundary conditions are
motivated by physical models incorporating the effect of frictions, as shown
in [20]. Oscillating models involving dynamic boundary conditions for networks
or more general polygonal domains have been considered, among others, by Ali
Mehmeti, cf. [1, Chapt. 4] and references therein, as well as by
Lagnese–Leugering–Schmidt in [26, § 2.7]. In fact, our setting can be adapted
to problems on networks, interface problems and domains alike, cf. examples
below.

Most recent papers deal with abstract methods based on the theories of
operator matrices and $C_{0}$-semigroups. These theories may have advantages
over more usual methods based on Hilbert space methods and energy estimates.
They allow more flexibility in treating non-dissipative systems. While most of
the above mentioned papers only treat undamped wave equations, aim of this
paper is to apply known methods in order to investigate a class of _damped_
problems. More precisely, we consider second order problems where the damping
effect can be observed in both the waveguide and its boundary.

While the first-order counterpart of this setting, i.e., _diffusion problems_
with dynamic boundary conditions, has been often discussed both on domains and
on ramified structures due to its relations to stochastic analysis (in
particular to the theory of Feller semigroups), comparatively less attention
has been devoted to _wave equations_ with dynamic boundary conditions. Though,
the connections between wave equations with further kinds of oscillatory
boundary conditions (which are well-known in the mathematical physics of
acoustic waves and mixed water/ice systems, cf. [25, 7, 8]) and dynamic ones
has been thoroughly shown in [19]. Wave equations with different kinds of
time-dependent boundary conditions have been recently considered by Nicaise
and coauthors, cf. [37] and references therein. A numeric approach to this
class of problems based on a Trotter–Kato-type result has been proposed in
[28]. See also the conference procee! dings [2].

We stress that our theory is formulated in the abstract context of Banach
spaces, whereas waves equations are usually discussed in a Hilbert space
framework. This is due to the fact that, by a celebrated result of Littman,
cf. [29], _undamped_ wave equations are well-posed in an $L^{p}$-setting if
and only if $p=2$ or the space dimension is 1. However, more recent results
indicate that this limitation does not apply to the _damped_ case, see e.g.
[14, 15].

##  2\. Mathematical framework

######  Assumption 2.1.

We impose the following throughout this paper.

  1. (1)

$X$, $Y$, and $\partial X$ are Banach spaces such that $Y\hookrightarrow X$.

  2. (2)

${A}:D({A})\subset X\to X$ and $C:D({C})\subset X\to X$ are linear operators.

  3. (3)

$L:D(A)\cap D(C)\to{\partial X}$ is a linear and surjective operator.

  4. (4)

$B_{1}:D(A)\to\partial X$ and $B_{2}:D(C)\to\partial X$ are linear operators.

  5. (5)

$B_{3}:D(B_{3})\subset{\partial X}\to{\partial X}$ and
$B_{4}:D(B_{4})\subset{\partial X}\to{\partial X}$ are linear and closed
operators.

Functions on the main waveguide will be throughout this paper vectors in a
Banach space $X$. We introduce a complete abstract second order problem

$\ddot{u}(t)=Au(t)+C\dot{u}(t),\qquad t{\geq 0}$

and equip such a problem with second order dynamic boundary conditions
represented by an equation

(2.1) $\ddot{w}(t)=B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}w(t)+B_{4}\dot{w}(t),\qquad
t{\geq 0},$

on another Banach space $\partial X$. Here the relation between the variables
$u$ and $w$ is expressed by

$w(t)=Lu(t)\qquad\hbox{and/or by}\qquad\dot{w}(t)=L\dot{u}(t),\qquad t\geq 0,$

where $L$ is some operator from $X$ to $\partial X$. We want to investigate
(analytic) well-posedness and asymptoptic behavior of such a system. To this
aim we re-write it in an abstract form and are eventually led to the complete
second order abstract Cauchy problem

(c${\mathcal{ACP}}^{2}$)
$\left\\{\begin{array}[]{rcl}\ddot{\mathfrak{u}}(t)&=&\mathcal{A}{\mathfrak{u}}(t)+\mathcal{C}\dot{\mathfrak{u}}(t),\qquad
t\geq 0,\\\
{\mathfrak{u}}(0)&=&{\mathfrak{f}}\in{\mathcal{X}},\qquad\dot{\mathfrak{u}}(0)={\mathfrak{g}}\in{\mathcal{X}},\end{array}\right.$

on the product space

${\mathcal{X}}:=X\times\partial X.$

Here

(2.2) $\mathcal{A}:=\begin{pmatrix}A&0\\\
B_{1}&B_{3}\end{pmatrix}\qquad\hbox{and}\qquad\mathcal{C}:=\begin{pmatrix}C&0\\\
B_{2}&B_{4}\end{pmatrix}$

are operator matrices on $\mathcal{X}$, and their domains will depend on _how
unbounded the damping term $C$ is with respect to the elastic term $A$_, as we
see next. Moreover, the new variable $\mathfrak{u}$ is to be understood as

${\mathfrak{u}}(t):=\begin{pmatrix}u(t)\\\ Lu(t)\end{pmatrix},\qquad t\geq 0.$

If we reduce the second order evolution problem in $({\rm
c}\mathcal{ACP}^{2})$ to a first order abstract Cauchy problem, our goal
becomes to discuss the well-posedness of

(${\mathbb{ACP}}$)
$\left\\{\begin{array}[]{rcl}\dot{\mathbb{u}}(t)&=&{\mathbb{A}}{\mathbb{u}}(t),\qquad
t\geq 0,\\\ {\mathbb{u}}(0)&=&{\mathbb{f}},\end{array}\right.$

in some suitable phase space, where $\mathbb{A}$ is the reduction matrix
defined by

(2.3) ${\mathbb{A}}:=\begin{pmatrix}0&I_{D({\mathcal{C}})}\\\
{\mathcal{A}}&{\mathcal{C}}\end{pmatrix}$

with suitable domain, and accordingly

${\mathbb{u}}(t)\equiv\begin{pmatrix}{\mathfrak{u}}(t)\\\
\dot{\mathfrak{u}}(t)\end{pmatrix},\quad t\geq
0,\qquad{\mathbb{f}}\equiv\begin{pmatrix}{\mathfrak{f}}\\\
{\mathfrak{g}}\end{pmatrix}.$

A setting of this kind permits to treat damped wave equations on domains or,
more generally, on networks or ramified structures.

######  Example 2.2.

Consider the initial value problems associated with a one dimensional damped
plate-like equation on a bounded interval

(2.4)
$\left\\{\begin{array}[]{rcll}\ddot{u}(t,x)&=&-u^{\prime\prime\prime\prime}(t,x)+\dot{u}^{\prime\prime}(t,x),&t\geq
0,\;x\in(0,1),\\\
u^{\prime\prime}(t,j)&=&(-1)^{j}u^{\prime}(t,j)-u(t,j),&t\geq 0,\;j=0,1,\\\
\ddot{u}(t,j)&=&(-1)^{j+1}u^{\prime\prime\prime}(t,j)+(-1)^{j}u^{\prime}(t,j)\\\
&&\qquad+(-1)^{j}\dot{u}^{\prime}(t,j)-u(t,j)-\dot{u}(t,j),&t\geq
0,\;j=0,1,\end{array}\right.$

or with a strongly damped wave equation on an _open book_ consisting of $N$
copies of a domain $\Omega\subset{\mathbb{R}}^{n}$

(2.5) $\left\\{\begin{array}[]{rcll}\ddot{u}_{j}(t,x)&=&\Delta(\alpha
u_{j}+\dot{u}_{j})(t,x),&t\geq 0,\;x\in\Omega,\;j=1,\ldots,N,\\\
u_{j}(t,z)&=&u_{\ell}(t,z)=:u(t,z),&t\geq
0,\;z\in\partial\Omega,\;j,\ell=1,\ldots,N,\\\
\ddot{u}_{j}(t,z)&=&-\frac{\partial}{\partial\nu}(\beta
u_{j}+\dot{u}_{j})(t,z)+\gamma u(t,z)+\delta\dot{u}(t,z),&t\geq
0,\;z\in\partial\Omega,\;j=1,\ldots,N,\end{array}\right.$

for $\alpha,\beta,\gamma,\delta\in\mathbb{C}$. Here the second equation
prescribes continuity along the _binding_ of the book.

The initial value problems associated to both systems can be reduced to
$(c{\mathcal{ACP}}^{2})$ on the Hilbert spaces
$L^{2}(0,1)\times{\mathbb{C}}^{2}$ and $L^{2}(\Omega)\times
L^{2}(\partial\Omega)$, respectively. Certain assumptions relating boundedness
of elastic and damping term are satisfied, and by known results on damped wave
equations we deduce analytic well-posedness.∎

In most usual examples (like in (2.4) and (2.5)), known energy estimates (cf.
[18, 5]) permit to apply known results on damped second order problems – c.f.
[36, 12, 13, 38, 34], [16, § XVIII.5.1], or [17, § VI.3], for methods based on
spectral theory, functional calculus, operator matrix theory, of forms.

All these methods require certain boundedness assumptions relating $A,C$ and
$B_{3},B_{4}$. Aim of this paper is to give sufficient conditions on
$A,C,L,B_{1},B_{2},B_{3},B_{4}$ that ensure that the reduction matrix
$\mathbb{A}$ generates a $C_{0}$-semigroup, independently of dissipativeness.
The theorems we are going to present in the remainder of this paper are of the
following form:

_If the damped wave equations with homogeneous boundary conditions associated
with (suitable restrictions of) the matrix operators $\mathcal{A}$ and
$\mathcal{C}$ are well-posed, then so is $(c\mathcal{ACP}^{2})$._

We will be specially concerned with investigatig further qualitative
properties (boundedness, compactness, almost periodicity…) enjoyed by such a
semigroup. In the following we will pay special attention to second order
problems that are governed by analytic semigroups, a feature often discussed
in applications, cf. [11, § 1]. Furthermore, a parabolic character directly
implies regularity results, and can thus be exploited in order to discuss
semilinear problems, e.g. by the techniques presented in [30]. Generation of
analytic semigroups in the context of damped wave equations with dynamic
boundary conditions has also been investigated in [39, 40] by different
methods.

Let us explain the plan of this paper. We have introduced in the Assumption
2.1.(1) a Banach space $Y$. Such a space $Y$ is in common applications somehow
related to the domain of the operator $C$ and to the phase space of the second
order problem – it was fact $Y=H^{2}(0,1)\cap H^{1}_{0}(0,1)$ and
$Y=H^{1}_{0}(\Omega)$ in (2.4) and (2.5), respectively. (In fact, in concrete
cases it will be a Sobolev space of the same order of the so-called _Kisyński
space_ of the wave equation, i.e., the first factor of the phase space.)
Depending on $Y$ and on the operator $L$, we need as in [31] to distinguish
three different cases: $L$ can be

– unbounded from $Y$ to $\partial X$,

– unbounded from $X$ to $\partial X$ but bounded from $Y$ to $\partial X$, or

– bounded from $X$ to $\partial X$.

In this paper _we only consider the first two cases_ , in Sections 3 and 4,
respectively. These occur, e.g., when we consider a wave equation on an
$L^{p}$-space and $L$ is the normal derivative (see Example 3.5) or the trace
operator (see Example 4.8), respectively. Our results should be compared with
those of [10, 39, 40]. Instead, the case of $L\in{\mathcal{L}}(X,\partial X)$
is typical for spaces $X$ where the point evaluation is a bounded operator.
The strongly damped case, i.e., the case of an operator $C$ that is “more
unbounded” than $A$, is technically slightly different and will be treated in
Sections 5.

Finally, in Section 6 we prove a technical lemma on the exponential stability
of semigroups generated by operator matrices. This seems to be new and of
independent interest.

##  3\. The damped case $L\not\in{\mathcal{L}}(Y,\partial X)$

Of concern in this section are second order abstract problems with dynamic
boundary conditions of the form

(AIBPV${}_{a}^{2}$)
$\left\\{\begin{array}[]{rcll}\ddot{u}(t)&=&Au(t)+C\dot{u}(t),&t{\geq 0},\\\
\ddot{w}(t)&=&B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}w(t)+B_{4}\dot{w}(t),&t{\geq
0},\\\ w(t)&=&Lu(t),&t{\geq 0},\\\ u(0)&=&f\in X,\qquad\;\;\dot{u}(0)=g\in
X,&\\\ x(0)&=&h\in\partial X,\qquad\dot{x}(0)=j\in\partial
X.&\end{array}\right.$

######  Assumption 3.1.

We complement the Assumptions 2.1 by the following.

  1. (1)

$\begin{pmatrix}A\\\ L\end{pmatrix}:D(A)\subset X\to X\times\partial X$ is
closed.

  2. (2)

$A_{0}:=A_{|{\rm ker}(L)}$ has nonempty resolvent set.

  3. (3)

$C$ is closed, $D(A)\subset D(C)$, and $[D(C)]$ is isomorphic to $Y$.

  4. (4)

$\partial Y$ is a Banach space, $[D(B_{4})]$ is isomorphic to $\partial Y$ and
$\partial Y\hookrightarrow\partial X$.

We denote by $[D(A)_{L}]$ the Banach space obtained by endowing $D(A)$ with
the graph norm of the closed operator $A\choose L$. Observe that by the Closed
Graph Theorem the embeddings
$[D(A_{0})]\hookrightarrow[D(A)_{L}]\hookrightarrow Y$ hold. Furthermore, for
$\lambda\in\rho(A_{0})$ we consider the _Dirichlet operators_
$D^{A,L}_{\lambda}$ associated with $A,L$, cf. [9, Lemma 2.3]. Such operators
are right inverses of $L$ for all $\lambda\in\rho(A_{0})$ and are such that
$AD_{\lambda}^{A,L}=\lambda D_{\lambda}^{A,L}$. Moreover, they are linear and
bounded from $\partial X$ to $[D(A)_{L}]$, cf. [31, Lemma 3.2].

The following can be verified by a direct matrix computation.

######  Lemma 3.2.

Consider the operator

(3.1) ${\mathbb{A}}:=\begin{pmatrix}0&0&I_{Y}&0\\\ 0&0&0&I_{\partial Y}\\\
A&0&C&0\\\ B_{1}&B_{3}&B_{2}&B_{4}\end{pmatrix}.$

with domain

(3.2) $D({\mathbb{A}}):=\left\\{\begin{pmatrix}u\\\ x\\\ v\\\
y\end{pmatrix}\in D(A)\times D(B_{3})\times D(C)\times D(B_{4}):Lu=x\right\\}$

on the Banach space

(3.3) ${\mathbb{X}}:=Y\times\partial Y\times X\times\partial X.$

Then the well-posedness of the first order abstract Cauchy problem
$(\mathbb{ACP})$ on $\mathbb{X}$ is equivalent to the well-posedness of $({\rm
AIBPV}_{a}^{2})$ on $X$ and $\partial X$.

We can identify a general function
${\mathbb{u}}:{\mathbb{R}}_{+}\to{\mathbb{X}}$ by

$\mathbb{u}(t)\equiv\begin{pmatrix}u(t)\\\ x(t)\\\ v(t)\\\
y(t)\end{pmatrix},\qquad t\geq 0.$

Hence, if ${\mathbb{u}}$ is a classical solution to $(\mathbb{ACP})$, so that
${\mathbb{u}}\in C^{1}({\mathbb{R}}_{+},{\mathbb{X}})\cap
C({\mathbb{R}}_{+},[D({\mathbb{A}})])$, then moreover $\dot{u}(t)=v(t)$,
$t\geq 0$, and $v\in C^{1}({\mathbb{R}}_{+},X)$, and we conclude that $u\in
C^{2}({\mathbb{R}}_{+},X)\cap C^{1}({\mathbb{R}}_{+},Y)$.

######  Lemma 3.3.

Let $\lambda\in\rho(A_{0})$. The operator matrix
$({\mathbb{A}},D({\mathbb{A}}))$ on $\mathbb{X}$ is similar to

(3.4) ${\mathbb{G}}:=\begin{pmatrix}0&I_{Y}&\vrule
height=14.0pt,depth=5.0pt&0&-D^{A,L}_{\lambda}\\\ \vskip-3.0pt\cr
A_{0}&C&\vrule height=14.0pt,depth=5.0pt&\lambda D^{A,L}_{\lambda}&0\\\
\hrule\cr 0&0&\vrule height=14.0pt,depth=5.0pt&0&I_{\partial Y}\\\
\vskip-3.0pt\cr B_{1}&B_{2}&\vrule
height=14.0pt,depth=5.0pt&B_{3}+B_{1}D^{A,L}_{\lambda}&B_{4}\\\ \end{pmatrix}$

with domain

$D({\mathbb{G}}):=D(A_{0})\times D(C)\times D(B_{3})\times Y)$

on the Banach space

${\mathbb{Y}}:=Y\times X\times\partial Y\times\partial X.$

###### Proof.

Let $\lambda\in\rho(A_{0})$. First of all, we consider an isomorphism of
$\mathbb{X}$ onto $\mathbb{Y}$ given by

${\mathbb{U}}_{\lambda}:=\begin{pmatrix}I_{Y}&-D^{A,L}_{\lambda}&0&0\\\
0&0&I_{X}&0\\\ 0&I_{\partial Y}&0&0\\\ 0&0&0&I_{\partial X}\\\
\end{pmatrix},\quad\hbox{with
inverse}\quad{\mathbb{U}}_{\lambda}^{-1}:=\begin{pmatrix}I_{Y}&&D_{\lambda}^{A,L}&0\\\
0&0&I_{\partial Y}&0\\\ 0&I_{X}&0&0\\\ 0&0&0&I_{\partial X}\\\ \end{pmatrix}.$

We want to compute
${\mathbb{G}}={\mathbb{U}}_{\lambda}{\mathbb{A}}{\mathbb{U}}^{-1}_{\lambda}$.
Observe that

$\displaystyle D({\mathbb{G}})$ $\displaystyle=$
$\displaystyle\\{{\mathbb{x}}\in{\mathbb{X}}:{\mathbb{U}}^{-1}_{\lambda}{\mathbb{x}}\in
D({\mathbb{A}})\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{pmatrix}u\\\
v\\\ x\\\ y\end{pmatrix}\in{\mathbb{X}}:\begin{pmatrix}u+D^{A,L}_{\lambda}x\\\
x\\\ v\\\ y\end{pmatrix}\\!\\!\\!\begin{array}[]{ll}&\in D(A)\times
D(B_{3})\times D(C)\times D(B_{4})\\\ &\qquad\hbox{and
}L(u+D^{A,L}_{\lambda}x)=x\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\begin{pmatrix}u\\\ v\\\ x\\\
y\end{pmatrix}\in{\mathbb{X}}:\begin{pmatrix}u+D^{A,L}_{\lambda}x\\\ x\\\ v\\\
y\end{pmatrix}\\!\\!\\!\begin{array}[]{ll}&\in D(A)\times D(B_{3})\times
D(C)\times D(B_{4})\\\ &\qquad\hbox{and }Lu=0\end{array}\right\\}$
$\displaystyle=$ $\displaystyle D(A_{0})\times D(C)\times D(B_{3})\times
D(B_{4}).$

Moreover,

$\displaystyle{\mathbb{U}}_{\lambda}{\mathbb{A}}{\mathbb{U}}^{-1}_{\lambda}\begin{pmatrix}u\\\
v\\\ x\\\ y\end{pmatrix}$ $\displaystyle=$
$\displaystyle{\mathbb{U}}_{\lambda}{\mathbb{A}}\begin{pmatrix}u+D^{A,L}_{\lambda}x\\\
x\\\ v\\\ y\end{pmatrix}$ $\displaystyle=$
$\displaystyle{\mathbb{U}}_{\lambda}\begin{pmatrix}v\\\ y\\\ A_{0}u+\lambda
D^{A,L}_{\lambda}x+Cv\\\
B_{1}(u+D^{A,L}_{\lambda}x)+B_{3}x+B_{2}v+B_{4}y\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}v-D^{A,L}_{\lambda}y\\\
A_{0}u+\lambda D^{A,L}_{\lambda}x+Cv\\\ y\\\
B_{1}(u+D^{A,L}_{\lambda}x)+B_{3}x+B_{2}v+B_{4}y\end{pmatrix}.$

This finally shows the claimed representation of the operator matrix
${\mathbb{G}}$. ∎

We are now in the position to prove the main result of this section.

######  Theorem 3.4.

Under the Assumptions 2.1 and 3.1, let
$B_{1}\in{\mathcal{L}}([D(A)_{L}],\partial X)$. Then the following assertions
hold.

  1. (1)

Assume that $B_{1}\in{\mathcal{L}}([D(A_{0})],\partial Y)$, or else that
$B_{1}\in{\mathcal{L}}(Y,\partial X)$, and moreover that
$B_{2}\in{\mathcal{L}}(Y,\partial Y)$, or else that
$B_{2}\in{\mathcal{L}}(X,\partial X)$. Then the operator matrix $\mathbb{A}$
generates a $C_{0}$-semigroup on $\mathbb{X}$ if and only if

(3.7) $\begin{pmatrix}0&I_{Y}\\\ A_{0}&C\end{pmatrix}\quad\hbox{with
domain}\quad D({A}_{0})\times Y$

and

(3.8) $\begin{pmatrix}0&I_{\partial Y}\\\
B_{3}&B_{4}\end{pmatrix}\quad\hbox{with domain}\quad D(B_{3})\times D(B_{4})$

generate $C_{0}$-semigroups on $Y\times X$ and $\partial Y\times\partial X$,
respectively.

  2. (2)

Let $B_{2}\in{\mathcal{L}}(Y,\partial X)$. Then the reduction matrices
introduced in (3.7)–(3.8) both generate analytic semigroups if and only if
$\mathbb{A}$ generates an analytic semigroup.

  3. (3)

Let $B_{1}=B_{2}=0$. If the semigroup generated by either of the matrices
defined in (3.7)–(3.8) is bounded and the other one is uniformly exponentially
stable, then the semigroup generated by ${\mathbb{A}}$ is bounded.

  4. (4)

Let $B_{1}=B_{2}=0$. Assume the semigroups generated by matrices in
(3.7)–(3.8) to be bounded. Let further the semigroup generated by the matrix
in (3.7) be analytic. If the matrices in (3.7)–(3.8) have no common purely
imaginary spectral values, then the semigroup generated by ${\mathbb{A}}$ is
bounded.

  5. (5)

Let $B_{1}\in{\mathcal{L}}(Y,\partial X)$ and
$B_{2}\in{\mathcal{L}}(X,\partial X)$. If both semigroups generated by
matrices in (3.7)–(3.8) are uniformly exponentially stable, then there exists
$\epsilon>0$ such that also the semigroup generated by ${\mathbb{A}}$ is
uniformly exponentially stable, provided that
$\|B_{1}\|_{{\mathcal{L}}(Y,\partial X)}+\|B_{2}\|_{{\mathcal{L}}(X,\partial
X)}<\epsilon$.

  6. (6)

The operator matrix $\mathbb{A}$ on $\mathbb{X}$ has compact resolvent if and
only if all the embeddings $[D({A}_{0})]\hookrightarrow Y\hookrightarrow X$
and $[D(B_{3})]\hookrightarrow\partial Y\hookrightarrow\partial X$ are
compact.

###### Proof.

Let $\lambda\in\rho(A_{0})$. By Lemma 3.3 instead of $\mathbb{A}$ on
$\mathbb{X}$ it suffices to investigate the similar operator matrix
$\mathbb{G}$ on $\mathbb{Y}$. We consider $\mathbb{G}$ as a $2\times 2$
operator matrix with diagonal domain. More precisely,

${\mathbb{G}}={\mathbb{G}}_{0}+{\mathbb{G}}_{1}:=\begin{pmatrix}{\bf A}&0\\\
{\bf C}&{{\bf D}}\end{pmatrix}+\begin{pmatrix}0&{\bf B}\\\
0&0\end{pmatrix},\qquad D(\mathbb{G})=D({\bf A})\times D({{\bf D}}),$

where the $2\times 2$ block-entries ${\bf A},{\bf B},{\bf C},{\bf D}$ are
defined as in (3.4). Also observe that by assumption the operator
$B_{1}D^{A,L}_{\lambda}$ is bounded from $\partial Y$ to $\partial X$ for all
$\lambda\in\rho(A_{0})$, so that we can discuss the generator property of the
reduction matrix introduced in (3.8) instead of $\bf D$. By the Assumptions
3.1 the Dirichlet operator $D^{A,L}_{\lambda}$ is bounded from $\partial X$ to
$Y$, hence the block entry $\bf B$ is bounded from $\partial Y\times\partial
X$ to $Y\times X$. Thus, by the bounded perturbation theorem we only have to
care about the lower triangular operator matrix ${\mathbb{G}}_{0}$.

(1) The off-diagonal block-entry $\bf C$ is bounded from $[D({\bf A})]$ to
$[D({{\bf D}})]$ or from $Y\times X$ to $\partial Y\times\partial X$. It
follows by a perturbation result by Desch–Schappacher or by the bounded
perturbation theorem, respectively, that ${\mathbb{G}}_{0}$ generates a
$C_{0}$-semigroup on $\mathbb{Y}$ if and only if both diagonal block-entries
${\bf A},{\bf D}$ of ${\mathbb{G}}_{0}$ generate $C_{0}$-semigroups on
$Y\times X$ and on $\partial Y\times\partial X$, respectively.

(2) The diagonal block-entries of ${\mathbb{G}}_{0}$ both generate analytic
semigroups. Moreover, the off-diagonal entry $\bf B$ is bounded from $[D({\bf
A})]$ to $\partial Y\times\partial X$. It follows by [35, Cor. 3.3] that
${\mathbb{G}}_{0}$ generates an analytic semigroup on $\mathbb{Y}$.

(3)–(4)–(5) These assertions follow directly from Proposition 6.1 below.

(6) Since $D({\mathbb{G}})=D({\bf A})\times D({\bf D})$, $\mathbb{G}$ has
compact resolvent if and only its diagonal block-entries have compact
resolvent. ∎

######  Example 3.5.

We discuss the initial value problem associated with

$\left\\{\begin{array}[]{rcll}\ddot{u}(t,x)&=&-\Delta^{2}u(t,x)+\Delta\dot{u}(t,x),&t{\geq
0},\;x\in\Omega\\\ \ddot{w}(t,z)&=&q_{1}(z)\Delta
u(t,z)+q_{2}(z)\dot{u}(t,z)\\\
&&\qquad-\Delta_{\partial\Omega}^{2}w(t,z)-w(t,z)+\Delta_{\partial\Omega}\dot{w}(t,z)-\dot{w}(t,z),&t{\geq
0},\;z\in\partial\Omega,\\\ w(t,z)&=&\frac{\partial\Delta
u}{\partial\nu}(t,z)-p(z)\Delta u(t,z),&t{\geq 0},\;z\in\partial\Omega,\\\
\frac{\partial{u}}{\partial\nu}(t,z)&=&p(z){u}(t,z),&t{\geq
0},\;z\in\partial\Omega,\\\
\frac{\partial\dot{u}}{\partial\nu}(t,z)&=&p(z)\dot{u}(t,z),&t{\geq
0},\;z\in\partial\Omega,\end{array}\right.$

similar to that discussed in [40, Exa. 4.3]. Here
$\Omega\subset{\mathbb{R}}^{n}$ is a bounded open domain with smooth boundary
$\partial\Omega$ and $p,q_{1},q_{2}\in L^{\infty}(\partial\Omega)$, $p<0$.
Observe that – whenever re-written as $({\rm c}\mathcal{ACP}^{2})$ – the
operator matrix $\mathcal{C}$ is in general neither self-adjoint, nor strictly
negative definite, and $\mathcal{A}\not=-{\mathcal{C}}^{2}$, thus it is not
possible to directly apply the results presented in [13], [16, § XVIII.5.1],
or [38, § 6.4].

In order to apply the results presented above, we consider

$Y:=\left\\{u\in H^{2}(\Omega):\frac{\partial
u}{\partial\nu}=pu_{|\partial\Omega}\right\\},\quad
X:=L^{2}(\Omega),\quad\partial Y:=H^{2}(\partial\Omega),\quad\partial
X:=L^{2}(\partial\Omega),$

and further

$C:=\Delta,\qquad D(C):=Y,$

i.e., $C$ is the Laplacian with Robin boundary conditions, and

$A:=-\Delta^{2},\qquad D(A):=\left\\{u\in
H^{\frac{7}{2}}(\Omega):\frac{\partial
u}{\partial\nu}=pu_{|\partial\Omega}\right\\}\subset D(C).$

Let

$Lu(z):=\frac{\partial\Delta u}{\partial\nu}(z)-p(z)\Delta
u(z)\qquad\hbox{for}\;u\in H^{\frac{7}{2}}(\Omega),\;z\in\partial\Omega.$

Such an operator is well defined in the sense of traces. Then by usual
boundary regularity results one sees that $-A_{0}=-A_{|\ker(L)}$ is the square
of $C$. The operator $C$ is self-adjoint and strictly negative definite, and
we obtain by [38, Thm. 6.4.3 and Thm. 6.4.4] that the operator matrix defined
in (3.7) generates an analytic, uniformly exponentially stable semigroup on
$Y\times X$. Let now

$(B_{1}u)(z):=q_{1}(z)\Delta u(z)\quad(B_{2}u)(z):=q_{2}(z)u(z),\qquad u\in
H^{2}(\Omega),\;z\in\partial\Omega.$

It is clear that $B_{1},B_{2}$ are bounded from $Y$ to $\partial X$ and from
$X$ to $\partial X$, respectively. Consider moreover the operators $B_{3}$ and
$B_{4}$ defined by

$\begin{array}[]{lll}&B_{3}:=-\Delta^{2}_{\partial\Omega}-I,&D(B_{3}):=H^{4}(\partial\Omega),\\\
&B_{4}:=\Delta_{\partial\Omega}-I,&D(B_{4}):=H^{2}(\partial\Omega),\end{array}$

where $\Delta_{\partial\Omega}$ denotes the Laplace–Beltrami operator, which
is self-adjoint and negative definite. By [13, Thm. 1.1] the operator matrix
defined in (3.8) generates a uniformly exponentially stable analytic semigroup
on $\partial Y\times\partial X$. If $\|q_{1}\|_{\infty}+\|q_{2}\|_{\infty}\to
0$, then by Theorem 3.4.(5) the solution $u$ converges to 0 in the energy
norm. By Theorem 3.4.(4), the semigroup governing the problem is also compact.
∎

##  4\. The damped case $L\in{\mathcal{L}}(Y,\partial X)$

The case of $L\in{\mathcal{L}}(Y,\partial X)$ introduces some technical
difficulties. In particular, we will show that our damped wave equations is
well-posed on a phase space that is _not_ a product space (as
$\mathbb{X}:=Y\times\partial Y\times X\times\partial X$ in Section 3 indeed
was). We thus slightly modify our setting.

######  Assumption 4.1.

We complement the Assumptions 2.1 by the following.

  1. (1)

$V$ is a Banach space such that $V\hookrightarrow Y$.

  2. (2)

$L$ can be extended to an operator that is bounded from $Y$ to $\partial X$,
which we denote again by $L$, and such that $\ker(L)=V$.

  3. (3)

$\begin{pmatrix}A\\\ L\end{pmatrix}:D(A)\subset X\to X\times\partial X$ is
closed.

  4. (4)

${A}_{0}:=A|_{D(A)\cap\ker(L)}$ has nonempty resolvent set.

  5. (5)

$C$ is bounded from $[D(A)_{L}]$ to $X$.

  6. (6)

$B_{1},B_{2}$ are bounded from $[D(A)_{L}]$ to $\partial X$.

  7. (7)

$B_{3}$ is bounded on $\partial X$.

######  Remark 4.2.

If $u$ is a classical solution to $({\rm AIBVP}^{2}_{a})$, then $u\in
C^{1}({\mathbb{R}}_{+},Y)$. Since by Assumption 4.1.(2) the operator $L$ is
bounded from $Y$ to $\partial X$, we obtain that $L$ and the derivation with
respect to time commute. In other words, if $w(t)=Lu(t)$ holds for all $t\geq
0$, then also $\dot{w}(t)=L\dot{u}(t)$ holds for all $t\geq 0$.

Thus, we are led to consider in this section a modified version of $({\rm
AIBVP}^{2}_{a})$, namely

(AIBPV${}_{b}^{2}$)
$\left\\{\begin{array}[]{rcll}\ddot{u}(t)&=&Au(t)+C\dot{u}(t),&t{\geq 0},\\\
\ddot{w}(t)&=&B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}w(t)+B_{4}\dot{w}(t),&t{\geq
0},\\\ w(t)&=&Lu(t),\qquad\dot{w}(t)=L\dot{u}(t),&t{\geq 0},\\\ u(0)&=&f\in
X,\qquad\;\;\dot{u}(0)=g\in X,&\\\ x(0)&=&h\in\partial
X,\qquad\dot{x}(0)=j\in\partial X.&\end{array}\right.$

As before, we can perform a first order reduction of such a problem, re-
writing it as $({\mathbb{ACP}})$.

We investigate $(\mathbb{ACP})$ on the non-diagonal Banach space
$\underline{\mathbb{X}}$ defined by

$\underline{\mathbb{X}}:=\left\\{\begin{pmatrix}u\\\ x\end{pmatrix}\in
Y\times\partial X:Lu=x\right\\}\times X\times\partial X\subset{\mathbb{X}},$

instead of $\mathbb{X}$ as in Section 3. A general function
${\mathbb{u}}:\mathbb{R}_{+}\to\underline{\mathbb{X}}$ is of the form

$\mathbb{u}(t)\equiv\begin{pmatrix}u(t)\\\ Lu(t)\\\ v(t)\\\
y(t)\end{pmatrix},\qquad t\geq 0.$

Observe that if $\mathbb{u}(\cdot)$ is a classical solution to the problem
$(\mathbb{ACP})$ in $\underline{\mathbb{X}}$, then by definition we obtain

$\frac{du}{dt}(\cdot)=v(\cdot)\qquad\hbox{and}\qquad\frac{d(Lu)}{dt}(\cdot)=y(\cdot)$

Again because $L$ and $\frac{d}{dt}$ commute for $u\in
C^{1}({\mathbb{R}}_{+},Y)$, we thus conclude that

$Lv(\cdot)=y(\cdot).$

######  Proposition 4.3.

Consider the operator matrix $\underline{\mathbb{A}}$ on
$\underline{\mathbb{X}}$ defined by

(4.1) $\underline{\mathbb{A}}=\begin{pmatrix}0&0&I_{Y}&0\\\ 0&0&0&I_{\partial
X}\\\ A&0&C&0\\\ B_{1}&B_{3}&B_{2}&B_{4}\end{pmatrix}.$

with domain

(4.2) $D(\underline{\mathbb{A}})=\left\\{\begin{pmatrix}u\\\ x\\\ v\\\
y\end{pmatrix}\in D(A)\times\partial X\times D(C)\times
D(B_{4}):Lu=x,\;Lv=y\right\\}.$

Then $\mathbb{u}\in C^{1}({\mathbb{R}}_{+},\underline{\mathbb{X}})$ is a
classical solution to the initial-value problem associated with

$\dot{\mathbb{u}}(t)={\mathbb{A}}\mathbb{u}(t),\qquad t{\geq 0},$

if and only if it is a classical solution to the initial-value problem
associated with

$\dot{\mathbb{u}}(t)=\underline{\mathbb{A}}\mathbb{u}(t),\qquad t{\geq 0}.$

######  Corollary 4.4.

The well-posedness (in the classical sense) of the first order abstract Cauchy
problem

($\underline{\mathbb{ACP}}$)
$\left\\{\begin{array}[]{rcl}\dot{\mathbb{u}}(t)&=&\underline{\mathbb{A}}{\mathbb{u}}(t),\qquad
t\geq 0,\\\ {\mathbb{u}}(0)&=&{\mathbb{f}},\end{array}\right.$

on $\mathbb{X}$ is equivalent to the well-posedness (in the classical sense)
of the $({\rm AIBPV}_{b}^{2})$ on $X$ and $\partial X$.

The following can be shown essentially like in the proof of [31, Lemma 5.2].
We denote by $D_{\lambda}^{A,L}$ the Dirichlet operators associated with $A,L$
introduced in Section 3.

######  Lemma 4.5.

The product Banach space

$\underline{\mathbb{V}}:=V\times X\times\partial X\times\partial X$

is isomorphic to $\underline{\mathbb{X}}$ via

$\underline{\mathbb{U}}_{\lambda}:=\begin{pmatrix}I_{Y}&-D_{\lambda}^{A,L}&0&0\\\
0&0&I_{X}&-D_{\lambda}^{A,L}\\\ 0&I_{\partial X}&0&0\\\ 0&0&0&I_{\partial
X}\\\ \end{pmatrix},\qquad\lambda\in\rho(A_{0}).$

The inverse of $\underline{\mathbb{U}}_{\lambda}$ is the operator matrix

$\underline{\mathbb{U}}^{-1}_{\lambda}:=\begin{pmatrix}I_{V}&&D_{\lambda}^{A,L}&0\\\
0&0&I_{\partial X}&0\\\ 0&I_{X}&0&D_{\lambda}^{A,L}\\\ 0&0&0&I_{\partial X}\\\
\end{pmatrix},\qquad\lambda\in\rho(A_{0}).$

In the remainder of this section we hence take $\lambda\in\rho(A_{0})$ and
investigate properties of the similar operator matrix
$\underline{\mathbb{U}}_{\lambda}\underline{\mathbb{A}}\underline{\mathbb{U}}^{-1}_{\lambda}$
on the product space $\mathbb{V}$.

A tedious but direct matrix computation, similar to that performed in the
proof of Lemma 3.3, yields the following.

######  Lemma 4.6.

Let $\lambda\in\rho(A_{0})$. Then the operator matrix $\underline{\mathbb{A}}$
on $\underline{\mathbb{X}}$ defined in (4.1)–(4.2) is similar to

(4.3) $\underline{\mathbb{G}}:=\begin{pmatrix}0&I_{V}&\vrule
height=14.0pt,depth=5.0pt&(*)\\\ \vskip-3.0pt\cr
A_{0}-D^{A,L}_{\lambda}B_{1}&C_{0}-D^{A,L}_{\lambda}B_{2}&\vrule
height=14.0pt,depth=5.0pt&&\\\ \hrule\cr 0&0&\vrule
height=14.0pt,depth=5.0pt&0&I_{\partial X}\\\ \vskip-3.0pt\cr
B_{1}&B_{2}&\vrule
height=14.0pt,depth=5.0pt&B_{3}+B_{1}D_{\lambda}^{A,L}&B_{4}+B_{2}D_{\lambda}^{A,L}\\\
\end{pmatrix}$

with domain

$D(\underline{\mathbb{G}}):=D(A_{0})\times D(C_{0})\times\partial X\times
D(B_{4})$

on the Banach space $\underline{\mathbb{V}}$. Here the upper-right block entry
$(*)$ is given by

$(*)=\begin{pmatrix}0&0\\\
-D_{\lambda}^{A,L}(B_{1}D_{\lambda}^{A,L}+B_{3})&-D_{\lambda}^{A,L}(B_{2}D_{\lambda}^{A,L}+B_{4})+(C-\lambda)D_{\lambda}^{A,L}\end{pmatrix}.$

The similarity transformation is performed by means of the operator matrix
$\underline{\mathbb{U}}_{\lambda}$ introduced in Lemma 4.5.

Observe that if $B_{4}\in{\mathcal{L}}(\partial X)$, then the lower-right
entry

(4.4) $\begin{pmatrix}0&I_{\partial X}\\\
B_{3}+B_{1}D_{\lambda}^{A,L}&B_{4}+B_{2}D_{\lambda}^{A,L}\end{pmatrix}$

in (4.3) is by assumption a bounded operator on $\partial X\times\partial X$.
The following parallels parallels Theorem 3.4.

######  Theorem 4.7.

Under the Assumptions 2.1 and 4.1 the following assertions hold.

  1. (1)

Let $B_{4}\in{\mathcal{L}}(\partial X)$. Then the operator matrix
$\underline{\mathbb{A}}$ generates a $C_{0}$-semigroup (resp., an analytic
semigroup) on $\underline{\mathbb{X}}$ if and only if

(4.5) $\begin{pmatrix}0&I_{V}\\\
{A}_{0}-D^{A,L}_{\lambda}B_{1}&C_{0}-D^{A,L}_{\lambda}B_{2}\end{pmatrix}\quad\hbox{with
domain}\quad D({A}_{0})\times V$

generates a $C_{0}$-semigroup (resp., an analytic semigroup) on $V\times X$
for some $\lambda\in\rho(A_{0})$.

  2. (2)

Let $B_{1}=B_{2}=0$ and $D_{\lambda}^{A,L}B_{4}\in{\mathcal{L}}(\partial X)$
for some $\lambda\in\rho(A_{0})$. If the semigroup generated by either of the
matrices defined in (4.4)–(4.5) is bounded and the other one is uniformly
exponentially stable, then the semigroup generated by $\underline{\mathbb{A}}$
is bounded.

  3. (3)

Let $B_{1}=B_{2}=0$. Assume the semigroups generated by matrices in
(4.4)–(4.5) to be bounded. Let further the semigroup generated by the matrix
in (4.4) be analytic. If the matrices in (4.4)–(4.5) have no common purely
imaginary spectral values, then the semigroup generated by ${\mathbb{A}}$ is
bounded.

  4. (4)

Let $B_{1}\in{\mathcal{L}}(V,\partial X)$ and
$B_{2}\in{\mathcal{L}}(X,\partial X)$. Assume both semigroups generated by
matrices in (4.4)–(4.5) to be uniformly exponentially stable. Then there is
$\epsilon>0$ such that the semigroup generated by $\underline{\mathbb{A}}$ is
uniformly exponentially stable whenever $\|B_{1}\|+\|B_{2}\|<\epsilon$.

  5. (5)

The operator matrix $\underline{\mathbb{A}}$ on $\underline{\mathbb{X}}$ has
compact resolvent if and only if both the embeddings
$[D({A}_{0})]\hookrightarrow V\hookrightarrow X$ are compact and ${\rm
dim}\;\partial X<\infty$.

###### Proof.

By Lemma 4.6 $\underline{\mathbb{A}}$ is a generator on
$\underline{\mathbb{X}}$ if and only if $\underline{\mathbb{G}}$ is a
generator on $\underline{\mathbb{Y}}$. Hence the operator matrix with diagonal
domain $\underline{\mathbb{G}}$ can be studied by means of the results in [35,
§ 3]. Observe that the upper-right block entry $(*)$ of
$\underline{\mathbb{G}}$ is a bounded operator from $\partial X\times\partial
X$ to $V\times X$, by assumption.

(1) Since $C\in{\mathcal{L}}([D(A)_{L}],X)$, the upper-right block-entry of
(4.3) is a bounded operator from $\partial X\times\partial X$ to $V\times X$.
Also the lower-left block-entry is bounded from $D(A_{0})\times V$ to
$\partial X\times\partial X$, and the claim follows by [35, Cor. 3.2].

(3)–(4)–(5) The claims follow by Proposition 6.1 below.

(6) The operator matrix $\underline{\mathbb{G}}$ has compact resolvent if and
only if its domain is compactly embedded in $\underline{\mathbb{X}}$, i.e., if
the embedding
$[D(\underline{\mathbb{G}})]=[D(A_{0})]\times[D(C_{0})]\times[\partial
X]\times[D(B_{4})]\hookrightarrow V\times X\times\partial X\times\partial X$
is compact. ∎

######  Example 4.8.

We discuss the initial value problem associated with

$\left\\{\begin{array}[]{rcll}\ddot{u}_{j}(t,x)&=&u^{\prime\prime}_{j}(t,x),&t\in{\mathbb{R}},\;x\in(0,1),\;j=1,\ldots,E,\\\
u_{j}(t,\mathsf{v}_{i})&=&u_{\ell}(t,\mathsf{v}_{i})=:d^{u}_{i}(t),&t\in\mathbb{R},\;j,\ell=1,\ldots,E,\;i=1\ldots,V,\\\
\ddot{u}(t,i)&=&\sum_{j=1}p_{ih}\phi_{hj}\frac{\partial
u_{j}}{\partial\nu}(t,h)+\sum_{h=1}^{V}m_{ih}d^{u}_{h}(t)+\sum_{h=1}^{V}n_{ih}d^{\dot{u}}_{h}(t),&t\in{\mathbb{R}},\;i=1,\ldots,V,\\\
\end{array}\right.$

on a network $G$ with $E$ edges and $V$ vertices. Here $M=(m_{ih})$,
$N=(n_{ih})$, and $P=(p_{ih})$ are $V\times V$ matrices. (We refer to [24, 33]
for the graph-theoretical notation as well as for references to this kind of
problems.) Let

$Y:=(W^{1,p}(0,1))^{E}\cap C(G),\quad
X:=(L^{p}(0,1))^{E},\quad\hbox{and}\quad\partial X:={\mathbb{C}}^{V},$

for any $1\leq p<\infty$. Furthermore, we set

$Au:=u^{\prime\prime},\qquad\hbox{for all}\;u\in D(A):=(W^{2,p}(0,1))^{E}\cap
C(G),\qquad C:=0.$

Thus, the damping effect only appears in the boundary conditions. Moreover we
consider an operator

$(B_{1}u)_{i}:=\sum_{j=1}\phi_{ij}\frac{\partial
u_{j}}{\partial\nu}(t,i)\qquad\hbox{for all}\;u\in D(B_{1}):=D(A)$

of Kirchhoff-type, and

$B_{2}:=0,\qquad B_{3}:=M,\qquad B_{4}:=N.$

Let

$Lu:=d^{u},\qquad D(L):=Y,$

so that

$V=(W^{1,p}_{0}(0,1))^{E}\qquad\hbox{and}\qquad D(A_{0})=(W^{2,p}(0,1)\cap
W^{1,p}_{0}(0,1))^{E},$

i.e., $A_{0}$ can be seen as a diagonal operator matrix consisting of second
derivatives on $E$ unconnected intervals, each equipped with Dirichlet
boundary conditions. Following the proof of [31, Prop. 7.1] one can show that
$A_{0}-D^{A,L}_{\lambda}B_{1}$ generates a cosine operator function on $X$ for
some $\lambda\in\rho(A_{0})$, hence the operator matrix in (4.5) generates a
$C_{0}$-group. Thus, $\underline{\mathbb{A}}$ generates a $C_{0}$-group on
$\underline{\mathbb{X}}$. This system is a generalisation of that considered
in [10, § 2] for $E=1$, $V=2$, and $B_{1}=B_{3}=0$.

Let $P=0$ and $M,N$ be negative definite. Then the matrix defined in (4.4) has
negative spectrum, hence it generates a semigroup that is uniformly
exponentially stable. Since moreover the group generated by the matrix in
(4.5) is bounded, we conclude by Theorem 4.7.(2) that the solution to the
problem is bounded and asymptotically almost periodic (for positive time). In
particular, the problem admits a unique classical (backward as well as
forward) solution.∎

We may sometimes interpret our dynamic boundary conditions as Wentzell-type
ones.

######  Proposition 4.9.

Let $\underline{\mathbb{A}}$ generate an analytic semigroup on $\mathbb{X}$.
Then the solution $u$ to $({\rm AIBVP}^{2}_{b})$ satisfies the abstract
Wentzell-type boundary conditions

(4.6)
$L\left(Au(t)+C\dot{u}(t)\right)=B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}Lu(t)+B_{4}L\dot{u}(t),\qquad
t>0.$

If further $C$ maps $D(A)$ into $Y$, then $u$ satisfies in fact

$LAu(t)+LC\dot{u}(t)=B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}Lu(t)+B_{4}L\dot{u}(t),\qquad
t>0.$

###### Proof.

By assumption $(\mathbb{ACP})$ is governed by an analytic semigroup, thus for
all initial data $\mathbb{f}\in\underline{\mathbb{X}}$ the orbit
${\mathbb{u}}(\cdot):=e^{\cdot\mathbb{A}}{\mathbb{f}}$ is of class
$C^{\infty}\big{(}(0,\infty);[D({\mathbb{A}})]\big{)}$. In particular, taking
the first coordinate $u$ of $\mathbb{u}$ and recalling that $u$ is by
definition the solution to $({\rm AIBVP}^{2}_{b})$, we deduce that

(4.7) $Au(t)+C\dot{u}(t)=\ddot{u}(t)\in D(A)\qquad\hbox{for all }t>0.$

By Assumption 2.1.(3) we can apply the operator $L$ to $Au(t)+C\dot{u}(t)$,
$t>0$. By Remark 4.2 $L$ commutes with the derivation with respect to time, so
that

(4.8)
$\ddot{w}(t)=L\ddot{u}(t)=L\big{(}Au(t)+C\dot{u}(t)\big{)}\qquad\hbox{for all
}t>0.$

Plugging (4.8) into (2.1) we finally obtain (4.6).

Let now $C$ map $D(A)$ into $Y$. Since also $\dot{u}(t)\in D(A)$, $t>0$, we
obtain that $C\dot{u}(t)\in Y$, $t>0$, and we conclude that

$Au(t)=\ddot{u}(t)-C\dot{u}(t)\in Y,\qquad t>0.$

Summing up, we can apply $L$ to each addend on the LHS of (4.7). ∎

######  Example 4.10.

We revisit the first system considered in Example 2.2. The associated initial
value problem system is governed by an analytic semigroup and Proposition
4.9.(4) applies. The solution $u$ satisfies $\frac{\partial^{k}u}{\partial
t^{k}}(t,\cdot)\in D(A)=H^{4}(0,1)$ for all $t>0$ and $k\in\mathbb{N}$, and in
particular $\dot{u}(t,\cdot),\ddot{u}(t,\cdot)\in H^{4}(0,1)$, $t>0$. If
follows that $\dot{u}^{\prime\prime}(t,\cdot)\in H^{2}(0,1)$ and
$u^{\prime\prime\prime\prime}(t,\cdot)=\dot{u}^{\prime\prime}(t,\cdot)-\ddot{u}(t,\cdot)\in
H^{2}(0,1)$ for all $t>0$.

Thus, for $t>0$ we can evaluate $u^{\prime\prime\prime\prime}(t,\cdot)$ and
$\dot{u}^{\prime\prime}(t,\cdot)$ at the endpoints of the interval $[0,1]$. We
conclude that the solution to the initial value problem associated with (2.4)
also satisfies

$\begin{array}[]{rl}&u^{\prime\prime\prime\prime}(t,j)-\dot{u}^{\prime\prime}(t,j)+(-1)^{j+1}u^{\prime\prime\prime}(t,j)+(-1)^{j}u^{\prime}(t,j)\\\
&\qquad\qquad+(-1)^{j}\dot{u}^{\prime}(t,j)-u(t,j)-\dot{u}(t,j)=0,\qquad
j=0,1,\;t>0,\end{array}$

a Wentzell-type boundary condition. ∎

##  5\. The strongly damped case

In this section we discuss the problem in a strongly damped setting, i.e., we
assume that $C$ is “more unbounded” than $A$, and modify our assumptions
accordingly. We treat both case $L\not\in{\mathcal{L}}(Y,\partial X)$ and
$L\in{\mathcal{L}}(Y,\partial X)$

More precisely, we consider a complete second order abstract initial-boundary
value problems with dynamic boundary conditions of the form

(AIBPV${}_{c}^{2}$)
$\left\\{\begin{array}[]{rcll}\ddot{u}(t)&=&Au(t)+C\dot{u}(t),&t{\geq 0},\\\
\ddot{w}(t)&=&B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}w(t)+B_{4}\dot{w}(t),&t{\geq
0},\\\ \dot{w}(t)&=&L\dot{u}(t),&t{\geq 0},\\\ u(0)&=&f\in
X,\qquad\;\;\dot{u}(0)=g\in X,&\\\ x(0)&=&h\in\partial
X,\qquad\dot{x}(0)=j\in\partial X.&\end{array}\right.$

Observe that the coupling relation expressed by the third equation is not the
same of $({\rm AIBPV}_{a}^{2})$ or $({\rm AIBPV}_{b}^{2})$.

######  Assumption 5.1.

We complement the Assumptions 2.1 by the following.

  1. (1)

$\begin{pmatrix}C\\\ L\end{pmatrix}:D(C)\subset X\to X\times\partial X$ is
closed.

  2. (2)

$C_{0}:=C_{|{\rm ker}(L)}$ has nonempty resolvent set.

  3. (3)

$A$ is closed, $D(C)\subset D(A)$, and $[D(A)]$ is isomorphic to $Y$.

  4. (4)

$\partial Y$ is a Banach space such that $[D(B_{4})]\hookrightarrow\partial
Y\hookrightarrow\partial X$.

As in Section 3, we denote by $[D(C)_{L}]$ the Banach space obtained by
endowing $D(C)$ with the graph norm of the $C\choose L$, and for
$\lambda\in\rho(C_{0})$ we consider the Dirichlet operators
$D^{C,L}_{\lambda}$ associated with $C,L$, which are bounded from $\partial X$
to $[D(C)_{L}]$. By the Closed Graph Theorem we further have that
$[D(C)_{L}]\hookrightarrow Y$.

######  Lemma 5.2.

Define the linear space

(5.1) $D_{d}({\mathbb{A}}):=\left\\{\begin{pmatrix}u\\\ x\\\ v\\\
y\end{pmatrix}\in D(A)\times D(B_{3})\times D(C)\times
D(B_{4}):Lv=y\right\\}.$

Consider the operator $\mathbb{A}$ with domain $D_{d}({\mathbb{A}})$ on the
Banach space $\mathbb{X}$, where $\mathbb{A}$ and $\mathbb{X}$ are defined as
in (3.1) and (3.3). Then the well-posedness of the first order abstract Cauchy
problem $(\mathbb{ACP})$ on $\mathbb{X}$ is equivalent to the well-posedness
of $({\rm AIBPV}_{c}^{2})$ on $X$ and $\partial X$.

With a proof similar to that of Lemma 3.3, one can see that the following
holds.

######  Lemma 5.3.

Let $\lambda\in\rho(C_{0})$. Then the operator matrix
$({\mathbb{A}},D_{d}({\mathbb{A}})$ on $\mathbb{X}$ is similar to

(5.2) ${\mathbb{H}}:=\begin{pmatrix}0&I_{Y}&\vrule
height=14.0pt,depth=5.0pt&0&-D^{C,L}_{\lambda}\\\ \vskip-3.0pt\cr
A-D^{C,L}_{\lambda}B_{1}&C_{0}-D^{C,L}_{\lambda}B_{2}&\vrule
height=14.0pt,depth=5.0pt&-D^{C,L}_{\lambda}B_{3}&D^{C,L}_{\lambda}(\lambda-
B_{3}D^{C,L}_{\lambda}-B_{4})\\\ \hrule\cr 0&0&\vrule
height=14.0pt,depth=5.0pt&0&I_{\partial Y}\\\ \vskip-3.0pt\cr
B_{1}&B_{2}&\vrule
height=14.0pt,depth=5.0pt&B_{3}&B_{4}+B_{2}D^{C,L}_{\lambda}\\\ \end{pmatrix}$

with domain

$D({\mathbb{H}}):=D(A)\times D(C_{0})\times D(B_{3})\times D(B_{4})$

on the Banach space

${\mathbb{Y}}:=Y\times X\times\partial Y\times\partial X.$

The similarity transformation is performed by means of the operator matrix

${\mathbb{V}}_{\lambda}:=\begin{pmatrix}I_{Y}&0&0&0\\\
0&0&I_{X}&-D^{C,L}_{\lambda}\\\ 0&I_{\partial Y}&0&0\\\ 0&0&0&I_{\partial
X}\\\ \end{pmatrix},$

which is an isomorphism from $\mathbb{X}$ onto $\mathbb{Y}$, for any
$\lambda\in\rho(C_{0})$.

######  Theorem 5.4.

Under the Assumptions 2.1 and 5.1 the following assertions hold.

  1. (1)

Assume that $B_{1}\in{\mathcal{L}}(Y,\partial X)$, and moreover that
$B_{2}\in{\mathcal{L}}([D(C_{0})],[D(B_{4})])\cap{\mathcal{L}}([D(C)_{L}],\partial
X)$ or else $B_{2}\in{\mathcal{L}}(X,\partial X)$. If
$D^{C,L}_{\lambda}B_{3}\in{\mathcal{L}}(\partial Y,X)$ and
$D^{C,L}_{\lambda}B_{4}\in{\mathcal{L}}(\partial X,X)$ for some
$\lambda\in\rho(C_{0})$, then $\mathbb{A}$ generates a $C_{0}$-semigroup on
$\mathbb{X}$ if and only if both $C_{0}-D^{C,L}_{\lambda}B_{2}$ and

(5.3) $\begin{pmatrix}0&I_{\partial Y}\\\ B_{3}&B_{4}\end{pmatrix}$

generate $C_{0}$-semigroups on $X$ and $\partial Y\times\partial X$,
respectively.

  2. (2)

Let $B_{1}\in{\mathcal{L}}(Y,\partial X)$ and
$B_{2}\in{\mathcal{L}}(X,\partial X)$. If for some $\lambda\in\rho(C_{0})$
both $C_{0}-D^{C,L}_{\lambda}B_{2}$ and the reduction matrix defined in (5.3)
generate analytic semigroups on $X$ and $\partial Y\times\partial X$,
respectively, then $\mathbb{A}$ generates an analytic semigroup on
$\mathbb{X}$.

  3. (3)

Let $B_{1}\in{\mathcal{L}}(Y,\partial X)$ and
$B_{2}\in{\mathcal{L}}([D(C)_{L}],\partial X)$. Assume that for some
$\lambda\in\rho(C_{0})$ $D^{C,L}_{\lambda}B_{3}\in{\mathcal{L}}(\partial Y,X)$
and $D^{C,L}_{\lambda}B_{4}\in{\mathcal{L}}(\partial X,X)$. If
$C_{0}-D^{C,L}_{\lambda}B_{2}$ and the reduction matrix defined in (5.3)
generate analytic semigroups on $X$ and $\partial Y\times\partial X$,
respectively, then $\mathbb{A}$ generates an analytic semigroup on
$\mathbb{X}$.

  4. (4)

Assume that $B_{1},B_{2}\in{\mathcal{L}}(Y,\partial X)$ and
$B_{3}\in{\mathcal{L}}(\partial Y,\partial X)$. Then $\mathbb{A}$ generates a
cosine operator function on $\mathbb{X}$ if and only both $C_{0}$ and $B_{4}$
generate cosine operator functions with associated phase spaces $Y\times X$
and $\partial Y\times\partial X$, respectively.

###### Proof.

Let $\lambda\in\rho(C_{0})$. By Lemma 5.3 instead of $\mathbb{A}$ on
$\mathbb{X}$ it suffices to investigate the similar operator matrix
$\mathbb{H}$ on $\mathbb{Y}$. We consider $\mathbb{H}$ as a $2\times 2$
operator matrix with diagonal domain, i.e.,

${\mathbb{H}}={\mathbb{H}}_{0}+{\mathbb{H}}_{1}:=\begin{pmatrix}{\bf A}&{\bf
B}\\\ {\bf C}&{{\bf D}}\end{pmatrix},\qquad D(\mathbb{H})=D({\bf A})\times
D({{\bf D}}),$

where the $2\times 2$ block-entries ${\bf A},{\bf B},{\bf C},{\bf D}$ are
defined as in (5.2).

(1) Under our assumptions we have
$B_{2}D^{C,L}_{\lambda}\in{\mathcal{L}}(\partial X)$, so that both
$\mathbf{A}$ and $\mathbf{D}$ generate $C_{0}$-semigroups on $Y\times X$ and
$\partial Y\times\partial X$, respectively. Now the off-diagonal block-entries
of $\mathbb{H}$ define an additive perturbation which is bounded either on
$[D({\mathbb{H}})]$ or on $\mathbb{X}$, and by [35, Cor. 3.2] the claim
follows.

(2)–(3) By assumption, both $\mathbf{A}$ and $\mathbf{D}$ generate analytic
semigroups on $Y\times X$ and $\partial Y\times\partial X$, respectively. Then
in (2) $\mathbf{C}$ is bounded from $Y\times X$ to $\partial Y\times\partial
X$ while $\mathbf{B}$ is bounded from $[D({\mathbf{D}})]$ to $Y\times X$, and
in (3) $\mathbf{B}$ is bounded from $\partial Y\times\partial X$ to $Y\times
X$ while $\mathbf{C}$ is bounded from $[D({\mathbf{A}})]$ to $\partial
Y\times\partial X$, and by [35, Cor. 3.3] the claim follows.

(4) By assumption, [32, Prop. 6.1] applies and $\bf A$ and $\bf D$ generate
cosine operator functions with associated phase spaces $(Y\times
Y)\times(Y\times X)$ and $(\partial Y\times\partial Y)\times(\partial
Y\times\partial X)$, respectively. Moreover, observe that ${\bf
B}\in{\mathcal{L}}([D({\bf D})],Y\times Y)$ and ${\bf
C}\in{\mathcal{L}}(Y\times Y,\partial Y\times\partial X)$. Thus, by [32, Prop.
3.2] $\mathbb{H}$ generates a cosine operator function on $\mathbb{X}$. ∎

Recall that any generator of a cosine operator function also generates an
analytic semigroup of angle $\frac{\pi}{2}$, cf. [4, Thm. 3.14.17].

Let us now modify our framework in order to deal with a setting where the
boundary operator $L$ is bounded from $Y$ to $\partial X$.

######  Assumption 5.5.

We complement the Assumptions 2.1 by the following.

  1. (1)

$V$ is a Banach space such that $V\hookrightarrow Y$.

  2. (2)

$L$ can be extended to an operator that is bounded from $Y$ to $\partial X$,
which we denote again by $L$, and such that $\ker(L)=V$.

  3. (3)

$\begin{pmatrix}C\\\ L\end{pmatrix}:D(C)\subset X\to X\times\partial X$ is
closed.

  4. (4)

${C}_{0}:=C|_{D(C)\cap\ker(L)}$ has nonempty resolvent set.

  5. (5)

$A$ is bounded from $[D(C)_{L}]$ to $X$.

  6. (6)

$B_{1},B_{2}$ are bounded from $[D(C)_{L}]$ to $\partial X$.

  7. (7)

$B_{3}$ is bounded on $\partial X$.

Under the Assumptions 5.5 Remark 4.2, Proposition 4.3, and Corollary 4.4 still
hold. Thus, we discuss the generator property of the same operator matrix
$(\underline{\mathbb{A}},D(\underline{\mathbb{A}}))$ on
$\underline{\mathbb{X}}$ introduced in Proposition 4.3. Moreover, also Lemma
4.5 remains valid up to replacing $\underline{\mathbb{U}}_{\lambda}$ therein
by

$\underline{\mathbb{V}}_{\lambda}:=\begin{pmatrix}I_{Y}&-D_{\lambda}^{C,L}&0&0\\\
0&0&I_{X}&-D_{\lambda}^{C,L}\\\ 0&I_{\partial X}&0&0\\\ 0&0&0&I_{\partial
X}\\\ \end{pmatrix},\qquad\lambda\in\rho(A_{0}).$

######  Lemma 5.6.

The operator matrix $\underline{\mathbb{A}}$ on $\underline{\mathbb{X}}$
defined in (4.1)–(4.2) is similar to

$\underline{\mathbb{H}}:=\begin{pmatrix}0&I_{V}&\vrule
height=14.0pt,depth=5.0pt&(*)\\\ \vskip-3.0pt\cr
A_{0}-D^{C,L}_{\lambda}B_{1}&C_{0}-D^{C,L}_{\lambda}B_{2}&\vrule
height=14.0pt,depth=5.0pt&&\\\ \hrule\cr 0&0&\vrule
height=14.0pt,depth=5.0pt&0&I_{\partial X}\\\ \vskip-3.0pt\cr
B_{1}&B_{2}&\vrule
height=14.0pt,depth=5.0pt&B_{3}+B_{1}D_{\lambda}^{C,L}&B_{4}+B_{2}D_{\lambda}^{C,L}\\\
\end{pmatrix}$

with domain

$D(\underline{\mathbb{G}}):=D(A_{0})\times D(C_{0})\times D(B_{3})\times
D(B_{4})$

on the Banach space $\underline{\mathbb{V}}$. Here the upper-right block entry
$(*)$ is given by

$(*)=\begin{pmatrix}0&0\\\
-D_{\lambda}^{C,L}(B_{1}D_{\lambda}^{C,L}+B_{3})&-D_{\lambda}^{C,L}(B_{2}D_{\lambda}^{C,L}+B_{4})+(\lambda-A)D_{\lambda}^{C,L}\end{pmatrix}.$

######  Theorem 5.7.

Under the Assumptions 2.1 and 5.5 the following assertions hold.

  1. (1)

Let $B_{4}\in{\mathcal{L}}(\partial X)$. Then the operator matrix
$\underline{\mathbb{A}}$ generates a $C_{0}$-semigroup (resp., an analytic
semigroup) on $\underline{\mathbb{X}}$ if and only if

(5.4) $\begin{pmatrix}0&I_{V}\\\
{A}_{0}-D^{C,L}_{\lambda}B_{1}&C_{0}-D^{C,L}_{\lambda}B_{2}\end{pmatrix}\quad\hbox{with
domain}\quad V\times D({C}_{0})$

generates a $C_{0}$-semigroup (resp., an analytic semigroup) on $V\times X$
for some $\lambda\in\rho(C_{0})$.

  2. (2)

Let $A\in{\mathcal{L}}(V,X)$, $B_{1},B_{2}\in{\mathcal{L}}(V,\partial X)$ and
$B_{4}\in{\mathcal{L}}(\partial X)$. Then $\underline{\mathbb{H}}$ generates a
cosine operator function on $\underline{\mathbb{X}}$ if and only if $C_{0}$
generates a cosine operator function with associated phase space $V\times X$.

###### Proof.

The assertion in (1) can be proved in a way similar to Theorem 4.1. To show
that (2) holds, observe that the lower-right block-entry of
$\underline{\mathbb{H}}$ is a bounded operator on $\partial X\times\partial X$
(hence the generator of a cosine operator function on $\partial
X\times\partial X$), and that by [32, Prop. 6.1] the upper-left block-entry of
$\mathbb{H}$ generates a cosine operator function with associated phase space
$(V\times V)\times(V\times X)$. Then, by assumption the lower-left block-entry
of $\underline{\mathbb{H}}$ is bounded from $V\times V$ to $\partial
X\times\partial X$ and the upper-right one is bounded from $\partial
X\times\partial X$ to $X\times X$. Thus, by [32, Prop. 3.2] also $\mathbb{H}$
generates a cosine operator function on $\mathbb{X}$. ∎

######  Remark 5.8.

Stability criteria like those stated in Theorems 3.4 and 4.7 could be easily
formulated also in the contexts of Theorems 5.4 and 5.7. However, little is
currently known about the asymptotical behavior of strongly damped systems,
thus such criteria could be hardly checked in concrete cases.

The following can be proved similarly to Proposition 4.9.

######  Proposition 5.9.

Let Theorem 5.7 apply. Then the solution $u$ to $({\rm AIBVP}^{2}_{b})$
satisfies abstract Wentzell-type boundary conditions

$L\left(Au(t)+C\dot{u}(t)\right)=B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}Lu(t)+B_{4}L\dot{u}(t),\qquad
t>0.$

If further $A$ maps $D(C)$ into $Y$, then $u$ satisfies in fact

$LAu(t)+LC\dot{u}(t)=B_{1}u(t)+B_{2}\dot{u}(t)+B_{3}Lu(t)+B_{4}L\dot{u}(t),\qquad
t>0.$

######  Example 5.10.

We consider the initial value problem associated with the system

$\left\\{\begin{array}[]{rcll}\ddot{u}(t,x)&=&\alpha
u^{\prime\prime}(t,x)+\dot{u}^{\prime\prime}(t,x),&t\geq 0,\;x\in(0,1),\\\
u(t,0)&=&\dot{u}(t,0)=0,&t\geq 0,\\\
\ddot{u}(t,1)&=&-\beta_{1}u^{\prime}(t,1)-\beta_{2}\dot{u}^{\prime}(t,1)+\beta_{3}u(t,1)+\beta_{4}\dot{u}(t,1),&t\geq
0,\\\ \end{array}\right.$

where $\alpha,\beta_{1},\beta_{2},\beta_{3},\beta_{4}\in\mathbb{C}$. A similar
problem has been discussed in [21, § 4] and [10, § 4], in a Hilbert space
setting only. Reformulate it as $({\mathbb{ACP}})$ by setting

$X:=L^{p}(0,1),\qquad Y:=\left\\{u\in
W^{2,p}(0,1):u(0)=0\right\\},\qquad{\partial X}:={\mathbb{C}},$

for $1\leq p<\infty$. We define the linear operators

$A:=\alpha\frac{d^{2}}{dx^{2}},\qquad C:=\frac{d^{2}}{dx^{2}},\qquad
D({A}):=D(C):=Y,$ $Lu:=u(1)\qquad\qquad\hbox{for all }u\in D(L):=Y,$
$B_{i}u:=-\beta_{i}u^{\prime}(1),\qquad\hbox{for all }u\in
D(B_{i}):=Y,\;i=1,2,$ $B_{i}:=\beta_{i},\qquad i=3,4.$

Thus, $\ker(L)=W^{2,p}(0,1)\cap W^{1,p}_{0}(0,1)$ and
$C_{0}:=C_{\arrowvert\ker(L)}$ is the second derivative with Dirichlet
boundary conditions, the generator of a cosine operator function on
$L^{p}(0,1)$. One sees moreover that all the Assumptions 2.1 and 5.5 are
satisfied. Since $B_{1},B_{2}\in{\mathcal{L}}(W^{2,p}(0,1),{\mathbb{C}})$, the
above initial-boundary value problem is governed by an analytic semigroup.
This semigroup yields a solution that satisfies Wentzell-type boundary
conditions for $t>0$. ∎

##  6\. A technical result

Consider the operator matrix

${\mathcal{H}}:=\begin{pmatrix}H&J\\\ K&L\end{pmatrix},\qquad
D({\mathcal{H}}):=D(H)\times D(L).$

on a product Banach space $E\times F$, where $H:E\to E$ and $L:F\to F$ are
closed operators. Our aim is to discuss the stability of the semigroup
generated by $\mathcal{H}$, in the spirit of [35]. In fact, if
$J\in{\mathcal{L}}(F,E)$ and $K\in{\mathcal{L}}(E,F)$, then $\mathcal{H}$
generates a $C_{0}$-semigroup on $E\times F$ if and only if $H$ and $L$
generate $C_{0}$-semigroups on $E$ and $F$, respectively, and in this case
$(e^{t\mathcal{H}})_{t\geq 0}$ is given by the Dyson–Phillips series

$\sum_{k=0}^{\infty}S_{k}(t),\qquad t\geq 0,$

where

$S_{0}(t):=\begin{pmatrix}e^{tH}&0\\\ 0&e^{tL}\end{pmatrix},\qquad t\geq 0,$

and

$S_{k}(t):=\int_{0}^{t}S_{0}(t-s)\begin{pmatrix}0&J\\\
K&0\end{pmatrix}S_{k-1}(s)ds,\qquad t\geq 0,\;k=1,2,\ldots.$

If we denote by $S_{k}^{(ij)}(t)$ the $(i,j)$-entry of the operator matrix
$S_{k}(t)$, $t\geq 0$, $1\leq i,j\leq 2$, then a direct matrix computation
shows that

$\displaystyle S_{k}(t)\begin{pmatrix}x\\\ y\end{pmatrix}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\begin{pmatrix}e^{(t-s)H}&0\\\
0&e^{(t-s)L}\end{pmatrix}\begin{pmatrix}JS_{k-1}^{(21)}(s)x+JS_{k-1}^{(22)}(s)y\\\
KS_{k-1}^{(11)}(s)x+KS_{k-1}^{(12)}(s)y\end{pmatrix}ds$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\begin{pmatrix}e^{(t-s)H}JS_{k-1}^{(21)}(s)x+e^{(t-s)H}JS_{k-1}^{(22)}(s)y\\\
e^{(t-s)L}KS_{k-1}^{(11)}(s)x+e^{(t-s)L}KS_{k-1}^{(12)}(s)y\end{pmatrix}ds,$

i.e., $(S_{k}(t))_{t\geq 0}$ can be expressed in terms of vector-valued
convolution,

(6.1) $S_{k}(t)=\begin{pmatrix}e^{\cdot H}*JS_{k-1}^{(21)}(\cdot)&e^{\cdot
H}*JS_{k-1}^{(22)}(\cdot)\\\ e^{\cdot L}*KS_{k-1}^{(11)}(\cdot)&e^{\cdot
L}*KS_{k-1}^{(12)}(\cdot)\end{pmatrix}(t),\qquad t\geq 0.$

By known results on vector-valued convolution we can now obtain the following.

######  Proposition 6.1.

Let $M_{1},M_{2}\geq 1$ and $\epsilon_{1},\epsilon_{2}\leq 0$ be constants
such that

(6.2) $\|e^{tH}\|\leq
M_{1}e^{\epsilon_{1}t}\qquad\hbox{and}\qquad\|e^{tL}\|\leq
M_{2}e^{\epsilon_{2}t},\qquad t\geq 0.$

Then the following assertions hold.

  1. (1)

Let $J=0$. If $\epsilon_{1}<0$ or $\epsilon_{2}<0$, then the semigroup
$(e^{t\mathcal{H}})_{t\geq 0}$ is bounded.

  2. (2)

Let $J=0$. If $(e^{tL})_{t\geq 0}$ is analytic and $\sigma(H)\cap\sigma(L)\cap
i{\mathbb{R}}=\emptyset$, then $(e^{t\mathcal{H}})_{t\geq 0}$ is bounded.

  3. (3)

Let both $\epsilon_{1}<0$ and $\epsilon_{2}<0$. Assume that

$M:=\frac{M_{1}M_{2}\|J\|\|K\|}{\epsilon_{1}\epsilon_{2}}<1.$

Then $(e^{t\mathcal{H}})_{t\geq 0}$ is uniformly exponentially stable.

###### Proof.

To begin with, one can prove by complete induction over $n$ that

$S_{2n}^{(12)}=S_{2n}^{(21)}=S_{2n+1}^{(11)}=S_{2n+1}^{(22)}=0,\qquad
n\in\mathbb{N}.$

(1) If $\epsilon_{1}<0$ and $K=0$, then one can check that

(6.3) $S^{(12)}_{1}=e^{\cdot H}*f:=e^{\cdot H}*Je^{\cdot L}$

and moreover $S^{21}_{1}=0$ as well as $S_{k}=0$ for all $k=2,3,\ldots$. We
first consider the case $\epsilon_{1}<0$. Then, by the Datko–Pazy theorem
$e^{\cdot H}x\in L^{1}({\mathbb{R}}_{+},E)$ for all $x\in E$. Since $e^{\cdot
L}y\in L^{\infty}({\mathbb{R}}_{+},F)$ for all $y\in F$, by the vector-valued
Young inequality (see [4, Prop. 1.3.5]) we see that $S^{(12)}_{1}y\in
L^{\infty}({\mathbb{R}}_{+},E)$ for all $y\in F$. The case $\epsilon_{2}<0$
can be treated similarly. Thus, we have shown that
$e^{\cdot\mathcal{H}}=S_{0}(\cdot)+S_{1}(\cdot)\in
L^{\infty}({\mathbb{R}}_{+},{\mathcal{L}}(E\times F))$.

(2) The Laplace transform $\hat{f}(\lambda)$ of $f$ defined in (6.3) is given
by $KR(\lambda,L)$ for all $\lambda$ with ${\rm Re}\lambda>0$. This yields
that the half-line spectrum ${\rm sp}(f)$ of $f$, defined as in [4, § 4.4], is
given by $\\{\eta\in{\mathbb{R}}:i\eta\in\sigma(H)\\}$. Thus, [4, Thm. 5.6.5]
yields that $S^{(12)}_{1}y\in L^{\infty}({\mathbb{R}}_{+},E)$ for all $y\in
F$, and again $e^{\cdot\mathcal{H}}\in
L^{\infty}({\mathbb{R}}_{+},{\mathcal{L}}(E\times F))$.

(3) We prove by complete induction over $n$ that the estimates

(6.4) $\|S_{2n}^{(11)}(\cdot)x\|_{L^{1}({\mathbb{R}}_{+},E)}\leq
M^{n}\frac{M_{1}}{|\epsilon_{1}|}\|x\|,\qquad x\in E,$ (6.5)
$\|S_{2n}^{(22)}(\cdot)y\|_{L^{1}({\mathbb{R}}_{+},F)}\leq
M^{n}\frac{M_{2}}{|\epsilon_{2}|}\|y\|,\qquad y\in F,$ (6.6)
$\|S_{2n+1}^{(12)}(\cdot)y\|_{L^{1}({\mathbb{R}}_{+},E)}\leq
M^{n}\frac{M_{1}M_{2}\|J\|}{\epsilon_{1}\epsilon_{2}}\|y\|,\qquad y\in F,$
(6.7) $\|S_{2n+1}^{(21)}(\cdot)x\|_{L^{1}({\mathbb{R}}_{+},F)}\leq
M^{n}\frac{M_{1}M_{2}\|K\|}{\epsilon_{1}\epsilon_{2}}\|x\|,\qquad x\in E,$

hold for all $k\in\mathbb{N}$. For $n=0$, one sees that (6.4)–(6.5) are
satisfied, since by (6.2) $\|e^{\cdot
H}x\|_{L^{1}}\leq-\frac{M_{1}}{\epsilon_{1}}\|x\|$ and $\|e^{\cdot
L}y\|_{L^{1}}\leq-\frac{M_{2}}{\epsilon_{2}}\|x\|$. Moreover,
$S_{1}^{(12)}(t)y=(e^{\cdot H}*Je^{\cdot L})(t)y$ by (6.1). By the Young
inequality we also obtain

$\|S_{1}^{(12)}(\cdot)y\|_{L^{1}}=\|e^{\cdot H}*Je^{\cdot
L}y\|_{L^{1}}\leq\|e^{\cdot H}\|_{L^{1}}\|J\|\|e^{\cdot
L}y\|_{L^{1}}\leq\frac{M_{1}M_{2}\|J\|}{\epsilon_{1}\epsilon_{2}}\|y\|.$

Likewise one can show that $\|S_{1}^{(21)}(\cdot)x\|_{L^{1}}=\|e^{\cdot
L}*Ke^{\cdot
H}x\|_{L^{1}}\leq\frac{M_{1}M_{2}\|K\|}{\epsilon_{1}\epsilon_{2}}\|x\|$, thus
the above inequalities hold for $n=0$. Assume now that they hold for $n$. Then
for one applies the Young inequality and obtains

$\displaystyle\|S_{2n+2}^{(11)}(\cdot)x\|_{L^{1}}$ $\displaystyle=$
$\displaystyle\|e^{\cdot H}*JS_{2n+1}^{(21)}(\cdot)x\|_{L^{1}}\leq\|e^{\cdot
H}\|_{L^{1}}\|J\|\|S_{2n+1}^{(21)}(\cdot)x\|_{L^{1}}$ $\displaystyle\leq$
$\displaystyle
M^{n}\frac{M_{1}M_{2}\|J\|\|K\|}{\epsilon_{1}\epsilon_{2}}\frac{M_{1}}{|\epsilon_{1}|}\|x\|=M^{n+1}\frac{M_{1}}{|\epsilon_{1}|}\|x\|.$

The remaining three estimates can be proven likewise, leading to

$\|S_{2n}(\cdot)\begin{pmatrix}x\\\
y\end{pmatrix}\|_{L^{1}}+\|S_{2n+1}(\cdot)\begin{pmatrix}x\\\
y\end{pmatrix}\|_{L^{1}}\leq M^{n}M_{0}\left\|\begin{pmatrix}x\\\
y\end{pmatrix}\right\|,\quad n\in{\mathbb{N}},\;\begin{pmatrix}x\\\
y\end{pmatrix}\in E\times F.$

where
$M_{0}:=\left(\frac{M_{1}}{|\epsilon_{1}|}+\frac{M_{2}}{|\epsilon_{2}|}+\frac{M_{1}M_{2}\|J\|}{\epsilon_{1}\epsilon_{2}}+\frac{M_{1}M_{2}\|K\|}{\epsilon_{1}\epsilon_{2}}\right)$.

We now prove the proposition’s claim. By assumption $M<1$, thus by the
dominated convergence theorem

$\displaystyle\int_{0}^{\infty}\|e^{t\mathcal{H}}\begin{pmatrix}x\\\
y\end{pmatrix}\|dt$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\left(\sum_{k=0}^{\infty}\|S_{k}(t)\begin{pmatrix}x\\\
y\end{pmatrix}\|\right)dt\leq\sum_{k=0}^{\infty}\|S_{k}(\cdot)\begin{pmatrix}x\\\
y\end{pmatrix}\|_{L^{1}({\mathbb{R}}_{+},E\times F)}$ $\displaystyle\leq$
$\displaystyle M_{0}\sum_{k=0}^{\infty}M^{k}\left\|\begin{pmatrix}x\\\
y\end{pmatrix}\right\|=\frac{M_{0}}{1-M}\left\|\begin{pmatrix}x\\\
y\end{pmatrix}\right\|,\qquad\begin{pmatrix}x\\\ y\end{pmatrix}\in E\times F.$

By the theorem of Datko–Pazy, this concludes the proof. ∎

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