# Uniqueness of some space dependent coefficients in a wave equation of
nonlinear acoustics

<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove uniqueness for some parameter identification problems
for the JMGT equation, a third order in time quasilinear PDE in nonlinear
acoustics. The coefficients to be recovered are the space dependent
nonlinearity parameter, sound speed, and attenuation parameter, and the
observation available is a single time trace of the acoustic pressure on the
boundary. This is a setting relevant to several ultrasound based tomography
methods. Our approach relies on the Inverse Function Theorem, which requires
to prove that the forward operator is a differentiable isomorphism in
appropriately chosen topologies and with an appropriate choice of the
excitation.

###### Key words and phrases:

nonlinearity parameter tomography, JMGT equation, nonlinear acoustics.

###### 1991 Mathematics Subject Classification:

Primary: 35R30; Secondary: 35B30, 35Q99.

This work was supported by the Austrian Science Fund FWF
http://dx.doi.org/10.13039/501100002428 under the grant P36318

Barbara Kaltenbacher

Department of Mathematics

Alpen-Adria-Universität Klagenfurt

Universitätsstraße 65-67, 9020 Klagenfurt, Austria

(Communicated by the associate editor name)

##  1\. Introduction

The consideration and even exploitation of nonlinearity in ultrasound imaging
has found much interest in the engineering and medical literature and is
recently also starting to become a topic of mathematical research in inverse
problems.

We consider, as one of the advanced models of nonlinear acoustics, the Jordan-
Moore-Gibson-Thompson (JMGT) equation [16, 30, 38]

$\tau u_{ttt}+u_{tt}-c_{0}^{2}\Delta u-\tau c_{0}^{2}\Delta
u+D[u]=\kappa(u^{2})_{tt}+r,$ (1)

where $u$ is the acoustic pressure, $c_{0}$ the speed of sound, $\tau$ the
relaxation time, and $\kappa$ $=\frac{\beta_{a}}{\varrho
c_{0}^{2}}=\tfrac{1}{\varrho c_{0}^{2}}(\tfrac{B}{2A}+1)$ contains the
nonlinearity parameter $\beta_{a}$ or $B/A$ along with the sound speed $c_{0}$
and the mass density $\varrho$. The JMGT equation can be written equivalently
as a second order wave type euation in terms of an auxiliary quantity
$u^{\tau}$

$\displaystyle u^{\tau}_{tt}-c_{0}^{2}\Delta
u^{\tau}+\tilde{D}[u^{\tau},u]=\kappa(u^{2})_{tt}+r,$ (2) $\displaystyle
u^{\tau}=\tau u_{t}+u$

The choice of the damping term $D[u]$ or $\tilde{D}[u^{\tau},u]$ is highly
relevant for the degree of ill-posedness of inverse problems related to the
PDE (1). In view of the fact that strong damping makes the forward problem
behave like a parabolic PDE, which renders the inverse problem severely ill-
posed, we use weak damping only. A term of the classical weak damping form
$D[u]=b_{0}u_{t}$ would not suffice to yield decay of the wave energy, though.
We therefore employ weak damping in terms of the auxiliary quantity $u^{\tau}$

$\tilde{D}[u^{\tau}]=b_{0}u^{\tau}_{t},\text{ that is, }D[u]=b_{0}(\tau
u_{t}+u)_{t}$ (3)

with $b_{0}\geq 0$ in this paper, while alternative (e.g., fractional) damping
terms [23] as relevant in ultrasonics might be studied in follow-up work.

This PDE contains several coefficients that are specific to the type of
tissue. They thus have to be regarded as function of space and therefore
provide a means of imaging. For example, in ultrasound tomography [2, 11, 12,
15, 28, 31], the sound speed $c_{0}=c_{0}(x)$ is the imaged quantity; in
nonlinearity parameter tomography, it is $\kappa=\kappa(x)$ [4, 7, 8, 14, 34,
39, 42, 43]. Also the attenuation coefficient $b_{0}=b_{0}(x)$ is known to
contain tissue specific information [9, 27, 36]. Mathematically, recovery of
these coefficients from the typical observations available in this context,
namely measurements of the acoustic pressure at some receiver array outside
the body, leads to coefficient identification problems in a PDE of the type
(1) from boundary observations of the acoustic pressure $u$.

The pressure data $h$ taken at the receivers can be written as a Dirichlet
trace on some manifold $\Sigma$ immersed in the domain $\Omega$ or attached to
its boundary $\Sigma\in\overline{\Omega}$

$h(t,x)=u(t,x),\quad(t,x)\in(0,T)\times\Sigma.$ (4)

$\Sigma$ models the transducer or hydrophone array and may as well just be a
subset of discrete points on a manifold.

We assume (1) to hold in a smooth and bounded domain
$\Omega\subseteq\mathbb{R}^{d}$, $d\in\\{1,2,3\\}$ and equip it with initial
conditions $u(t=0)=u_{0}$, $u_{t}(t=0)=u_{1}$, $u_{tt}(t=0)=u_{2}$, as well as
homogeneous impedance boundary conditions

$\gamma_{0}u+\gamma_{1}\partial_{\nu}u=0\mbox{ on }(0,T)\times\partial\Omega$
(5)

with $\gamma_{0},\gamma_{1}\in L^{\infty}(\partial\Omega)$,
$\gamma_{0},\gamma_{1}\geq 0$,
$\gamma_{0}+\gamma_{1}\geq\underline{\gamma}>0$; the case of vanishing
$\gamma_{1}$ or $\gamma_{0}$ represents Dirichlet or Neumann conditions,
respectively. The space- and time-dependent source term $r$ in (1) models
excitation, for example by a piezoelectric transducer array.

We refer to [1, 22, 23, 40] for results related to the identification of the
nonlinearity coefficient $\kappa$ alone in a classical model of nonlinear
acoustics, the Westervelt equation (basically (1) with vanishing relaxation
time $\tau=0$). In [1] its uniqueness from the whole Neumann-Dirichlet map
(instead of the single measurement (4)) is shown; [40] provides a uniqueness
and conditional stability result for the linearized problem of identifying
$\kappa$ in a higher order model of nonlinear acoustics in place of the
Westervelt equation. In [22, 23] we have proven injectivity of the linearized
forward operator mapping $\kappa$ to $h$ in the Westervelt equation with
classical strong damping and also with some fractional damping models as
relevant in ultrasonics.

A proof of injectivity of the linearized forward operator for the simultaneous
recovery of $\kappa(x)$ and $c_{0}(x)$ in the Westervelt equation in dimension
$d\in\\{1,2,3\\}$ from measurements with two excitations can be found in [24],
where it enables to apply a frozen Newton method and to show its convergence.
For reconstruction results in 1-d we refer to [24] and in 2-d (based on a
multiharmonic expansion for the Westervelt equation) to [26].

A similar linearized uniqueness proof for JMGT in place of Westervelt will
serve as a basis for proving local uniqueness of $\kappa$ by means of the
Inverse Function Theorem here. A considerable part of this paper is devoted to
establishing the required well-definedness and differentiability of the
forward operator. Beyond this, the aim of this paper is to study simultaneous
identification of $\kappa$ and $c_{0}$ or $b_{0}$ as space variable functions.
Indeed, we will show that $\kappa(x)$ is locally unique even for unknown
$c_{0}(x)$ and also provide results on simultaneous identifiability of
$\kappa(x)$ and $c_{0}(x)$ or of $\kappa(x)$ and $b_{0}(x)$.

###  1.1. The inverse problem

The pointwise observation setting (4) is an idealized one and will be extended
to a more general observation operator $\mathcal{C}_{\Sigma}$, allowing
generalization to, e.g., locally averaging observations, cf. (7). Also the PDE
model will be adapted to take into account the fact that for sufficiently
small pressure amplitudes, nonlinearity can be neglected, cf. (6).

Therewith we consider identification of the space dependent coefficients
$\kappa(x)$, $c_{0}(x)$, $b_{0}(x)$, in the attenuated and switched JMGT
equation in pressure form

$\displaystyle\tau u_{ttt}+\bigl{(}(1-2\kappa\sigma
u)u_{t}\bigr{)}_{t}+c^{2}\mathcal{A}_{c}u+\tau
c^{2}\mathcal{A}_{c}u_{t}+b_{0}(\tau u_{t}+u)_{t}$ $\displaystyle=r\quad t>0$
(6) $\displaystyle u(0)=u_{0},\quad u_{t}(0)=u_{1},\quad u_{tt}(0)$
$\displaystyle=u_{2},$

from observations

$h(t)=\mathcal{C}_{\Sigma}u(t),\quad t\in(0,T).$ (7)

with some linear and bounded observation operator
$\mathcal{C}_{\Sigma}:V\mapsto Y$, mapping from a space $V$ of $x$-dependent
functions to some data space $Y$. Typical examples are

$\displaystyle\mathcal{C}_{\Sigma}:H^{s}(\Omega)\to L^{2}(\Sigma)$
$\displaystyle v\mapsto\textup{tr}_{\Sigma}v$
$\displaystyle\mathcal{C}_{\Sigma}:C(\overline{\Omega})\to\mathbb{R}^{N}$
$\displaystyle v\mapsto\Bigl{(}v(x_{i})\Bigr{)}_{i\in\\{1,\ldots,N\\}}$
$\displaystyle\mathcal{C}_{\Sigma}:L^{2}(\Omega)\to\mathbb{R}^{N}$
$\displaystyle
v\mapsto\Bigl{(}\int_{\Omega}\eta_{i}(x)v(x)\,dx\Bigr{)}_{i\in\\{1,\ldots,N\\}}$

with the trace operator $\textup{tr}_{\Sigma}:H^{s}(\Omega)\to L^{2}(\Sigma)$,
$s>\frac{1}{2}$, measurement locations $x_{i}\in\overline{\Omega}$ and
$\eta_{i}\in L^{2}(\Omega)$ locally supported around $x_{i}$. In each of these
examples, applicability of $\mathcal{C}_{\Sigma}$ to $u$ sets different
spatial regularity requirements on the PDE solution.

In equation (6), $c>0$ is the constant mean wave speed, and we make use of the
combined elliptic operator

$\mathcal{A}_{c}=-\frac{c_{0}(x)^{2}}{c^{2}}\Delta\ \textup{ with }\ c_{0},\
\frac{1}{c_{0}}\ \in L^{\infty}(\Omega)$ (8)

that contains the possibly spatially varying coefficient $c_{0}(x)>0$ and is
equipped with the boundary conditions (5). To achieve self-adjointness of
$\mathcal{A}_{c}$, we use the weighted $L^{2}$ inner product with weight
function $c^{2}/c_{0}^{2}(x)$, that is,
$\|v\|_{L^{2}_{c^{2}/c_{0}^{2}}(\Omega)}=\int_{\Omega}\frac{c^{2}}{c_{0}^{2}(x)}\,v^{2}(x)\,dx$.
We can think of $c_{0}$ as being spatially variable and of
$\frac{c_{0}}{c}\sim 1$ to be normalized, while the magnitude of the wave
speed is given by the constant $c$. Due to compactness of its inverse, the
operator $\mathcal{A}_{c}$ has an eigensystem
$(\lambda_{j},(\varphi_{j}^{k})_{k\in K^{\lambda_{j}}})_{j\in\mathbb{N}}$
(where $K^{\lambda_{j}}$ is an enumeration of the eigenspace corresponding to
$\lambda_{j}$). This allows for a diagonalization of the operator as
$(\mathcal{A}_{c}v)(x)=\sum_{j=1}^{\infty}\lambda_{j}\sum_{k\in
K^{\lambda_{j}}}\langle v,\varphi_{j}^{k}\rangle\varphi_{j}^{k}(x)$.

######  Remark 1.

Taking into account spatial variability of the mass density, (1) would need to
be written as
$\tfrac{\tau}{\lambda(x)}u_{ttt}+\frac{1}{\lambda(x)}u_{tt}-\nabla\cdot(\frac{1}{\varrho(x)}\nabla
u)-\tau\nabla\cdot(\frac{1}{\varrho(x)}\nabla
u_{t})+D[u]=\kappa(u^{2})_{tt}+r$ with $u$ being the pressure, $\lambda$ the
bulk modulus, $\varrho$ the mass density, and $c_{0}=\frac{\lambda}{\varrho}$
the sound speed, (cf., e.g., [3, 28] for the linear case). Usually
$x$-dependence of $\varrho$ is not taken into account; Note that our general
setting with a spatial differential operator $\mathcal{A}_{c}$ would be able
to cover this dependency as well.

While $\kappa$, $c_{0}$, $b_{0}$, are positive, possibly space dependent
coefficients, the relaxation time $\tau\geq 0$ is constant and the switching
factor $\sigma$ $\in[0,1]$ depends on the magnitude of the pressure and in
particular vanishes for small $u$, see (12) for its precise definition.

It was shown in, e.g., [22, 23], that essential information on the space-
dependent coefficients is contained in the residues at the poles of the
Laplace transformed observations $\widehat{h}$. This, in its turn, via the
elementary identity

$\displaystyle\textup{Res}(\widehat{h},p)$ $\displaystyle=\lim_{z\to
p}(z-p)\widehat{h}(z)=\lim_{z\to p}\widehat{h^{\prime}-ph}(z)+h(0)$ (9)
$\displaystyle=\lim_{T\to\infty}\lim_{z\to
p}\int_{0}^{T}e^{-izt}(h^{\prime}(t)-ph(t))\,dt+h(0)$
$\displaystyle=\lim_{T\to\infty}\int_{0}^{T}\frac{d}{dt}\Bigl{(}e^{-ipt}h(t)\Bigr{)}\,dt+h(0)=\lim_{T\to\infty}e^{-ipT}h(T),$

(provided absolute convergence holds, which allows us to interchange the
integrals), leads us to study the asymptotics of solutions as time tends to
infinity.

Correspondingly, in the uniqueness proofs of Section 3 we will make use of the
forward operator $\mathbb{F}$ that takes $\kappa$ to the residues of the
Laplace transform $\widehat{\mathcal{C}_{\Sigma}u}$ at the poles of the
Laplace transformed observation $\widehat{h}$. In order to be able to take
Laplace transforms, throughout Section 3 we will assume to have observations
on the whole positive timeline, which in case of $r$ being analytic with
respect to time (for example, just vanishing) from a time instance
$\underline{T}$ on, follows by analytic continuation of the Fourier components
of $u$, which will be shown to satisfy linear ODEs from a time instance
$T_{*}$ on in Section 3.

Continuous Fréchet differentiability of the forward operator is an essential
ingredient of the Inverse Function Theorem, that we plan to employ in our
uniqueness proof. Concerning well-posedness of the JMGT equation we can
largely rely on existing results in the literature, see, e.g. [5, 6, 17, 18,
21, 29, 32, 33, 35, 37], in particular [19] for the undamped case, which is
relevant for the weak (rather than strong) damping we are employing here.
Indeed, a proof of Fréchet differentiability would fail for the Westervelt
equation in the absence of strong damping, due to the loss of regularity
arising in that case, see, e.g., [10, 20]. While well-posedness in the above
mentioned references is inherently local in time in the above cited
references, the switching allows us to prove global in time well-posedness of
(13) in Section 2, which is essential for the evaluation of residues according
to (9).

Section 3 then contains several uniqueness results that rely on the previously
shown differentiability and a proof of the linearized forward operator being
an isomorphism in appropriately chosen topologies:

  * •

local uniqueness of $\kappa$;

  * •

linearized uniqueness of $\kappa$ and $c_{0}$;

  * •

linearized uniqueness of $\kappa$ and $b_{0}$;

from the single boundary observation (7).

### Notation

Below we will abbreviate $H^{2}_{\diamondsuit}(\Omega)=H_{0}^{1}(\Omega)\cap
H^{2}(\Omega)$ and make use of the spaces $\dot{H}^{s}(\Omega)$ induced by the
norm

$\|v\|_{H^{s}(\Omega)}=\Bigl{(}\sum_{j=1}^{\infty}\lambda_{j}^{s}\sum_{k\in
K^{\lambda_{j}}}|\langle v,\varphi_{j}^{k}\rangle|^{2}\Bigr{)}^{1/2}$ (10)

with the eigensystem $(\lambda_{j},\varphi_{j})$ of the operator
$\mathcal{A}_{c}$.

Moreover, the Bochner-Sobolev spaces $L^{p}(0,T;Z)$, $H^{q}(0,T;Z)$ with $Z$
some Lebesgue or Sobolev space and $T$ a finite or infinite time horizon will
be used. By $L^{p}(X)$, $W^{s,p}(X)$ we abbreviate the Bochner spaces over the
whole positive real line $L^{p}(0,\infty;X)$, $W^{s,p}(0,\infty;X)$.

$\mathcal{B}_{\rho}(x_{0})^{X}=\\{x\in X\,:\,\|x\|_{X}\leq\rho\\}$ denotes the
closed ball with radius $\rho>0$ and center $x_{0}$ in the normed space $X$.

We denote the Laplace transform of a function $v\in L^{1}(0,\infty)$ by
$\widehat{v}(z)=\int_{0}^{\infty}e^{-zt}v(t)\,dt$ for all $z\in\mathbb{C}$
such that this integral exists.

The ordinary ${L^{2}(\Omega)}$ inner product and norm will be denoted by
$|\cdot|_{L^{2}(\Omega)}$, $(\cdot,\cdot)_{L^{2}(\Omega)}$, while the weigthed
$L^{2}$-inner product with weight $\tfrac{c^{2}}{c_{0}^{2}}$ (or more
generally, the inner product in which $\mathcal{A}_{c}$ is symmetric) will be
abbreviated by $\langle,\cdot,\cdot\rangle$.

Generic constants will be denoted by $C$ (possibly indicating in parentheses
dependence on certain quantities). Some specific constants are
$C_{\textup{PF}}$ as appearing in the Poincaré-Friedrichs inequality
$|v|_{L^{2}(\Omega)}\leq C_{\textup{PF}}|\nabla|_{L^{2}(\Omega)}$, $v\in
H_{0}^{1}(\Omega)$, and the norm $C_{H^{s},L^{p}}^{\Omega}$ of the embedding
$H^{s}(\Omega)\to L^{p}(\Omega)$.

##  2\. The forward problem

In this section we analyze the forward problem: first of all, in Section 2.1,
the parameter-to-state map $S:(\kappa,c_{0}^{2},b_{0})\mapsto u$; then, in
Section 2.2 the total forward operator, which we define by mapping the state
$u$ obtained from $S$ into the observations (7) and then further into the
residues at the poles of $\widehat{h}$. The latter, along with an
appropriately defined topology, will be the right setting for proving
uniqueness of $\kappa$ by means of the Inverse Function Theorem.

###  2.1. Well-definedness and differentiability of the parameter-to-state map

With a smooth cutoff function $\chi:[0,\infty)\to[0,1]$ such that for some
thresholds $0<\underline{m}<\overline{m}<\infty$

$\chi([0,\underline{m}])=\\{0\\},\quad\chi([\bar{m},\infty))=\\{1\\},\quad\chi\in
W^{1,\infty}(0,\infty),\quad\chi^{\prime}\geq 0\mbox{ a.e.}$ (11)

and

$\sigma(t)=\chi(|u^{\tau}(t)|_{L^{2}(\Omega)}^{2}).$ (12)

we consider (6), that is

$\displaystyle\tau u_{ttt}+\bigl{(}(1-2\kappa\sigma
u)u_{t}\bigr{)}_{t}+c^{2}\mathcal{A}_{c}u+\tau
c^{2}\mathcal{A}_{c}u_{t}+b_{0}(\tau u_{t}+u)_{t}$ $\displaystyle=r\quad t>0$
(13) $\displaystyle u(0)=u_{0},\quad u_{t}(0)=u_{1},\quad u_{tt}(0)$
$\displaystyle=u_{2},$

where for simplicity of exposition the operator
$\mathcal{A}_{c}=-\frac{c_{0}^{2}}{c^{2}}\Delta$ is equipped with homogeneous
Dirichlet boundary conditions (while the more general case (5) follows
analogously). Moreover, in (13) we have positive space-dependent coefficients
$\kappa$, $c_{0}$, $b_{0}$, constants $\tau>0$, $c>0$, as well as the time
dependent function $\sigma$ as defined in (12). The wave energy

$\mathcal{E}_{0}[u^{\tau}](t):=\frac{1}{2}\Bigl{(}|u^{\tau}_{t}(t)|_{L^{2}(\Omega)}^{2}+c^{2}|\nabla
u^{\tau}(t)|_{L^{2}(\Omega)}^{2}\Bigr{)}$ (14)

of $u^{\tau}=\tau u_{t}+t$ and therewith $|u^{\tau}(t)|_{L^{2}(\Omega)}^{2}$
will be shown to decay to zero as $t\to\infty$. Thus, the nonlinearity will be
switched off after a certain time instance $T_{*}$ and therefore the solution
follows a linear weakly damped wave equation from that time instance on, so
that we have full control over the pole locations and residues of the Laplace
transform of its solution.

An important observation here is that $T_{*}$ can be steered by the magnitude
of $b_{0}$; more precisely, $T_{*}$ can be made smaller by making $b_{0}$
larger, while the local existence time $T$ of (13) does not decrease with
increasing $b_{0}$. This allows us to combine the local in time nonlinear
existence interval with the time span of linear wave propagation to obtain
global in time well-posedness and differentiability results for (13).

We first of all state an adaption of existing local in time well-posedness
results for (13) to the situation of spatially variable coefficients and weak
damping, see subsections 2.1.1, 2.1.2.

Additionally, in subsection 2.1.3, we obtain exponential decay of the wave
energy $\mathcal{E}_{0}[u^{\tau}]$ due to the persistent weak damping, which
then implies that nonlinearity is switched off after some time instance
$T_{*}$ and $u$ follows a linear weakly damped wave equation. This is
essential for controlling its residues that according to (9) are governed by
the asymptotics as $t\to\infty$.

Finally, in subsection 2.1.4, we prove continuous differentiability of the
forward operator in a norm dictated by the uniqueness proof.

Throughout this section we assume $\Omega$ to be a bounded $C^{1,1}$ domain so
that we can make use of elliptic regularity according to, e.g., [13, Theorem
2.4.2.5].

####  2.1.1. Energy estimates and well-posedness of the linearization

We start by analyzing a linear version of (13)

$\displaystyle\tau
u_{ttt}+\bigl{(}(1-\bar{\sigma}\mathfrak{a})u_{t}\bigr{)}_{t}+c^{2}\mathcal{A}_{c}u+\tau
c^{2}\mathcal{A}_{c}u_{t}+b_{0}(\tau u_{t}+u)_{t}-\mu u_{t}-\eta u$
$\displaystyle=\mathfrak{r}\quad t>0$ (15) $\displaystyle u(0)=u_{0},\quad
u_{t}(0)=u_{1},\quad u_{tt}(0)$ $\displaystyle=u_{2},$

with

$\displaystyle\bar{\sigma}\in
W^{1,\infty}(0,\infty),\quad\bar{\sigma}([0,\infty))\subseteq[0,1].$ (16)

Later on, in Theorem 2.1 we will use Banach’s Contraction Principle to prove
convergence of a fixed point iteration defined by $u_{n+1}$ solving (15) with
$\bar{\sigma}=\chi(|u_{n}(t)|_{L^{2}(\Omega)}^{2})$, and $\mathfrak{a}=2\kappa
u_{n}$ towards the unique solution of (13).

By adapting the proof of [19, Proposition 3.1] we obtain

######  Proposition 1.

Let $T>0$, $c_{0},\frac{1}{c_{0}}$ $b_{0}\in L^{\infty}(\Omega)\cap
W^{1,3}(\Omega)$, $b_{0}\geq 0$, $\mu\in
L^{\infty}(0,T;H^{2}_{\diamondsuit}(\Omega))$, $\eta\in
L^{\infty}(0,T;H_{0}^{1}(\Omega))$, $\bar{\sigma}\in W^{1,\infty}(0,T)$.

Then there exists $\rho_{\mathfrak{a}}>0$ such that for any $\mathfrak{a}\in
L^{\infty}(0,T;L^{\infty}(\Omega))\cap L^{\infty}(0,T;W^{1,3}(\Omega))\cap
W^{1,\infty}(L^{2}(\Omega))$ satisfying

$\displaystyle\|\mathfrak{a}\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}+\|\nabla\mathfrak{a}\|_{L^{\infty}(0,T;L^{3}(\Omega))}+\|(\bar{\sigma}\mathfrak{a})_{t}\|_{L^{\infty}(0,T;L^{2}(\Omega))}$
(17) $\displaystyle+\|\nabla
b_{0}\|_{L^{3}(\Omega)}+\|\mu\|_{L^{\infty}(0,T;H^{2}(\Omega)}+\|\nabla\eta\|_{L^{\infty}(0,T;L^{2}(\Omega)}\leq\rho_{\mathfrak{a}},$

and any $\mathfrak{r}\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap
L^{\infty}(0,T;L^{2}(\Omega))$, $(u_{0},u_{1},u_{2})\in
H^{2}_{\diamondsuit}(\Omega)\times H^{2}_{\diamondsuit}(\Omega)\times
H_{0}^{1}(\Omega)$ the initial boundary-value problem (15) has a unique
solution

$u\in U_{T}=W^{3,\infty}(0,T;{L^{2}(\Omega)})\cap
W^{2,\infty}(0,T;H_{0}^{1}(\Omega))\cap
W^{1,\infty}(0,T;H^{2}_{\diamondsuit}(\Omega))$ (18)

on $(0,T)$. Furthermore, this solution satisfies

$\displaystyle\|u_{ttt}(t)\|_{L^{2}}^{2}+\|\nabla
u_{tt}(t)\|_{L^{2}}^{2}+\|\Delta u_{t}(t)\|_{L^{2}}^{2}+\|\Delta
u(t)\|_{L^{2}}^{2}$ (19)
$\displaystyle\leq\tilde{C}(\tau)(1+T)\Bigl{(}\|u^{\tau}_{tt}(t)\|_{L^{2}}^{2}+\|\nabla
u^{\tau}_{t}(t)\|_{L^{2}}^{2}+\|\Delta u^{\tau}(t)\|_{L^{2}}^{2}\Bigr{)}$
$\displaystyle\leq
C(\tau)e^{K(\tau)(\rho_{\mathfrak{a}}^{2}+1)T}\Bigl{(}\|\nabla
u_{2}\|_{L^{2}}^{2}+\|\Delta u_{1}\|_{L^{2}}^{2}+\|\Delta
u_{0}\|_{L^{2}}^{2}+\|\nabla\mathfrak{r}\|_{L^{2}(L^{2})}^{2}\Bigr{)},$

for all $t\in(0,T)$, where the constants $C(\tau)$ and $K(\tau)$ tend to
$+\infty$ as $\tau\to 0^{+}$, but are independent of $\rho_{\mathfrak{a}}$,
$b_{0}$, and the final time $T$.

###### Proof.

The proof relies on an energy estimate that is obtained by testing (15), that
is, in terms of $u^{\tau}=\tau u_{t}+u$,

$\displaystyle
u^{\tau}_{tt}+c^{2}\mathcal{A}_{c}u^{\tau}+\tilde{b}_{0}u^{\tau}_{t}-\tilde{\mu}u^{\tau}-\tilde{\eta}u=\mathfrak{r},$
(20) $\displaystyle u^{\tau}(0)=\tau u_{1}+u_{0},\quad u^{\tau}_{t}(0)=\tau
u_{2}+u_{1},$ $\displaystyle\text{where
$\tilde{b}_{0}=b_{0}-\tfrac{1}{\tau}\bar{\sigma}\mathfrak{a}$,
$\tilde{\mu}=\tfrac{1}{\tau}(\mu+(\bar{\sigma}\mathfrak{a})_{t}-\tfrac{1}{\tau}\bar{\sigma}\mathfrak{a})$,
$\tilde{\eta}=\eta-\tilde{\mu}$},$

with $-\Delta u^{\tau}_{t}=-\Delta(\tau u^{\tau}_{t}+u^{\tau})_{t}$ and
integrating by parts with respect to space in the first, third and fourth
term, which results in the spatial differentiability requirements
$\nabla\mathfrak{a}(t)$, $\nabla b_{0}$, $\nabla\mathfrak{a}_{t}(t)$ $\in
L^{3}(\Omega)$ with sufficiently small norm.
The additional term $b_{0}(\tau u_{t}+u)_{t}$ as compared to [19, Proposition
3.1] then clearly yields a nonnegative contribution to the left hand side. ∎

####  2.1.2. Well-posedness of the switched JMGT equation

Analogously to [19, Theorem 4.1], Banach’s Contraction Principle yields local
in time well-posedness of the nonlinear problem.

In view of the requirements (17) with $\mathfrak{a}=2\kappa u$, we first of
all consider the coefficient space

$X_{T}=(L^{\infty}(\Omega)\cap W^{1,3}(\Omega))^{3}\text{ for
$(\kappa,c_{0}^{2},b_{0})$}.$ (21)

######  Theorem 2.1.

Assume that $T>0$, $c>0$, $b>0$, $\rho>0$.

Then there exists $\rho_{0}>0$ such that for any
$(\kappa,c_{0}^{2},b_{0})\in\mathcal{B}_{\rho}^{X_{T}}(0,c,b)$ and any
excitation $r\in L^{2}(0,T;H_{0}^{1}(\Omega))\cap
L^{\infty}(0,T;L^{2}(\Omega))$ as well as initial data $(u_{0},u_{1},u_{2})\in
H^{2}_{\diamondsuit}(\Omega)\times H^{2}_{\diamondsuit}(\Omega)\times
H_{0}^{1}(\Omega)$ with

$\|\mathfrak{r}\|_{L^{2}(0,T;H_{0}^{1}(\Omega))\cap
L^{\infty}(0,T;L^{2}(\Omega))}^{2}+|\nabla u_{2}|_{L^{2}(\Omega)}^{2}+|\Delta
u_{1}|_{L^{2}(\Omega)}^{2}+|\Delta u_{0}|_{L^{2}(\Omega)}^{2}\leq\rho_{0},$
(22)

there exists a unique solution $u\in U_{T}$ (as defined in (18)) of (13) and
this solution satisfies the energy estimate (19).

####  2.1.3. Exponential decay of the wave energy

By testing (6), that is,

$\displaystyle u^{\tau}_{tt}-c_{0}^{2}\Delta
u^{\tau}+\tilde{b}_{0}u^{\tau}_{t}-\tilde{\mu}u_{t}=r,$ $\displaystyle
u^{\tau}=\tau u_{t}+u\,,$ $\displaystyle\text{where
$\tilde{b}_{0}=b_{0}-\tfrac{1}{\tau}2\kappa\sigma u$,
$\tilde{\mu}=2\kappa(\tfrac{1}{\tau}\sigma u-(\sigma u)_{t})$},$

with $u^{\tau}_{t}+\theta u^{\tau}=(\tau u_{t}+u)_{t}+\theta(\tau
u_{t}+u)_{t}$ in place of $-\Delta u^{\tau}_{t}$, we obtain the energy
identity

$\displaystyle\tfrac{1}{2}\tfrac{d}{dt}\Bigl{[}|u^{\tau}_{t}|_{L^{2}(\Omega)}^{2}+|c_{0}\nabla
u^{\tau}|_{L^{2}(\Omega)}^{2}+\theta|\tilde{b}_{0}^{1/2}\,u^{\tau}|_{L^{2}(\Omega)}^{2}+\theta(u^{\tau}_{t},u^{\tau})_{L^{2}(\Omega)}\Bigr{]}$
(23)
$\displaystyle+|(\tilde{b}_{0}-\theta)^{1/2}\,u^{\tau}_{t}|_{L^{2}(\Omega)}^{2}+\theta|c_{0}\nabla
u^{\tau}|_{L^{2}(\Omega)}^{2}$ $\displaystyle=(r-\tilde{\mu}u_{t}-\nabla
u^{\tau}\cdot\nabla c_{0}^{2},u^{\tau}_{t}+\theta u^{\tau})_{L^{2}(\Omega)},$

provided $b_{0}\geq\theta+\tfrac{2}{\tau}\kappa\sigma u$. Here, by Young’s
Inequality

$|\theta(u^{\tau}_{t},u^{\tau})_{L^{2}(\Omega)}|\leq\tfrac{1}{2}|u^{\tau}_{t}|_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\theta|\tilde{b}_{0}^{1/2}\,u^{\tau}|_{L^{2}(\Omega)}^{2}$

since $\theta\leq\tilde{b}_{0}$ and

$|(\nabla u^{\tau}\cdot\nabla c_{0}^{2},u^{\tau}_{t}+\theta
u^{\tau})_{L^{2}(\Omega)}|\leq\tfrac{1}{2}|(\tilde{b}_{0}-\theta)^{1/2}\,u^{\tau}_{t}|_{L^{2}(\Omega)}^{2}+\tfrac{1}{2}\theta|c_{0}\nabla
u^{\tau}|_{L^{2}(\Omega)}^{2}$

if $4C_{PF}\|\tfrac{1}{c_{0}}\|_{L^{\infty}(\Omega)}\|\nabla
c_{0}\|_{L^{\infty}(\Omega)}<1$,
$\tfrac{1}{\tilde{b}_{0}-\theta}\leq\tilde{\epsilon}\leq\tfrac{\theta(1-4C_{PF}\|\tfrac{1}{c_{0}}\|_{L^{\infty}(\Omega)}\|\nabla
c_{0}\|_{L^{\infty}(\Omega)})}{4\|\nabla c_{0}\|_{L^{\infty}(\Omega)}^{2}}$
for some $\tilde{\epsilon}>0$.

This allows us to prove exponential decay of $\mathcal{E}_{0}[u^{\tau}](t)$
and thus of $|u^{\tau}(t)|_{L^{2}(\Omega)}^{2}$.

######  Corollary 1.

Under the assumptions of Theorem 2.1 with

$\tfrac{2}{\tau}\|\kappa\|_{L^{\infty}(\Omega)}C_{H^{2}\to
L^{\infty}}C(\tau)e^{K(\tau)(\rho_{\mathfrak{a}}^{2}+1)T}\rho_{0}+\rho\leq\tfrac{b}{4},$
(24)

and

$4C_{PF}\|\tfrac{1}{c_{0}}\|_{L^{\infty}(\Omega)}\|\nabla
c_{0}\|_{L^{\infty}(\Omega)}\leq\tfrac{1}{2}$ (25)

there exist $\alpha=\alpha(b)>0$, $C>0$ such that for all
$(\kappa,c_{0}^{2},b_{0})\in\mathcal{B}_{\rho}^{X}(0,c,b)$, $r\in
L^{2}(0,T;H_{0}^{1}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))$,
$(u_{0},u_{1},u_{2})\in H^{2}_{\diamondsuit}(\Omega)\times
H^{2}_{\diamondsuit}(\Omega)\times H_{0}^{1}(\Omega)$ with (22), the estimate

$e^{\alpha
t}\mathcal{E}_{0}[u^{\tau}](t)\leq\mathcal{E}_{0}[u^{\tau}](0)+C\alpha\|e^{\alpha/2\,\cdot}r\|_{L^{2}(0,t;{L^{2}(\Omega)})}^{2}\quad\text{
for all }t\in(0,T)$ (26)

holds with some constant $C>0$ independent of $T$ and $b$.

In particular, for
$T_{*}=T_{*}(b):=\tfrac{1}{\alpha}\ln(\tfrac{C_{\textup{PF}}}{\underline{m}}\rho_{1})$
and any $\kappa$, $c_{0}$, $b_{0}$, $r$, $u_{0}$, $u_{1}$ satisfying the above
and

$\mathcal{E}_{0}[u](0)+C\alpha\|e^{\alpha/2\,\cdot}r\|_{L^{2}(0,t;{L^{2}(\Omega)})}^{2}\leq\rho_{1}^{2},$

we have $\sigma[u](t)=0$ for all $t\geq T_{*}$.

Here $\alpha(b)\to\infty$ and $T_{*}(b)\to 0$ as $b\to\infty$.

###### Proof.

Setting $\bar{\sigma}=\chi(|u^{\tau}(t)|_{L^{2}(\Omega)}^{2})$,
$\mathfrak{a}=2\kappa u$, and using (24), we obtain
$\tilde{b}_{0}\geq\tfrac{3b}{4}$. Thus, with (25), choosing $\theta:=b/2$, and
applying Young’s inequality, we obtain from (23) that

$\tfrac{d}{dt}\mathcal{E}_{0}[u^{\tau}](t)+\tfrac{b}{4}\mathcal{E}_{0}[u^{\tau}](t)\leq
Cb|r(t)|_{L^{2}(\Omega)}^{2},$

with $C$ independent of $b$. Hence

$\displaystyle\tfrac{d}{dt}\Bigl{[}e^{(b/4)t}\mathcal{E}_{0}[u^{\tau}](t)\Bigr{]}=e^{(b/4)t}\Bigl{\\{}\tfrac{d}{dt}\mathcal{E}_{0}[u^{\tau}](t)+\tfrac{b}{4}\mathcal{E}_{0}[u^{\tau}](t)\Bigr{\\}}\leq
Cbe^{(b/4)t}|r(t)|_{L^{2}(\Omega)}^{2}$

After integration over $[0,t]$, this implies the assertion with
$\alpha(b)=b/4$. ∎

Hence, for fixed $T$, $\rho_{0}$, $\rho_{1}$, $\rho>0$, choosing $b$
sufficiently large, we can achieve that nonlinearity is switched off at
$T_{*}<T$ and the PDE turns into a linear one (with still spatially variable
coefficients) from $T_{*}$ on, which allows to conclude its global in time
well-posedenss. Thus, the assertions of Theorem 2.1 and of Proposition 1 with
$\mathfrak{a}=2\kappa\sigma u$ and $\sigma$ as in (12) remain valid with $T$
replaced by $\infty$ and $U_{T}$ replaced by $U_{\infty}$.

####  2.1.4. Differentiability of the parameter-to-state map

Unter the assumptions

$\displaystyle r\in L^{2}(H^{1}(\Omega))\cap L^{\infty}(L^{2}(\Omega)),\
e^{b/8\,\cdot}r\in L^{2}(0,t;{L^{2}(\Omega)}),$ (27) $\displaystyle
u_{0},u_{1}\in H^{2}_{\diamondsuit}(\Omega),u_{2}\in H_{0}^{1}(\Omega)\text{
with \eqref{smalldata},}$

the parameter-to-state map

$S:\mathcal{B}^{X}_{\rho}(0)\to
U_{\infty},\quad(\kappa,c_{0}^{2},b_{0})\mapsto u\ \text{ where $u$ solves
\eqref{eqn:JMGT_init_D_cutoff}}$ (28)

is well defined as a mapping from $X$ to $U:=U_{\infty}$ (cf. (18)), due to
the above extension of Theorem 2.1 to the whole positive time line. Slightly
increasing the requirement coefficient regularity as compared to (21) in view
of (25) by setting

$\displaystyle X=(L^{\infty}(\Omega)\cap W^{1,3}(\Omega))\times
W^{1,\infty}(\Omega)\times(L^{\infty}(\Omega)\cap W^{1,3}(\Omega))\text{ for
$(\kappa,c_{0}^{2},b_{0})$},$ (29) $\displaystyle
X=\Bigl{(}L^{\infty}(\Omega)\cap W^{1,3}(\Omega)\Bigr{)}^{2}\text{ for
$(\kappa,b_{0})$}.$

we separately consider the option of regarding $c_{0}^{2}$ as fixed and
varying $\kappa$ and $b_{0}$ only, which in fact will make a difference in the
spaces in which we will obtain differentiability.

Its formal derivative (which we will show to be the Fréchet derivative in
Proposition 2) is given by
$S^{\prime}(\kappa,c_{0}^{2},b_{0})(\underline{d\kappa},\underline{dc}_{0}^{2},\underline{db}_{0})=v$
where $v$ solves

$\displaystyle\tau
v_{ttt}+\bigl{(}(1-2\kappa\sigma[u]u)v_{t}\bigr{)}_{t}-c_{0}^{2}\Delta v-\tau
c_{0}^{2}\Delta v_{t}+b_{0}(\tau v_{tt}+v_{t})=\mathfrak{r}(v)$ (30)
$\displaystyle\text{with
}\mathfrak{r}(v):=2\bigl{(}\kappa(\sigma[u]v+\sigma^{\prime}[u][v])u+\underline{d\kappa}\sigma[u]u)u_{t}\bigr{)}_{t}+\underline{dc}_{0}^{2}\Delta
u^{\tau}-\underline{db}_{0}u^{\tau}_{t}$

with homogeneous initial and boundary conditions. Here and in the following we
explicitely indicate dependence of $\sigma$ on $u$ by the notation $\sigma[u]$
and have

$\sigma^{\prime}[u][v](t)=\chi^{\prime}(|u^{\tau}(t)|_{L^{2}(\Omega)}^{2})(u^{\tau}(t),v^{\tau}(t))_{L^{2}(\Omega)}\
\text{ with }v^{\tau}=\tau v_{t}+v.$

Well-posedeness of (30) can be shown by a contraction argument for the
(affinely linear) fixed point operator $\mathcal{T}:\tilde{v}\mapsto v$
solving (30) with $\mathfrak{r}(\tilde{v})$ in place of
$\mathfrak{r}(v)$.111Note that Banach’s Contraction Principle in this affinely
linear setting boils down to the use of the Neumann series. To this end, we
use the fact that

$\|\mathfrak{r}(\tilde{v}_{1})-\mathfrak{r}(\tilde{v}_{2})\|_{L^{2}({L^{2}(\Omega)})}\leq\tilde{C}R\|\tilde{v}_{1}-\tilde{v}_{2}\|_{\mathcal{E}},$

for some constant $\tilde{C}>0$, and

$\|v\|_{\mathcal{E}}:=|\sqrt{\mathcal{E}_{0}[\tau
v_{t}+v]}|_{L^{\infty}(0,\infty)}$

which by an estimate analogous to (26) yields

$\|\mathcal{T}\tilde{v}_{1}-\mathcal{T}\tilde{v}_{2}]\|_{\mathcal{E}}\leq
C\tilde{C}R\|\tilde{v}_{1}-\tilde{v}_{2}]\|_{\mathcal{E}},$

thus, by smallness of $R$, contractivity of $\mathcal{T}$.

Note that since the term $\underline{dc}_{0}^{2}\Delta u^{\tau}$ is not
necessarily contained in $L^{2}(H^{1}(\Omega))$, which would be needed for an
application of Proposition 1, we had to resort to the weaker topology induced
by the wave energy, that is,

$U_{\textup{lo}}=\\{u\in L^{2}({L^{2}(\Omega)})\,:\,\tau u_{t}+u\in
W^{1,\infty}(L^{2}(\Omega))\cap L^{\infty}(H_{0}^{1}(\Omega))\\}$ (31)

and thus obtain existence and uniqueness of
$v=S^{\prime}(\kappa,c_{0}^{2},b_{0})(\underline{d\kappa},\underline{dc}_{0}^{2},\underline{db}_{0})$
in $U_{\textup{lo}}$ for $(\kappa,c_{0}^{2},b_{0})\in X$,
$(\underline{d\kappa},\underline{dc}_{0}^{2},\underline{db}_{0})\in
L^{\infty}(\Omega)^{3}$.

When considering the parameter-to-state-map $S(\kappa,b_{0})$ needed in the
recovery of $\kappa$ and $b_{0}$ only (while keeping $c_{0}^{2}$ fixed), this
term is not present and (19) yields

$\|\mathcal{T}\tilde{v}_{1}-\mathcal{T}\tilde{v}_{2}]\|_{U}\leq
C\tilde{C}R\|\tilde{v}_{1}-\tilde{v}_{2}]\|_{U},$

hence existence and uniqueness of
$v=S^{\prime}(\kappa,b_{0})(\underline{d\kappa},\underline{db}_{0})$ in the
solution space $U$ for $(\underline{d\kappa},\underline{db}_{0})\in
L^{\infty}(\Omega)^{2}$.

Analogously we obtain a Lipschitz estimate of $S$

$\displaystyle\|S(\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-S(\kappa,c_{0}^{2},b_{0})\|_{U_{\textup{lo}}}\leq
L\|(\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-(\kappa,c_{0}^{2},b_{0})\|_{L^{\infty}(\Omega)^{3}},$
(32) $\displaystyle\|S(\tilde{\kappa},\tilde{b}_{0})-S(\kappa,b_{0})\|_{U}\leq
L\|(\tilde{\kappa},\tilde{b}_{0})-(\kappa,b_{0})\|_{L^{\infty}(\Omega)^{2}},$

which allows to extend $S$ to a (Lipschitz continuous) mapping
$S:\mathcal{B}_{\rho}(0)^{\tilde{X}}\to U_{\textup{lo}}$ or $U$, respectively,
where 222Note that $L^{\infty}(\Omega)\cap W^{1,3}(\Omega)$ is not dense in
$L^{\infty}(\Omega)$ for otherwise density of $C(\Omega)$ in $W^{1,3}(\Omega)$
would imply $C(\Omega)=L^{\infty}(\Omega)$.

$\tilde{X}=\left(\overline{L^{\infty}(\Omega)\cap
W^{1,3}(\Omega)}^{L^{\infty}}\right)^{3}\text{ with
}\|\cdot\|_{\tilde{X}}=\|\cdot\|_{L^{\infty}(\Omega)^{3}}.$ (33)

To prove that $S^{\prime}$ is indeed the Fréchet derivative of $S$, we
consider the PDE that is satisfied (along with homogeneous initial and
boundary conditions) by the first order Taylor remainder
$w=S(\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-S(\kappa,c_{0}^{2},b_{0})-S^{\prime}(\kappa,c_{0}^{2},b_{0})((\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-(\kappa,c_{0}^{2},b_{0}))=\tilde{u}-u-v$,
namely

$\tau w_{ttt}+\bigl{(}(1-2\kappa\sigma[u]u)w_{t}\bigr{)}_{t}-c_{0}^{2}\Delta
w-\tau c_{0}^{2}\Delta w_{t}+b_{0}(\tau w_{t}+w)=\mathfrak{r}(\tilde{u},u,v)$
(34)

where $\tilde{u}=S(\tilde{\kappa})$, $u=S(\kappa)$,
$v=S^{\prime}(\kappa)(\tilde{\kappa}-\kappa)$,

$\displaystyle\mathfrak{r}(\tilde{u},u,v)$ (35)
$\displaystyle:=2\Bigl{(}\bigl{(}\tilde{\kappa}\sigma[\tilde{u}]\tilde{u}-\kappa\sigma[u]u\bigr{)}\tilde{u}_{t}-\bigl{(}\kappa(\sigma[u]v+\sigma^{\prime}[u][v]u)+(\tilde{\kappa}-\kappa)\sigma[u]u\bigr{)}u_{t}\Bigr{)}_{t}$
$\displaystyle\quad+c_{0}^{2}\Delta
w^{\tau}-(\tilde{c}_{0}^{2}\Delta\tilde{u}^{\tau}-c_{0}^{2}\Delta
u^{\tau}-c_{0}^{2}\Delta v^{\tau}-(\tilde{c}_{0}^{2}-c_{0}^{2})\Delta
u^{\tau})$
$\displaystyle\quad+b_{0}w^{\tau}_{t}-(\tilde{b}_{0}\tilde{u}^{\tau}_{t}-b_{0}u^{\tau}_{t}-b_{0}v^{\tau}_{t}-(\tilde{b}_{0}-b_{0})u^{\tau}_{t})$
$\displaystyle=2\Bigl{(}(\tilde{\kappa}-\kappa)(\sigma[\tilde{u}]\tilde{u}-\sigma[u]u)\tilde{u}_{t}+\kappa(\sigma[\tilde{u}]\tilde{u}-\sigma[u]u)(\tilde{u}-u)_{t}$
$\displaystyle\qquad\qquad\quad+\kappa\bigl{(}\sigma[\tilde{u}]\tilde{u}-\sigma[u]u-\sigma[u]v-\sigma^{\prime}[u][v]u\bigr{)}u_{t}\Bigr{)}_{t}$
$\displaystyle\quad-(\tilde{c}_{0}^{2}-c_{0}^{2})\Delta(\tilde{u}^{\tau}-u^{\tau})-(\tilde{b}_{0}-b_{0})(\tilde{u}^{\tau}_{t}-u^{\tau}_{t})$

and

$\displaystyle\sigma[\tilde{u}]\tilde{u}-\sigma[u]u=(\sigma[\tilde{u}]-\sigma[u])\tilde{u}+\sigma[u](\tilde{u}-u)$
$\displaystyle\sigma[\tilde{u}]\tilde{u}-\sigma[u]u-\sigma[u]v-\sigma^{\prime}[u][v]u$
$\displaystyle=(\sigma[\tilde{u}]-\sigma[u])(\tilde{u}-u)+\sigma[u]w+(\sigma[\tilde{u}]-\sigma[u]-\sigma^{\prime}[u][\tilde{u}-u])u+\sigma^{\prime}[u][w])u.$

The energy estimates (26), (19) yield $\|w\|_{U_{\textup{lo}}}$
$=O(\|(\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-(\kappa,c_{0}^{2},b_{0})\|_{L^{\infty}(\Omega)^{3}}^{2})$
or $\|w\|_{U}$
$=O(\|(\tilde{\kappa},\tilde{b}_{0})-(\kappa,b_{0})\|_{L^{\infty}(\Omega)^{2}}^{2})$
and analogously we obtain a Lipschitz estimate of $S^{\prime}$

$\displaystyle\|S^{\prime}(\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-S^{\prime}(\kappa,c_{0}^{2},b_{0})\|_{L(\tilde{X},U_{\textup{lo}})}\leq
L\|(\tilde{\kappa},\tilde{c}_{0}^{2},\tilde{b}_{0})-(\kappa,c_{0}^{2},b_{0})\|_{L^{\infty}(\Omega)^{3}},$
(36)
$\displaystyle\|S^{\prime}(\tilde{\kappa},\tilde{b}_{0})-S^{\prime}(\kappa,b_{0})\|_{L(\tilde{X},U)}\leq
L\|(\tilde{\kappa},\tilde{b}_{0})-(\kappa,b_{0})\|_{L^{\infty}(\Omega)^{2}}.$

Altogether we have proven

######  Proposition 2.

Assume that $c>0$, $\rho>0$, $b>0$, the latter being sufficiently large.

Then there exists $\rho_{0}>0$ such that for any $r$, $(u_{0},u_{1},u_{2})$
satisfying (27), the operator $S$ is well-defined as a mapping
$\mathcal{B}_{\rho}^{X}(0,c^{2},b)\to U=U_{\infty}$, cf. (18). If additionally
$\chi\in W^{2,\infty}(0,\infty)$ then $S$ satisfies the Lipschitz estimate
(32) and can be extended to a (Lipschitz continuous) mapping
$S:\mathcal{B}_{\tilde{\rho}}^{\tilde{X}}(0,c^{2},b)\to U_{\textup{lo}}$,
which is Fréchet differentiable with Lipschitz continuous derivative (36). The
assertions remain valid with $U_{\textup{lo}}$ replaced by $U$ if $S$ is
considered as a function of $(\kappa,b_{0})$ only.

Since Fréchet differentiability remains valid under strengthening of the
topology in preimage space, the assertions of Proposition 2 remain valid if we
simply stay with the stronger space (29) in place of (33).

###  2.2. Well-definedness and differentiability of the forward operator for
fixed $c_{0}(x)$ and constant $b_{0}(x)=b$

As a preparation for the use of the Inverse Function Theorem for proving
uniqueness of $\kappa$, we will now study the operator $\mathbb{F}$ that maps
$(\kappa,c_{0}^{2},b_{0})$ to the residues of
$\widehat{\mathcal{C}_{\Sigma}u}$ at the poles of $\widehat{h}$.

In our uniqueness proof, in order to take residues at poles $p$ of the Laplace
transformed solution to the PDE and its linearization, we need sufficiently
fast decay as time tends to infinity according to the identity (9)

$\textup{Res}(\widehat{f},p)=\lim_{T\to\infty}e^{-ipT}f(T).$

Note that the poles we consider will be single and their real parts will be
negative, so that $|e^{-pt}|$ is exponentially increasing as $t\to\infty$. In
particular, for an integrable function with finite support $[0,T]$ in time,
the residue vanishes (which is clear in view of the fact that the Laplace
transform of a finitely supported function has no poles).

In view of this, we will study the large time behavior of solutions. Due to
the exponential decay according to Corollary 1 and the switching by means of
$\chi$, from $T_{*}$ on, the solution $u$ to (13) coincides with the (time
shifted) solution $u_{*}$ of a linear PDE

$u(t-T_{*})=u_{*}(t)\quad t>T_{*},$ (37)

where

$\displaystyle
u^{\tau}_{*tt}+c^{2}\mathcal{A}_{c}u^{\tau}_{*}+b_{0}u^{\tau}_{*t}$
$\displaystyle=r_{*}\quad t>0$ (38) $\displaystyle u^{\tau}_{*}(0)=\tau
u_{*1}+u_{*0},\quad u^{\tau}_{*t}(0)$ $\displaystyle=\tau u_{*2}+u_{*1}$
$\displaystyle u^{\tau}=\tau u_{t}+u$

In order to be able to apply separation of variables, we assume the (possibly
space dependent) sound speed $c_{0}(x)$ to be fixed and $b_{0}$ to be constant
$b_{0}(x)\equiv\bar{b}$. 333Sligtly more generally, we could assume the three
operators id, $\mathcal{A}_{c}$, $b_{0}\cdot$ to be simultaneously
diagonizable By means of an eigensystem $(\lambda_{j},(\varphi_{j}^{k})_{k\in
K^{\lambda_{j}}})_{j\in\mathbb{N}}$ (where $K^{\lambda_{j}}$ is an enumeration
of the eigenspace corresponding to $\lambda_{j}$) of $\mathcal{A}_{c}$, we can
then write $u^{\tau}_{*}(x,t)=\sum_{j=1}^{\infty}\sum_{k\in
K^{\lambda_{j}}}u_{*j}^{\tau k}(t)\varphi_{j}^{k}(x)$ with $u_{*j}^{\tau
k}(t)=\langle u^{\tau}_{*}(t),\varphi_{j}^{k}\rangle$ solving the relaxation
equation

$\displaystyle{u_{*j}^{\tau k}}^{\prime\prime}+c^{2}\lambda_{j}u_{*j}^{\tau
k}+\bar{b}{u_{*j}^{\tau k}}^{\prime}$ $\displaystyle=r_{*j}^{k}\quad t>0$
$\displaystyle u_{*j}^{\tau k}(0)=u_{*j0}^{\tau k},\quad{u_{*j}^{\tau
k}}^{\prime}(0)$ $\displaystyle=u_{*j1}^{\tau k},$

whose Laplace transformed solutions are given by

$\widehat{u}_{*j}^{\tau
k}(z)=\frac{\widehat{r}_{*j}^{k}(z)+u_{*j1}^{k}+(z+\bar{b})u_{*j0}^{k}}{\omega_{\lambda_{j}}(z)},\
\textup{ with }\omega_{\lambda}(z)=z^{2}+\bar{b}z+c^{2}\lambda\,.$ (39)

The poles and residues of the resolvent functions
$\frac{1}{\omega_{\lambda_{j}}(z)}$ compute explicitly as

$\displaystyle p_{j}^{\pm}$
$\displaystyle=-\frac{\bar{b}}{2}\pm\imath\sqrt{c^{2}\lambda_{j}-\frac{\bar{b}^{2}}{4}}\,,$
(40) $\displaystyle\textup{Res}(\tfrac{1}{\omega_{\lambda_{j}}};p_{j}^{+})$
$\displaystyle=\lim_{z\to
p_{j}^{+}}\frac{z-p_{j}^{+}}{(z-p_{j}^{+})(z-p_{j}^{-})}=\frac{1}{\imath\sqrt{4c^{2}\lambda_{j}-\bar{b}^{2}}}\,.$

In particular, we have the following two essential properties

######  Lemma 2.2.

The poles of $\frac{1}{\omega_{\lambda}}$ differ for different $\lambda$.

######  Lemma 2.3.

The residues of the poles of $\frac{1}{\omega_{\lambda}}$ do no vanish.

These properties remain valid for more general damping models involving
fractional derivatives, see [25, Lemmas 11.4, 11.5].

######  Remark 2.

From the formula for the poles and residues it becomes evident why strong
damping needs to be switched off eventually. Indeed, with

$u_{*tt}+c^{2}\mathcal{A}_{c}u_{*}+b_{0}u_{*t}+b_{1}\mathcal{A}_{c}u_{*t}$

in place of (38) we would have

$\displaystyle p_{j}^{\pm}$
$\displaystyle=-\frac{b_{0}+b_{1}\lambda_{j}}{2}\pm\sqrt{\frac{(b_{0}+b_{1}\lambda_{j})^{2}}{4}-c^{2}\lambda_{j}}\,,$
$\displaystyle\textup{Res}(\tfrac{1}{\omega_{\lambda_{j}}};p_{j}^{+})$
$\displaystyle=\frac{1}{2\sqrt{\frac{(b_{0}+b_{1}\lambda_{j})^{2}}{4}-c^{2}\lambda_{j}}}\,.$

in place of (40). Since $\lambda_{j}\to\infty$ as $j\to\infty$, the poles
$p^{+}_{j}$ (up to finitely many) would be real and negative; in case
$b_{1}>0$ their real parts would tend to $-\infty$ as $j\to\infty$.

The explicit representation (40) together with (37), (9) allows us to compute
the residues of the Laplace transformed components $u_{j}^{\tau k}=\langle
u^{\tau},\varphi_{j}^{k}\rangle$ at the poles $p_{j}$ as follows. Since

$\displaystyle u_{j}^{\tau k}(t)={\rm 1\\!\\!I}_{[0,T_{*})}(t)u_{j}^{\tau
k}(t)+{\rm 1\\!\\!I}_{[T_{*},\infty)}(t)u_{*j}^{\tau
k}(t-T_{*})=:u_{[0,T_{*})}+u_{[T_{*},\infty)}$ $\displaystyle\textup{where
}\widehat{u_{[0,T_{*})}}(z)=0\textup{ and
}\widehat{u_{[T_{*},\infty)}}(z)=\int_{T_{*}}^{\infty}e^{-zt}u_{*j}^{\tau
k}(t-T_{*})\,dt$ $\displaystyle\hskip
85.35826pt=\int_{0}^{\infty}e^{-z(s+T_{*})}u_{*j}^{\tau
k}(s)\,ds=e^{-zT_{*}}\widehat{u}_{*j}^{k}(z)$

due to (9), (39), (40), we have

$\displaystyle\textup{Res}(\widehat{u_{j}^{\tau k}};p_{j}^{+})$
$\displaystyle=0+e^{-p_{j}^{+}T_{*}}\textup{Res}(\widehat{u}_{*j}^{\tau
k};p_{j}^{+})=\frac{e^{-p_{j}^{+}T_{*}}(\widehat{r}_{*j}^{k}(p_{j}^{+})+u_{*j1}^{\tau
k}+(p_{j}^{+}+\bar{b})u_{*j0}^{\tau
k})}{\imath\sqrt{4c^{2}\lambda_{j}-\bar{b}^{2}}}$ (41)

In here, have

$\displaystyle|e^{-p_{j}^{+}T_{*}}\widehat{r}_{*j}^{k}(p_{j}^{+})|$
$\displaystyle=|e^{-p_{j}^{+}T_{*}}\int_{0}^{\infty}e^{-p_{j}^{+}s}r_{j}^{k}(s+T_{*})\,ds|=|\int_{T_{*}}^{\infty}e^{-p_{j}^{+}t}r_{j}^{k}(t)\,dt|$
(42)
$\displaystyle\leq\int_{T_{*}}^{\infty}e^{-\Re(p_{j}^{+})t}|r_{j}^{k}(t)|\,dt=\|e^{(\bar{b}/2)\cdot}r_{j}^{k}|_{[T^{*},\infty)}\|_{L^{1}(0,\infty)}\,.$

Moreover with $u_{*j0}=\langle u(T_{*}),\varphi_{j}^{k}\rangle$,
$u_{*j1}=\langle u_{t}(T_{*}),\varphi_{j}^{k}\rangle$ according to (37), we
have

$\displaystyle|e^{-p_{j}^{+}T_{*}}(u_{*j1}^{\tau k}+p_{j}^{+}u_{*j0}^{\tau
k})|$ $\displaystyle\leq e^{-\Re(p_{j}^{+})T_{*}}\bigl{(}|\langle
u^{\tau}_{t}(T_{*}),\varphi_{j}^{k}\rangle|+|p_{j}^{+}+\bar{b}|\,|\langle
u^{\tau}(T_{*}),\varphi_{j}^{k}\rangle|\bigr{)}$ $\displaystyle\leq
e^{(\bar{b}/2)T_{*}}\bigl{(}|\langle
u^{\tau}_{t}(T_{*}),\varphi_{j}^{k}\rangle|+c\sqrt{2\lambda_{j}}\|\langle
u^{\tau}(T_{*}),\varphi_{j}^{k}\rangle|\bigr{)}$ $\displaystyle\leq
2e^{(\bar{b}/2)T_{*}}\sqrt{\mathcal{E}_{0}[\langle
u^{\tau},\varphi_{j}^{k}\rangle\varphi_{j}^{k}](T_{*})}.$

This contains the wave energy of $u^{\tau}$ as it has evolved nonlinearly up
to $T_{*}$ according to (13). To estimate it, we employ Corollary 1 from
section 2.1.

Thus with

$\left|\imath\sqrt{4c^{2}\lambda-\bar{b}^{2}}\right|\geq
c\sqrt{\lambda}\quad\textup{ for all }\lambda\geq\frac{\bar{b}^{2}}{3c^{2}}$
(43)

in (41) we have shown the following.

######  Proposition 3.

Under the conditions of Theorem 2.1, Corollary 1, the solution to (13) exists
for all $t>0$ and the generalized Fourier coeffcients of its Laplace transform
satisfy the estimate

$\displaystyle\sum_{j=1}^{\infty}\lambda_{j}\sum_{k\in
K^{\lambda_{j}}}\textup{Res}(\widehat{u_{j}^{\tau k}};p_{j}^{+})^{2}$
$\displaystyle\leq
Ce^{(\bar{b}-\alpha)T_{*}}\Bigl{(}\mathcal{E}_{0}[u](0)+\|e^{\alpha/2\,\cdot}r\|_{L^{2}(0,T_{*};{L^{2}(\Omega)})}\Bigr{)}+\|e^{(\bar{b}/2)\cdot}r|_{[T^{*},\infty)}\|_{L^{1}({L^{2}(\Omega)})}^{2}$

on $u_{j}^{\tau k}(t)=\langle u^{\tau}(t),\varphi_{j}^{k}\rangle$. Here $C>0$,
$\alpha$, $T_{*}$ can be chosen independent of the individual inital data and
excitation but depending only on the radii $\rho$, $\rho_{0}$, $\rho_{1}$ in
Corollary 1.

To prove differentiability with respect to $\kappa$ at any point
$(\kappa,c_{0}^{2},\bar{b})\in\mathcal{B}_{\rho}^{X}(0,c^{2},b)$, with
$\nabla\bar{b}=0$, we write the forward operator in two alternative ways:

$\mathbb{F}=\mathcal{R}_{c_{0}^{2},\bar{b}}\circ\mathcal{C}_{\Sigma}\circ
S=\mathbb{A}_{c_{0}^{2},\bar{b}}\circ S_{*}+\vec{a}_{c_{0}^{2},\bar{b}}$

where $\mathcal{C}_{\Sigma}$ and $S$ are defined as in (7), (28) and

$\mathcal{R}_{c_{0}^{2},\bar{b}}:h\mapsto\Bigl{(}(\textup{Res}(\widehat{h};p_{m}^{+}))_{m\in\mathbb{N}},(\textup{Res}(\widehat{h};p_{m}^{-}))_{m\in\mathbb{N}}\Bigr{)}$
$S_{*}:X\to H^{2}(\Omega)\times
H_{0}^{1}(\Omega),\quad(\kappa,c_{0}^{2},b_{0})\mapsto(u^{\tau}(T_{*}),u^{\tau}_{t}(T_{*})),$
(44)

(where attainment of $(u^{\tau}(T_{*}),u^{\tau}_{t}(T_{*}))$ in
$H^{2}(\Omega)\times H_{0}^{1}(\Omega)$ can be justified analogously to
attainment of initial data in the well-posedness proof of JMGT, see e.g.,
[20]),

$\displaystyle\vec{a}_{c_{0}^{2},\bar{b}}=\Bigl{(}$
$\displaystyle(\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{+})\widehat{r}_{*m}(p_{m}^{+})B_{\lambda_{m}}[(1,\ldots,1)])_{m\in\mathbb{N}},$
$\displaystyle\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{-})\widehat{r}_{*m}(p_{m}^{-})B_{\lambda_{m}}[(1,\ldots,1)])_{m\in\mathbb{N}}\Bigr{)}$

where
$\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{+})=-\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{-})=\frac{1}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}$

$\displaystyle\mathbb{A}_{c_{0}^{2},\bar{b}}:(u^{\tau}_{*0},u^{\tau}_{*1})\mapsto\Bigl{(}$
$\displaystyle(\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{+})B_{\lambda_{m}}[(u_{*m1}^{\tau
k}+(p_{j}^{+}+\bar{b})u_{*m0}^{\tau k})_{k\in
K^{\lambda_{m}}})_{m\in\mathbb{N}},$
$\displaystyle\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{-})B_{\lambda_{m}}[(u_{*m1}^{\tau
k}+(p_{j}^{+}+\bar{b})u_{*m0}^{\tau k})_{k\in
K^{\lambda_{m}}}])_{m\in\mathbb{N}}\Bigr{)}$
$B_{\lambda}:\mathbb{R}^{|K^{\lambda}|}\to
E^{\Sigma}_{\lambda}:=\mbox{span}((\mathcal{C}_{\Sigma}\varphi_{k})_{k\in
K^{\lambda}}),\quad(\beta_{k})_{k\in K^{\lambda}}\mapsto\sum_{k\in
K^{\lambda}}\beta_{k}\mathcal{C}_{\Sigma}\varphi_{k}.$ (45)

Here the operators $\mathcal{R}_{c_{0}^{2},\bar{b}}$,
$\mathbb{A}_{c_{0}^{2},\bar{b}}$, $\vec{a}_{c_{0}^{2},\bar{b}}$ and also
$B_{\lambda_{m}}$ depend on $c_{0}^{2},\bar{b}$ via dependence of the
eigenfunctions and poles $p_{m}^{\pm}$ on $c_{0}^{2},\bar{b}$.

To resolve the contributions within each eigenspace in our uniqueness proof,
as in [25] we impose a linear independence assumption on the eigenspace
$E^{\Sigma}_{\lambda}$ of each eigenvalue $\lambda$

$\left(\sum_{k\in K^{\lambda}}\beta_{k}\varphi_{k}(x)=0\ \mbox{ for all
}x\in\Sigma\right)\ \Longrightarrow\ \left(\beta_{k}=0\mbox{ for all }k\in
K^{\lambda}\right)\,.$ (46)

That is, for each eigenvalue $\lambda$ of $\mathcal{A}_{c}$ with
eigenfunctions $(\varphi_{k})_{k\in K^{\lambda}}$, the eigenfunctions, when
restricted to the observation manifold $\Sigma$ (or more generally, when
$\mathcal{C}_{\Sigma}$ is applied) keep their linear independence. This is a
similar to usual conditions imposed in coefficient identification problems in
wave type equations that combine assumptions on the observation geometry with
a non-trapping condition on the sound speed. We refer to, e.g., [23] for a
discussion.

This implies that for each eigenvalue $\lambda$, the mapping $B_{\lambda}$
defined by (45) is bijective and (as a mapping between finite dimensional
spaces) its inverse is bounded. We use this to define the norm in image space
as

$\|(\widehat{h}^{+},\widehat{h}^{-})\|_{\mathbb{Y}}^{2}:=\sum_{m=1}^{\infty}\lambda_{m}^{2}\|B_{\lambda_{m}}^{-1}\mbox{Proj}^{Y}_{E^{\Sigma}_{\lambda_{m}}}\widehat{h}_{m}^{+}\|_{\mathbb{R}^{K^{\lambda_{m}}}}^{2}+\sum_{m=1}^{\infty}\lambda_{m}^{2}\|B_{\lambda_{m}}^{-1}\mbox{Proj}^{Y}_{E^{\Sigma}_{\lambda_{m}}}\widehat{h}_{m}^{-}\|_{\mathbb{R}^{K^{\lambda_{m}}}}^{2}.$

With the so defined space $\mathbb{Y}$, the operator
$\mathbb{A}_{c_{0}^{2},\bar{b}}:H^{2}(\Omega)\times
H_{0}^{1}(\Omega)\to\mathbb{Y}$ is linear and bounded and therefore, due to
differentiability of $S_{*}$ (which is shown analogously to differentiability
of $S$), the forward operator
$\mathbb{F}:\mathcal{B}_{\rho}^{X}(0,c^{2},b)\to\mathbb{Y}$ is Fréchet
differentiable with respect to $\kappa$.

In our uniqueness proof we will use a slightly modified definition of the
forward operator, namely

$\widetilde{\mathbb{F}}=\widetilde{\mathcal{R}}_{c_{0}^{2},\bar{b}}\circ\mathcal{C}_{\Sigma}\circ
S=\widetilde{\mathbb{A}}_{c_{0}^{2},\bar{b}}\circ
S_{*}+\widetilde{\vec{a}}_{c_{0}^{2},\bar{b}}$

where still $\mathcal{C}_{\Sigma}$, $S$, $S_{*}$ are defined as in (7), (28),
(44) and, with two given time dependent functions $f_{1}$, $f_{2}$,

$\displaystyle\widetilde{\mathcal{R}}_{c_{0}^{2},\bar{b}}:h\mapsto\Bigl{(}$
$\displaystyle(\widehat{f}_{1}(p_{m}^{-})\textup{Res}(\widehat{h};p_{m}^{+})+\widehat{f}_{1}(p_{m}^{+})\textup{Res}(\widehat{h};p_{m}^{-}))_{m\in\mathbb{N}},$
$\displaystyle(\widehat{f}_{2}(p_{m}^{-})\textup{Res}(\widehat{h};p_{m}^{+})+\widehat{f}_{2}(p_{m}^{+})\textup{Res}(\widehat{h};p_{m}^{-}))_{m\in\mathbb{N}}\Bigr{)}$
$\displaystyle\widetilde{\vec{a}}_{c_{0}^{2},\bar{b}}=\Bigl{(}$
$\displaystyle(\tfrac{1}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}(\widehat{f}_{1}(p_{m}^{-})\widehat{r}_{*m}(p_{m}^{+})-\widehat{f}_{1}(p_{m}^{+})\widehat{r}_{*m}(p_{m}^{-}))B_{\lambda_{m}}[(1,\ldots,1)])_{m\in\mathbb{N}},$
$\displaystyle(\tfrac{1}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}(\widehat{f}_{2}(p_{m}^{-})\widehat{r}_{*m}(p_{m}^{+})-\widehat{f}_{2}(p_{m}^{+})\widehat{r}_{*m}(p_{m}^{-}))B_{\lambda_{m}}[(1,\ldots,1)])_{m\in\mathbb{N}}\Bigr{)}$
$\displaystyle\widetilde{\mathbb{A}}_{c_{0}^{2},\bar{b}}:(u_{*0},u_{*1})\mapsto$
$\displaystyle\Bigl{(}(\tfrac{\widehat{f}_{1}(p_{m}^{-})-\widehat{f}_{1}(p_{m}^{+})}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}B_{\lambda_{m}}[(u_{*m1}^{\tau
k}+\bar{b}u_{*m0}^{\tau k})_{k\in
K^{\lambda_{m}}}]+\tfrac{\widehat{f}_{1}(p_{m}^{-})p_{m}^{+}-\widehat{f}_{1}(p_{m}^{+})p_{m}^{-}}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}B_{\lambda_{m}}[(u_{*m0}^{\tau
k})_{k\in K^{\lambda_{m}}}])_{m\in\mathbb{N}},$
$\displaystyle\quad+(\tfrac{\widehat{f}_{2}(p_{m}^{-})-\widehat{f}_{2}(p_{m}^{+})}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}B_{\lambda_{m}}[(u_{*m1}^{\tau
k}+\bar{b}u_{*m0}^{\tau k})_{k\in
K^{\lambda_{m}}}]+\tfrac{\widehat{f}_{2}(p_{m}^{-})p_{m}^{+}-\widehat{f}_{2}(p_{m}^{+})p_{m}^{-}}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}B_{\lambda_{m}}[(u_{*m0}^{\tau
k})_{k\in K^{\lambda_{m}}}])_{m\in\mathbb{N}}\Bigr{)}$

Analogously to above, we can conclude the following differentiability result.

######  Corollary 2.

Under the assumptions of Proposition 2, and with
$(\widehat{f}_{1}(p_{m}^{+})-\widehat{f}_{1}(p_{m}^{-}))_{m\in\mathbb{N}}$,
$(\widehat{f}_{2}(p_{m}^{+})-\widehat{f}_{2}(p_{m}^{-}))_{m\in\mathbb{N}}$
$\in\ell^{\infty}$, the operator $\widetilde{\mathbb{F}}$ is well-defined as a
mapping $\mathcal{B}_{\rho}^{X}\to\mathbb{Y}$. If additionally $\chi\in
W^{2,\infty}(0,\infty)$ then $\widetilde{\mathbb{F}}$ satisfies a Lipschitz
estimate and can be extended to a (Lipschitz continuous) mapping
$\widetilde{\mathbb{F}}:\mathcal{B}_{\tilde{\rho}}^{\tilde{X}}(0,c^{2},b)\to\mathbb{Y}$,
which is Fréchet differentiable with Lipschitz continuous derivative.

##  3\. Uniqueness

In this section we will prove

  1. (i)

uniqueness of $\kappa$ (while $c_{0}$ does not necessarily need to be known
for this purpose; the essential condition in which $c_{0}$ is implicitly
involved is (46));

  2. (ii)

linearized uniqueness simultaneously of $(\kappa,c_{0}^{2})$;

  3. (iii)

linearized uniqueness simultaneously of $(\kappa,b_{0})$;

from the boundary observations (7).

A crucial step for this purpose will be to show that for every
$(\kappa,c_{0}^{2},\bar{b})\in\mathcal{B}_{\rho}^{X}(0,c^{2},b)$ with
$\nabla\bar{b}=0$, the operator

$D\widetilde{\mathbb{F}}(0,c_{0}^{2}):=\left(\begin{array}[]{c}\partial_{\kappa}\widetilde{\mathbb{F}}(0,c_{0}^{2},\bar{b})\\\
d_{c_{0}^{2}}\widetilde{\mathbb{F}}(\kappa,c_{0}^{2},\bar{b})\end{array}\right)=\left(\begin{array}[]{c}\widetilde{\mathcal{R}}_{c_{0}^{2},\bar{b}}\circ\mathcal{C}_{\Sigma}\circ\partial_{\kappa}S(0,c_{0}^{2},\bar{b})\\\
\widetilde{\mathcal{R}}_{c_{0}^{2},\bar{b}}\circ\mathcal{C}_{\Sigma}\circ\partial_{c_{0}^{2}}S(0,c_{0}^{2},\bar{b})\end{array}\right)$

is an isomorphism between $\mathbb{X}_{1}\times\mathbb{X}_{2}$ and
$\mathbb{Y}$. Here $\partial_{\kappa}\widetilde{\mathbb{F}}$ is the (partial)
Fréchet derivative according to the results from the previous section, while
$d_{c_{0}^{2}}\widetilde{\mathbb{F}}$ is just a formal linearization in
$c_{0}^{2}$ direction. On one hand, this implies that also
$\partial_{\kappa}\widetilde{\mathbb{F}}(0,c_{0}^{2},\bar{b}):\mathbb{X}_{1}\to\mathbb{Y}$
is a isomorphism and thus, by the Inverse Function Theorem, (i) follows. On
the other hand, we can directly conclude linearized uniqueness (ii) in the
sense that
$(\partial_{\kappa}\widetilde{\mathbb{F}}(0,c_{0}^{2},\bar{b}),d_{c_{0}^{2}}\widetilde{\mathbb{F}}(0,c_{0}^{2},\bar{b}))$
and thus
$(\mathcal{C}_{\Sigma}\circ\partial_{\kappa}S(0,c_{0}^{2},\bar{b}),\mathcal{C}_{\Sigma}\circ\partial_{c_{0}^{2}}S(0,c_{0}^{2},\bar{b})$
is injective. Similarly to the latter, one can also prove (iii).

###  3.1. An isomorphism property of the linearized forward operator

Fixing $c_{0}$, $\bar{b}$, we denote by $(p_{m}^{\pm})_{m\in\mathbb{N}}$ the
sequence of poles according to (40). Note that these are precisely the poles
of the Laplace transformed data $\widehat{h}$, provided $c_{0}$, $\bar{b}$ are
the actual coefficients of the PDE for which this data has been taken, which
we assume to hold in this section.

As in the linearized injectivity proof of [23] we use an excitation $r$ that
for $\kappa=0$ leads to a space-time separable solution
$u^{0}(x,t)=\phi(x)\psi(t)$ of (13), namely

$\displaystyle
r(x,t):=\phi(x){\psi^{\tau}}^{\prime\prime}(t)+c^{2}(\mathcal{A}_{c}\phi)(x)\psi^{\tau}+\bar{b}\phi(x){\psi^{\tau}}^{\prime}(t)$
(47) $\displaystyle\text{where }\psi^{\tau}=\tau\psi^{\prime}+\psi$

with some given functions $\phi(x)$, $\psi(t)$ such that

$\displaystyle\phi\in\mathcal{D}(\mathcal{A}_{c}),\quad\tfrac{1}{\phi},\,\tfrac{1}{\Delta\phi}\in\dot{H}^{s}(\Omega)\text{
for some }s>\tfrac{d}{2}$ (48)
$\displaystyle\psi(0)=\psi^{\prime}(0)=\psi^{\prime\prime}(0)=0$
$\displaystyle\nu_{m}:=\widehat{f}_{1}(p_{m}^{-})\widehat{f}_{2}(p_{m}^{+})-\widehat{f}_{1}(p_{m}^{+})\widehat{f}_{2}(p_{m}^{-})\not=0\quad\mbox{
for all }m\in\mathbb{N}$ $\displaystyle\text{ for
}f_{1}(t):=\tau\psi^{\prime}(t)+\psi(t),\quad
f_{2}(t):=\chi(|\phi|_{L^{2}(\Omega)}^{2}(\tau\psi^{\prime}(t)+\psi(t))^{2})(\psi^{2})^{\prime\prime}(t).$

With this, the linearization of (13) at $\kappa=0$, $c_{0}^{2}$,
$b_{0}=\bar{b}$ becomes

$\displaystyle\underline{du}^{\tau}_{tt}+c^{2}\mathcal{A}_{c}\underline{du}^{\tau}+\bar{b}\underline{du}^{\tau}_{t}$
$\displaystyle=f_{2}(t)\underline{d\kappa}(x)\phi^{2}(x)-f_{1}(t)\underline{dc}_{0}^{2}\Delta\phi(x)\quad\mbox{
in }\Omega\times(0,T)$
$\displaystyle\underline{du}(0)=0,\quad\underline{du}^{\tau}(0)=0,\quad\underline{du}^{\tau}_{t}(0)$
$\displaystyle=0\quad\mbox{ in }\Omega$
$\displaystyle\underline{du}^{\tau}=\tau\underline{du}_{t}+\underline{du}.$

Moreover, using separation of variables and the relaxation functions we can
write

$\widehat{\underline{du}^{\tau}}(x,z)=\sum_{j=1}^{\infty}\tfrac{1}{\omega_{\lambda_{j}}(z)}\sum_{k\in
K^{\lambda_{j}}}\Bigl{(}\widehat{f}_{2}(z)\langle\underline{d\kappa}\phi^{2},\varphi_{j}^{k}\rangle+\widehat{f}_{1}(z)\langle\underline{dc}^{2}_{0}(-\Delta\phi),\varphi_{j}^{k}\rangle\Bigr{)}\varphi_{j}^{k}(x).$
(49)

Due to Lemma 2.2, taking the residues at the poles singles out the
contributions of the individual eigenspaces to this sum, that is, after
evaluation by the observation operator

$\displaystyle\textup{Res}(\widehat{\mathcal{C}_{\Sigma}\underline{du}};p_{m}^{\pm})$
(50)
$\displaystyle=\textup{Res}(\tfrac{1}{\omega_{\lambda_{m}}};p_{m}^{\pm})\sum_{k\in
K^{\lambda_{m}}}\Bigl{(}\widehat{f_{2}}(p_{m}^{\pm})\langle\underline{d\kappa}\phi^{2},\varphi_{m}^{k}\rangle+\widehat{f_{1}}(p_{m}^{\pm})\langle\underline{dc}^{2}_{0}(-\Delta\phi),\varphi_{m}^{k}\rangle\Bigr{)}\mathcal{C}_{\Sigma}\varphi_{m}^{k}.$

This allows us to exploit some cancellations and write the linearization of
the forward operator at $\kappa=0$, $c_{0}^{2}=0$, $b_{0}=\bar{b}$ as

$\displaystyle
D\widetilde{\mathbb{F}}(0,c_{0}^{2})(\underline{d\kappa},\underline{dc}_{0}^{2})=\Bigl{(}$
$\displaystyle(\tfrac{\nu_{m}}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}B_{\lambda_{m}}[(\langle\underline{d\kappa}\phi^{2},\varphi_{m}^{k}\rangle)_{k\in
K_{\lambda_{m}}}])_{m\in\mathbb{N}},$ (51)
$\displaystyle(-\tfrac{\nu_{m}}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}B_{\lambda_{m}}[(\langle\underline{dc}^{2}_{0}(-\Delta\phi),\varphi_{m}^{k}\rangle)_{k\in
K_{\lambda_{m}}}])_{m\in\mathbb{N}}\Bigr{)}$

with $\nu_{m}$ as in (48). As a consequence of Lemma 2.3 and (48), the factors
$\frac{\nu_{m}}{\imath\sqrt{4c^{2}\lambda_{m}-\bar{b}^{2}}}$ appearing here do
not vanish. Under condition

$|\nu_{m}|^{2}\geq c_{\nu}\lambda_{m}^{s-1},\quad m\in\mathbb{N}\text{ for
some }s>\frac{d}{2}$ (52)

we have

$\displaystyle\|D\widetilde{\mathbb{F}}(0,c_{0}^{2})(\underline{d\kappa},\underline{dc}_{0}^{2})\|_{\mathbb{Y}}$
(53)
$\displaystyle=\sum_{m\in\mathbb{N}}\tfrac{\lambda_{m}^{2}|\nu_{m}|^{2}}{{4c^{2}\lambda_{m}-\bar{b}^{2}}}\sum_{k\in
K_{\lambda_{m}}}\Bigl{(}|\langle\underline{d\kappa}\phi^{2},\varphi_{m}^{k}\rangle|^{2}+|\langle\underline{dc}^{2}_{0}(-\Delta\phi),,\varphi_{m}^{k}\rangle|^{2}\Bigr{)}$
$\displaystyle\geq\tfrac{c_{\nu}}{c^{2}}\Bigl{(}\|\underline{d\kappa}\phi^{2}\|_{\dot{H}^{s}(\Omega)}^{2}+\|\underline{dc}^{2}_{0}(-\Delta\phi)\|_{\dot{H}^{s}(\Omega)}^{2}\Bigr{)}$

and thus, $D\widetilde{\mathbb{F}}(0,c_{0}^{2})$ is an isomorphism between
$\mathbb{X}_{1}\times\mathbb{X}_{2}$ and $\mathbb{Y}$, where

$\mathbb{X}_{1}:=\\{\underline{d\kappa}\in L^{\infty}(\Omega)\,:\
\underline{d\kappa}\phi^{2}\in\dot{H}^{s}(\Omega)\\},\quad\mathbb{X}_{2}:\\{\underline{dc}_{0}^{2}\in
L^{\infty}(\Omega)\,:\
\underline{dc}^{2}_{0}(-\Delta\phi)\in\dot{H}^{s}(\Omega)\\},$

so that $\mathbb{X}_{1}\times\mathbb{X}_{2}\times\\{\bar{b}\\}\subseteq X$.

###  3.2. Linearized uniqueness of $(\kappa,c_{0}^{2})$ for constant
$b_{0}(x)\equiv\bar{b}$

From the isomorphism property of $D\widetilde{\mathbb{F}}(0,c_{0}^{2})$, we
conclude its injectivity and thus injectivity of
$(\mathcal{C}_{\Sigma}\circ\partial_{\kappa}S(0,c_{0}^{2},\bar{b}),\mathcal{C}_{\Sigma}\circ\partial_{c_{0}^{2}}S(0,c_{0}^{2},\bar{b})$.

###  3.3. Linearized uniqueness of $(\kappa,b_{0})$ for fixed $c_{0}(x)$

Similarly to the above isomorphism proof, also injectivity of
$(\mathcal{C}_{\Sigma}\circ\partial_{\kappa}S(0,c_{0}^{2},\bar{b}),\mathcal{C}_{\Sigma}\circ\partial_{b_{0}}S(0,c_{0}^{2},\bar{b})$
can be shown. To this end, we keep (47) and replace
$f_{1}(t):=\tau\psi^{\prime}(t)+\psi(t)$ in (48) by
$f_{1}(t):=\tau\psi^{\prime\prime}(t)+\psi^{\prime}(t)$. With this, the
linearization of (13) at $\kappa=0$, $c_{0}^{2}$, $b_{0}=\bar{b}$ becomes

$\displaystyle\underline{du}^{\tau}_{tt}+c^{2}\mathcal{A}_{c}\underline{du}^{\tau}+\bar{b}\underline{du}^{\tau}_{t}$
$\displaystyle=f_{2}(t)\underline{d\kappa}(x)\phi^{2}(x)+f_{1}(t)\underline{db}_{0}\phi(x)\quad\mbox{
in }\Omega\times(0,T)$
$\displaystyle\underline{du}(0)=0,\quad\underline{du}^{\tau}(0)=0,\quad\underline{du}^{\tau}_{t}(0)$
$\displaystyle=0\quad\mbox{ in }\Omega$
$\displaystyle\underline{du}^{\tau}=\tau\underline{du}_{t}+\underline{du}.$

and so we just have to replace $\underline{dc}^{2}_{0}(-\Delta\phi)$ by
$\underline{db}_{0}\phi$ in (49) and (50), (51), (53).

###  3.4. Local uniqueness of $\kappa$

The fact that the operator $D\widetilde{\mathbb{F}}(0,c_{0}^{2})$ is an
isomorphism between $\mathbb{X}_{1}\times\mathbb{X}_{2}$ and $\mathbb{Y}$ also
implies that
$\partial_{\kappa}\widetilde{\mathbb{F}}(0,c_{0}^{2},\bar{b}):\mathbb{X}_{1}\to\mathbb{Y}$
is a isomorphism. This together with the differentiability result Corollary 2
allows us to apply the Inverse Function Theorem (see, e.g., [41, Section 4.8])
and conclude that
$\mathbb{F}(\cdot,c_{0}^{2},\bar{b}):\mathbb{X}_{1}\to\mathbb{Y}$ is locally
bijective (with a continuous inverse between these spaces) and thus, due to
completeness of the eigensystem and our assumption that $\phi\not=0$ in
$\Omega$ the full forward map $\kappa\to\mathcal{C}_{\Sigma}u$ is locally
injective. More precisely, we even have local well-posedness of the inverse
problem in the $\mathbb{X}_{1}$-$\mathbb{Y}$ topologies.

######  Theorem 3.1.

Assume that the conditions of Proposition 2 are satisfied, that (46) holds,
$r$ takes the form (47) with $\phi$, $\psi$ satisfying (48), (52), and that
$c_{0}(x)$, $b_{0}(x)\equiv\bar{b}$ are the actual sound speed and damping
coefficients. Then there exists $\rho>0$ such that for any
$\kappa,\tilde{\kappa}\in B_{\rho}^{\mathbb{X}_{1}}(0)$, equality of the
observations
$\mathcal{C}_{\Sigma}S(\kappa,c_{0}^{2},\bar{b})(t)=\mathcal{C}_{\Sigma}S(\tilde{\kappa},c_{0}^{2},\bar{b})(t)$,
$t>0$ implies $\kappa=\tilde{\kappa}$.

This does not require $c_{0}(x)$ nor $\mathcal{A}_{c}$ to be known, just the
conditions (47), (48), (46) to be satisfied by the (possibly unknown) operator
$\mathcal{A}_{c}$ and its eigensystem.

Existence of an excitation $r$ according to (47) such that
$e^{\alpha/2\cdot}r\in L^{2}({L^{2}(\Omega)})$ and (48) holds might be hard to
reconcile with (52), in particular in dimensions larger than one, where (52)
requires growth of $\nu_{n}$. However, we can use the result to recover an
arbitrary number $M$ of generalized Fourier components of $\kappa\phi^{2}$,
assuming only (47), $e^{\alpha/2\cdot}r\in L^{2}({L^{2}(\Omega)})$ and (48) to
hold, by setting
$c_{\nu}:=\max\\{|\nu_{m}|^{-2}\lambda_{m}^{s-1}\,:\,m\in\\{1,\ldots,M\\}\\}>0$
for a given $s>\frac{d}{2}$ and working in the space
$\mathbb{X}_{1M}:=\\{\underline{d\kappa}\in L^{\infty}(\Omega)\,:\
\underline{d\kappa}\phi^{2}\in\text{span}(\varphi_{j}^{k})_{j\in\\{1,\ldots,M\\},\,k\in
K^{\lambda_{j}}}\\}\subseteq\dot{H}^{s}(\Omega)$.

######  Corollary 3.

Assume that the conditions of Proposition 2 are satisfied, that (46) holds,
$r$ takes the form (47) with $\phi$, $\psi$ satisfying (48), and that
$c_{0}(x)$, $b_{0}(x)\equiv\bar{b}$ are the actual sound speed and damping
coefficients. Then there exists $\rho>0$ such that for any
$\kappa,\tilde{\kappa}\in B_{\rho}^{\mathbb{X}_{1M}}(0)$, equality of the
observations
$\mathcal{C}_{\Sigma}S(\kappa,c_{0}^{2},\bar{b})(t)=\mathcal{C}_{\Sigma}S(\tilde{\kappa},c_{0}^{2},\bar{b})(t)$,
$t>0$ implies $\kappa=\tilde{\kappa}$.

## Acknowledgment

This work was supported by the Austrian Science Fund FWF
http://dx.doi.org/10.13039/501100002428 under the grant P36318.

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