# The Westervelt–Pennes–Cattaneo model: local well-posedness and singular
limit for vanishing relaxation time

Imen Benabbas† and  Belkacem Said-Houari‡

###### Abstract.

In this work, we investigate a mathematical model of nonlinear ultrasonic
heating based on a coupled system of the Westervelt equation and the
hyperbolic Pennes bioheat equation (Westervelt–Pennes–Cattaneo model). Using
the energy method together with a fixed point argument, we prove that our
model is locally well-posed and does not degenerate under a smallness
assumption on the pressure data in the Westervelt equation. In addition, we
perform a singular limit analysis and show that the Westervelt–Pennes–Fourier
model can be seen as an approximation of the Westervelt–Pennes–Cattaneo model
as the relaxation parameter tends to zero. This is done by deriving uniform
bounds of the solution with respect to the relaxation parameter.

###### Key words and phrases:

ultrasonic heating, Westervelt’s equation, nonlinear acoustics, Pennes bioheat
equation, Cannateo model, HIFU

###### 2010 Mathematics Subject Classification:

35L70, 35K05

†AMNEDP Laboratory, Faculty of Mathematics, USTHB<EMAIL_ADDRESS>
‡Department of Mathematics, College of Sciences, University of Sharjah, P. O.
Box: 27272, Sharjah, United Arab Emirates<EMAIL_ADDRESS>
##  1\. Introduction

We are interested in the analysis of a nonlinear thermo-acoustic system
modeling the propagation of ultrasonic waves, such as the high intensity
focused ultrasound (HIFU), trough thermo-viscous fluids. With various
applications including medical and industrial use in lithotripsy,
thermotherapy, ultrasound cleaning and sonochemistry [31, 9, 7, 18, 10], the
behavior of high-intensity focused ultrasound (HIFU) and the mathematical
models describing it are receiving a great deal of attention from researchers.
For instance, in medical procedures, the focused ultrasound is used to
generate localized heating that can destroy the targeted region. Indeed, this
technique is proving its success in the treatment of both benign and malignant
tumors. Due to the high frequency of the sound waves, the nonlinear effect of
their propagation cannot be eluded and it is well-established in nonlinear
acoustics that Westervelt equation [27] takes this effect into consideration.

In this paper, we consider a coupled system of the Westervelt for the pressure
and a hyperbolic Pennes equation for the temperature. More precisely, we
consider the system

(1.1) $\left\\{\begin{aligned} &p_{tt}-c^{2}(\bar{\Theta})\Delta p-b\Delta
p_{t}=K(\bar{\Theta})\left(p^{2}\right)_{tt},\qquad&\text{in}\
\Omega\times(0,T),\\\
&\rho_{\textup{a}}C_{\textup{a}}\bar{\Theta}_{t}+\nabla\cdot
q+\rho_{\textup{b}}C_{\textup{b}}W(\bar{\Theta}-\Theta_{\textup{a}})=\mathcal{Q}(p_{t}),\qquad&\text{in}\
\Omega\times(0,T),\\\ &\tau
q_{t}+q+\kappa_{\textup{a}}\nabla\bar{\Theta}=0,\qquad&\text{in}\
\Omega\times(0,T).\end{aligned}\right.$

Here the acoustic pressure and the temperature fluctuations are denoted
respectively by $p$ and $\bar{\Theta}$; $c$ is the speed of sound and $b>0$ is
the sound diffusivity. The function $K(\bar{\Theta})$ is allowed to depend
$\bar{\Theta}$ and it is given by

$K(\bar{\Theta})=\frac{\beta_{\text{acous}}}{\rho c^{2}(\bar{\Theta})},$

where $\rho$ is the mass density and $\beta_{\text{acous}}$ is the parameter
of nonlinearity. The source term in the second equation $\mathcal{Q}(p_{t})$
represents the acoustic energy absorbed by the tissue. The medium parameters
$\rho_{\textup{a}},C_{\textup{a}}$ and $\kappa_{\textup{a}}$ stand,
respectively, for the ambient density, the ambient heat capacity and thermal
conductivity of the tissue. The additional term
$\rho_{\textup{b}}C_{\textup{b}}W(\bar{\Theta}-\Theta_{\textup{a}})$ accounts
for the heat loss due to blood circulation, with
$\rho_{\textup{b}},C_{\textup{b}}$ being the density and specific heat
capacity of blood, and $W$ expressing the tissue’s volumetric perfusion rate
measured in milliliters of blood per milliliter of tissue per second.

The second and the third equations in (1.1) consist the hyperbolic version of
the Pennes equation (1.5), where the heat equation is given by the Cattaneo
(or the Cattaneo–Maxwell law). We supplement (1.1) with the initial conditions

(1.2) $p|_{t=0}=p_{0},\quad
p_{t}|_{t=0}=p_{1},\quad\bar{\Theta}|_{t=0}=\bar{\Theta}_{0},\quad
q|_{t=0}=q_{0}$

and Dirichlet boundary conditions

(1.3)
$p|_{\partial\Omega}=0,\qquad\bar{\Theta}|_{\partial\Omega}=\Theta_{\textup{a}},$

with $\Theta_{\textup{a}}$ denoting the ambient temperature, that is typically
taken in the human body to be $37^{\circ}C$; see [4].

When $c$ and $K$ are constants, the first equation in (1.1) reduces to the the
Westervelt equation for the pressure $p(x,t)$:

(1.4) $p_{tt}-c^{2}\Delta p-b\Delta p_{t}=K\left(p^{2}\right)_{tt}.$

Equation (1.4) is widely used in acoustics and describes the propagation of
sound waves in a fluid medium and can be derived from the
Navier–Stokes–Fourier model by assuming the Fourier law of heat conduction and
by taking into account the thermoviscous effects. See [27], [5] and [16,
Chapter 5] for more details. Significant progress have been made recently
toward the understanding of the solutions to Westervelt equation and their
behavior; see [13, 3, 12, 19, 15, 26] and the references therein. The results
in [13, 19, 26] investigated, local well-posedness, global well-posedness and
asymptotic behavior of solutions, for different types of boundary conditions
and in various functional settings. Particularly, in [19, 26] the authors
relied on maximal regularity in $L^{p}$-spaces to obtain the existence of a
unique solution with low regularity assumptions on the initial data. This is
feasible thanks to the strong damping represented by the term $-b\Delta p_{t}$
when $b>0$, that lends to the Westervelt equation its parabolic character.

Combining the second and the third equation equations in (1.1), we obtain the
hyperbolic Pennes equation (see [11, Eq. 3] and [29, Eq. 7])

(1.5)
$\displaystyle\tau\rho_{\textup{a}}C_{\textup{a}}\bar{\Theta}_{tt}+(\rho_{\textup{a}}C_{\textup{a}}+\tau\rho_{\textup{b}}C_{\textup{b}}W)\bar{\Theta}_{t}+\rho_{\textup{b}}C_{\textup{b}}W(\bar{\Theta}-\Theta_{\textup{a}})-\kappa_{\textup{a}}\Delta\bar{\Theta}$
$\displaystyle=$
$\displaystyle\,\mathcal{Q}(p_{t})+\tau\partial_{t}\mathcal{Q}(p_{t}).$

The terms in (1.5) that appear with the $\tau$ prefactor arise from the use of
the Cattaneo law of heat condition [2]:

(1.6) $\tau q_{t}+q+\kappa_{\textup{a}}\nabla\bar{\Theta}=0$

which is a modified version of the classical Fourier law of heat conduction:

(1.7) $q+\kappa_{\textup{a}}\nabla\bar{\Theta}=0.$

Equation (1.7) implies instantaneous thermal energy deposition in the medium.
That is any temperature disturbance causes an instantaneous perturbation in
the temperature at each point in the medium. The Cattaneo law was introduced
to overcome this drawback of the infinite speed of thermal signals in the
Fourier law. The idea is to introduce a time lag into the relationship between
the heat flux and the temperature gradient, which results in the term $\tau
q_{t}$ in (1.6), where $\tau$ is the relaxation time parameter.

Using (1.6) models the heat propagation as a damped wave equation. This
phenomenon is known as second sound effect and it is experimentally observed
in materials at a very low temperature, where the heat seems to propagate as a
thermal wave, which is the reason for this name (see the review paper [24]).

For $\tau=0$, equation (1.5) becomes

(1.8)
$\rho_{\textup{a}}C_{\textup{a}}\bar{\Theta}_{t}-\kappa_{\textup{a}}\Delta\bar{\Theta}+\rho_{\textup{b}}C_{\textup{b}}W(\bar{\Theta}-\Theta_{\textup{a}})=\mathcal{Q}(p_{t}).$

Equation (1.8) is the parabolic Pennes equation which is a bioheat transfer
equation that is widely used for studying heat transfer in biological systems.
It takes into account the heat transfer by conduction in the tissues and the
convective heat transfer due to blood perfusion. See [23] for the derivation
of (1.8).

In [21] Nikolić and Said-Houari considered the Westervelt–Pennes–Fourier
system, which is the coupling between the first equation in (1.1) and (1.8).
That is, they investigated the system

(1.9) $\left\\{\begin{aligned} &p_{tt}-c^{2}(\bar{\Theta})\Delta p-b\Delta
p_{t}=k(\bar{\Theta})\left(p^{2}\right)_{tt},\\\
&\rho_{\textup{a}}C_{\textup{a}}\bar{\Theta}_{t}-\kappa_{\textup{a}}\Delta\bar{\Theta}+\rho_{\textup{b}}C_{\textup{b}}W(\bar{\Theta}-\Theta_{\textup{a}})=\mathcal{Q}(p_{t})\end{aligned}\right.$

in $\Omega\times(0,T)$ with Dirichlet–Dirichlet boundary conditions. They
proved a local well-posedness of (1.9) using the energy method together with a
fixed point argument. The work in [21] was followed by [22], where the authors
proved a global existence and asymptotic behavior of the solution of (1.9)
under a smallness assumption on the initial data. Using the maximal regularity
estimate for parabolic systems, Wilke in [28] improves slightly the regularity
assumptions in [21] and also considered the case $b=b(\bar{\Theta})$.

In this paper, we consider the Westervelt–Pennes–Cattaneo system (1.1) and
investigate the local well-posedness and the singular limit as the time
relaxation parameter $\tau$ tends to zero. To state and prove our result and
to lighten the notation, we put

(1.10)
$m=\rho_{\textup{a}}C_{\textup{a}}\qquad\text{and}\qquad\ell=\rho_{\textup{b}}C_{\textup{b}}W,$

and make the change of variables $\Theta=\bar{\Theta}-\Theta_{a}$ in the
temperature and denote by

(1.11) $k(\Theta)=K(\Theta+\Theta_{\textup{a}})=\frac{1}{\rho
c^{2}(\Theta+\Theta_{\textup{a}})}\beta_{\text{acou}}\quad\text{and}\quad
h(\Theta)=c^{2}(\Theta+\Theta_{\textup{a}})$

to get the following system

(1.12a) $\left\\{\begin{aligned} &p_{tt}-h(\Theta)\Delta p-b\Delta
p_{t}=k(\Theta)\left(p^{2}\right)_{tt},\qquad&\text{in}\ \Omega\times(0,T)\\\
&m\Theta_{t}+\nabla\cdot q+\ell\Theta=\mathcal{Q}(p_{t}),\qquad&\text{in}\
\Omega\times(0,T)\\\ &\tau
q_{t}+q+\kappa_{\textup{a}}\nabla\Theta=0,\qquad&\text{in}\
\Omega\times(0,T)\end{aligned}\right.$ complemented with homogeneous boundary
conditions (1.12b) $p|_{\partial\Omega}=0,\qquad\Theta|_{\partial\Omega}=0$
and the initial conditions (1.12c) $p|_{t=0}=p_{0},\quad
p_{t}|_{t=0}=p_{1},\quad\Theta|_{t=0}=\Theta_{0}:=\bar{\Theta}_{0}-\Theta_{\textup{a}},\quad
q|_{t=0}=q_{0}.$

As in [21], here the medium parameters $c$ and $K$ in the Westervelt equation
are not constant. They are taken explicitly dependent on the temperature, in
order to account for the fact that the heating generated by the ultrasound
waves affects their speed of propagation and the position of the focal region:
a phenomenon that is known as thermal lensing [4, 8]. Precisely, we assume
this dependence to be polynomial, in agreement with the experimentally
observed behavior documented in [1]. Also, for simplicity, we assume that the
function $\mathcal{Q}$ has the form

(1.13)
$\mathcal{Q}(p_{t})=\mathcal{Q}(p_{t})=\frac{2b}{\rho_{\textup{a}}C_{\textup{a}}^{4}}(p_{t})^{2}$

although our proof works for quite general $\mathcal{Q}(p_{t})$ satisfying
Assumption 2 in [21].

To establish well-posedness, we carry out an energy analysis for the
linearization of the underlying system, which will allow applying a fixed-
point argument to work out the existence of a unique solution to the nonlinear
problem (1.12). In doing so, we encounter two main challenges. On the one
hand, we have to take into account the interplay between the pressure and the
temperature owing to the coupling of their respective equations. The
temperature-dependent of the coefficients $h(\Theta)$ and $k(\Theta)$ in the
pressure equation necessitates that they should be kept bounded. That is to
say, the function $\Theta$ needs to be in $L^{\infty}((0,T)\times\Omega)$.
Moreover, note that the first equation in (1.12) is quasilinear. To see this,
it suffices to write the term on the right as
$k(\Theta)(p^{2})_{tt}=2k(\Theta)pp_{tt}+2k(\Theta)(p_{t})^{2}$. This brings
about the risk of degeneracy of the term $(1-2k(\Theta)p)p_{tt}$, which is
usually avoided by imposing a smallness constraint on the acoustic pressure
[13]. Therefore, we are led to work with higher-order energies for both the
pressure and the temperature. One key ingredient in obtaining the a priori
estimates for these energies is the Sobolev embedding theorem, especially the
continuous embedding $H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$. On the
other hand, obtaining uniform energy bound with respect to $\tau$ for the heat
conduction system requires careful attention. Further, considering that in
practical situations, $\tau$ takes very small values, it is interesting to
investigate the behavior of the system when $\tau$ goes to zero. In order to
do this, we adopt the approach in [14, 20]. Even though the models under
consideration in these works involve single equations, the method is quite
constructive and could be applied to the coupled problem at hand. The key
ideas are to first derive estimates that are uniform with respect to $\tau$,
which will justify taking the limit as $\tau$ tends to zero; then we make use
of the compactness of Sobolev embeddings to show that as $\tau$ tends to zero,
the solution of (1.12) converges to the solution of the system corresponding
to $\tau=0$.

Our paper is organized as follows. The main results are stated in Section 2.
In Section 3, we collect some theoretical results that will prove useful in
the sequel, and we state the general assumptions on the coefficients in system
(1.12). Sections 4 is devoted to the energy analysis of the hyperbolic Pennes
equation (the Cattaneo system), while Section 5 treats the linearized
Westervelt equation. In Section 6, we prove the local well-posedness of the
nonlinear problem (1.12). Finally, in Section 7, we perform the singular limit
analysis and show that the solution of the Westervelt–Pennes–Cattaneo model
converges to the solution of the Westervelt–Pennes–Fourier system when the
time relaxation vanishes.

##  2\. Main results

In this section, we state the main results of this paper and give the strategy
of the proof. In order to give context to the main results stated below, we
begin by specifying the functional setup adopted in this work. We define

(2.1) $\mathcal{X}:=X_{p}\times X_{\Theta}\times X_{q},$

where the spaces $X_{p},X_{\Theta}$ and $X_{q}$ are given, respectively as

(2.2) $\displaystyle X_{p}=$ $\displaystyle\,\Big{\\{}p\in
L^{\infty}(0,T;H^{3}(\Omega)\cap H^{1}_{0}(\Omega)),$ $\displaystyle\quad
p_{t}\in L^{\infty}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
L^{2}(0,T;H^{3}(\Omega)\cap H^{1}_{0}(\Omega)),$ $\displaystyle\quad p_{tt}\in
L^{\infty}(0,T;H^{1}_{0}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)),$ $\displaystyle\quad p_{ttt}\in
L^{2}(0,T;L^{2}(\Omega))\Big{\\}};$ $\displaystyle X_{\Theta}=$
$\displaystyle\,\\{\Theta\in L^{\infty}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)),\Theta_{t}\in L^{\infty}(0,T;H^{1}_{0}(\Omega)),$
$\displaystyle\quad\Theta_{tt}\in L^{\infty}(0,T;L^{2}(\Omega))\\};$
$\displaystyle X_{q}=$ $\displaystyle\,\\{q\in
L^{\infty}(0,T;(H^{1}(\Omega))^{d});q_{t},q_{tt}\in
L^{2}(0,T;(L^{2}(\Omega))^{d})\\}.$

Our first result ensures local in time well-posedness, which is uniform with
respect to the relaxation parameter $\tau$.

######  Theorem 2.1.

Let $T>0$ and $\bar{\tau}>0$ be a fixed small constant. Let
$\tau\in(0,\bar{\tau}]$ and assume that

(2.3) $\displaystyle(p_{0},p_{1})$ $\displaystyle\in H^{3}(\Omega)\cap
H^{1}_{0}(\Omega)\times H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$
$\displaystyle(\Theta_{0},q_{0})$ $\displaystyle\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\times(H^{1}(\Omega))^{d}.$

There exists $\delta=\delta(T)>0$ such that if

(2.4) $\|p_{0}\|_{H^{3}}+\|p_{1}\|_{H^{2}}+\|p_{tt}(0)\|_{H^{1}}\leq\delta$

then system (1.12) has a unique solution $(p,\Theta,q)\in\mathcal{X}$.

Let us now outline the main steps in the proof of Theorem 2.1 and list some
comments about the above result.

  1. 1.

We consider a linearization of the underlying model (1.12) that will see the
Westervelt equation decoupled from the Cattaneo system for heat transfer. This
allows us to treat the first equation in (1.12) separately from the second and
third equations. However, the decoupling of the linearized system does not
allow us to transfer the damping induced by the damped wave equation for the
temperature to the linearized Westervelt equation.

We use Galerkin approximations to prove the existence of a unique solution for
linearized Cattaneo system (4.1), together with uniform energy estimates with
respect to $\tau$. Next, motivated by [21], we often rely on Sobolev embedding
theorem to conduct an energy analysis yielding the well-posedness of the
linearized Westervelt equation in a finite time horizon $T>0$. In addition, we
derive some energy bounds that will be useful in the analysis of the nonlinear
problem.

  2. 2.

Having all the necessary ingredients, we can tackle the nonlinear coupled
problem (1.12) by defining the solution of the system (1.12) as the fixed
point of a carefully defined mapping. Bringing together the already
established energy estimates and using the Banach fixed-point theorem, we
prove well-posedness for the nonlinear problem and show that the solution does
not degenerate under a smallness assumption on the initial data of the
acoustic pressure.

  3. 3.

Note that the smallness condition in Theorem 2.1 is imposed only on the
pressure data and not on the temperature data. This seems necessary to avoid
the degeneracy of the Westervelt equation.

  4. 4.

Using the same method, we can also treat the case $b=b(\Theta)$. Our
assumption on $b$ to be constant is just to avoid technicalities in the proof.
Also, the condition $b>0$ is crucial in our analysis. It is an important open
problem to study the case $b=0$.

In the following theorem, we state a convergence of the solution of the
Westervelt–Pennes–Cattaneo model to the solution of Westervelt–Pennes–Fourier
model as the relaxation time $\tau$ tends to zero and as a consequence, we
recover at the limit, the well-posedness result in [21]. To facilitate
relating the limit to the solution of Westervelt–Pennes–Fourier model, in this
part we use the wave equation (1.5) instead of Cattaneo’s system. We denote by
$\Theta_{1}^{\tau}:=\Theta^{\tau}_{t}(0,x)$.

######  Theorem 2.2.

Given $T>0$ and $\tau\in(0,\bar{\tau}]$. Let the initial data

(2.5) $\displaystyle(p_{0}^{\tau},p_{1}^{\tau})\in H^{3}(\Omega)\cap
H^{1}_{0}(\Omega)\times H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$
$\displaystyle(\Theta_{0}^{\tau},\Theta_{1}^{\tau})\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega),$

satisfy the assumptions of Theorem 2.1. Then, the familly of solutions
$(p^{\tau},\Theta^{\tau})_{\tau\in(0,\bar{\tau}]}$ converges weakly (see
(7.7), (7.5)) to the solution $(p,\Theta)\in X_{p}\times X_{\Theta}$ of the
Westervelt–Pennes–Fourier system:

(2.6) $\left\\{\begin{aligned} &(1-2k(\Theta)p)p_{tt}-h(\Theta)\Delta
p-b\Delta p_{t}=2k(\Theta)(p_{t})^{2},\qquad&\text{in}&\
\Omega&\times(0,T),\\\
&m\Theta_{t}+\ell\Theta-\kappa_{\textup{a}}\Delta\Theta=\mathcal{Q}(p_{t}),\qquad&\text{in}&\
\Omega&\times(0,T),\\\
&(p,p_{t})|_{t=0}=(p_{0},p_{1}),\quad\Theta|_{t=0}=\Theta_{0},\end{aligned}\right.$

with homogeneous Dirichlet conditions (1.12b).

It is essential to note that the main difficulty in proving Theorem 2.2 lies
in obtaining energy estimates that are uniform with respect to $\tau$. In
other words, our goal is to prevent the constants from becoming infinitely
large as $\tau$ approaches zero in the estimates. This justifies the process
of passing to the limit when the parameter $\tau$ tends to zero. The
uniformity in $\tau$ is not a requirement for the well-posedness result in
Theorem 2.1, but it is essential in the proof of Theorem 2.2. Lastly, we
remark that as the estimates are dependent on the time $T>0$, the existence of
solutions is local in nature. However, we can expect to reach global well-
posedness by taking the approach in [22].

##  3\. Preliminaries and assumptions

In this section, we introduce a few notations and collect some helpful
embedding results and inequalities that we will repeatedly use in the proofs.

###  3.1. Notation

Throughout the paper, we assume that $\Omega\subset\mathbb{R}^{d}$, where
$d\in\\{1,2,3\\}$, is a bounded and smooth domain. We denote by $T>0$ the
final propagation time. The letter $C$ denotes a generic positive constant
that does not depend on time, and can have different values on different
occasions. We often write $f\lesssim g$ where there exists a constant $C>0$,
independent of parameters of interest such that $f\leq Cg$. We often omit the
spatial and temporal domain when writing norms; for example,
$\|\cdot\|_{L^{p}L^{q}}$ denotes the norm in $L^{p}(0,T;L^{q}(\Omega))$.

###  3.2. Inequalities and embedding results

In the upcoming analysis, we shall employ the continuous embeddings
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$, $H^{1}(\Omega)\hookrightarrow
L^{6}(\Omega)$ and $H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$. In
particular, using Poincaré’s inequality we obtain for $v\in H^{1}_{0}(\Omega)$
(see [25, Theorem 7.18])

(3.1) $\displaystyle\text{if}\ d>2,\quad\|v\|_{L^{p}}\leq C\|\nabla
v\|_{L^{2}}\quad\text{for}\quad 2\leq p\leq\frac{2d}{d-2},$
$\displaystyle\text{if}\ d=2,\quad\|v\|_{L^{p}}\leq C\|\nabla
v\|_{L^{2}}\quad\text{for}\quad 2\leq p<\infty.$

Moreover, taking into account the boundedness of the operator
$(-\Delta)^{-1}:L^{2}(\Omega)\rightarrow H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$,
we find the inequality

$\qquad\|v\|_{L^{\infty}}\leq C_{1}\|v\|_{H^{2}}\leq C_{2}\|\Delta
v\|_{L^{2}}.$

We will also call on the 1D-embedding $H^{1}(0,T,L^{2}(\Omega))\hookrightarrow
C(0,T;L^{2}(\Omega))$. In fact, this last embedding combined with Poincaré’s
inequality, yields for all $v\in H^{1}(0,T,L^{2}(\Omega))$

(3.2) $\displaystyle\max_{t\in[0,T]}\|v(t)\|_{L^{2}}$ $\displaystyle\leq
C(\|v\|_{L^{2}L^{2}}+\|v_{t}\|_{L^{2}L^{2}})$ $\displaystyle\leq C(\|\nabla
v\|_{L^{2}L^{2}}+\|v_{t}\|_{L^{2}L^{2}}),$

where the constant $C>0$ depends only on $T$ (see, e.g. [6, Theorem 2, p.
286]).
We recall Young’s $\varepsilon$-inequality

(3.3) $xy\leq\varepsilon x^{n}+C(\varepsilon)y^{m},\quad\text{where}\quad\
x,y>0,\quad 1<m,n<\infty,\quad\frac{1}{m}+\frac{1}{n}=1,$

and $C(\varepsilon)=(\varepsilon n)^{-m/n}m^{-1}$.
Further, we will make use of Ladyzhenskaya’s inequality for $u\in
H^{1}(\Omega)$

(3.4) $\|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1-d/4}\|u\|_{H^{1}}^{d/4},\qquad
1\leq d\leq 4.$

We state a version of Gronwall’s inequality which will be utilized in the
proofs and has been provided in [21].

######  Lemma 3.1.

Let $I=[0,t]$ and let $\alpha,\beta:I\rightarrow\mathbb{R}$ be locally
integrable functions. Given $u,v:I\rightarrow\mathbb{R}$ such that $v$ is non-
negative and integrable and $u$ is in $C^{1}(I)$. We assume that

(3.5) $u^{\prime}(t)+v(t)\leq\alpha(t)u(t)+\beta(t),\ \text{for}\ t\in I,\quad
u(0)=u_{0}.$

Then, it holds that

(3.6) $u(t)+\int_{0}^{t}v(s)\,\textup{d}s\leq
u_{0}e^{A(t)}+\int_{0}^{t}\beta(s)e^{A(t)-A(s)}\,\textup{d}s,$

where

(3.7) $A(t)=\int_{0}^{t}\alpha(s)\,\textup{d}s.$

###  3.3. Assumptions

In accordance with the perceived polynomial growth of the speed of sound
$c=c(\Theta)$, we make the following assumptions on the functions $h$ and $k$.
We assume that $h\in C^{2}(\mathbb{R})$ and there exists $h_{1}>0$ such that

(H1) $h(s)\geq h_{1},\quad\forall s\in\mathbb{R}.$ Moreover, assume that there
exist $\gamma_{1}>0$ and $C>0$, such that (H2) $|h^{\prime\prime}(s)|\leq
C(1+|s|^{\gamma_{1}}),\quad\forall s\in\mathbb{R}.$ Using Taylor’s formula, we
also have (H3) $|h^{\prime}(s)|\leq C(1+|s|^{1+\gamma_{1}}),\quad\forall
s\in\mathbb{R}.$

Since the function $k$ is related to the speed of sound by the formula (1.11),
it follows that

(K1) $|k(s)|\leq k_{1}:=\frac{\beta_{\text{acous}}}{\rho h_{1}}.$ Further, we
have (3.9a) $\displaystyle|k^{\prime\prime}(s)|$ $\displaystyle\lesssim
k_{1}^{2}|h^{\prime\prime}(s)|+k_{1}^{3}|h^{\prime}(s)|^{2}\lesssim
k_{1}^{2}(1+|s|^{\gamma_{1}})+k_{1}^{3}(1+|s|^{1+\gamma_{1}})^{2},$ which by
using Taylor’s formula, implies that there exists $\gamma_{2}>0$, such that
(K2)
$|k^{\prime}(s)|\lesssim(1+|s|^{1+\gamma_{2}}),\qquad|k^{\prime\prime}(s)|\lesssim(1+|s|^{\gamma_{2}}).$

##  4\. The hyperbolic bioheat equation

In this section, we consider the hyperbolic heat equation

(4.1) $\left\\{\begin{aligned} &m\Theta_{t}+\nabla\cdot
q+\ell\Theta=f,\qquad&\text{in}\ \Omega\times(0,T),\\\ &\tau
q_{t}+q+\kappa_{\textup{a}}\nabla\Theta=0,\qquad&\text{in}\
\Omega\times(0,T),\end{aligned}\right.$

together with the initial conditions in (1.12c) and the boundary condition in
(1.12b). Our main goal is to prove a priori estimates under minimal
assumptions on the initial data $\Theta_{0}$ and $q_{0}$ and on the the source
term $f$.

In order to state and prove our main result, we define the total energy
associated to (4.1) as

(4.2) $E^{\tau}[\Theta,q](t):=\sum_{k=0}^{2}E_{k}[\Theta,q](t),\quad t\geq 0$

where the energies $E_{k},k=0,1,2$ are given by

(4.3)
$E_{k}[\Theta,q](t):=\frac{1}{2}\Big{(}m\kappa_{\textup{a}}\|\partial_{t}^{k}\Theta(t)\|^{2}_{L^{2}}+\tau\|\partial_{t}^{k}q(t)\|^{2}_{L^{2}}\Big{)}.$

We also define the associated dissipation

(4.4) $D[\Theta,q](t):=\sum_{k=0}^{2}D_{k}[p,\Theta](t)$

with

(4.5)
$D_{k}[\Theta,q](t):=\ell\kappa_{\textup{a}}\|\partial_{t}^{k}\Theta(t)\|^{2}_{L^{2}}+\|\partial_{t}^{k}q(t)\|^{2}_{L^{2}},\quad
k=0,1,2.$

Since, the coefficients in (1.12a) depends on $\Theta$, we should establish
some higher order estimates on $\Theta$ which will allow us to control the
$L^{\infty}$-norm of $\Theta$ through the Sobolev embedding theorem. Hence, we
introduce the following energy in terms of $\Theta$ only

(4.6)
$\mathcal{E}[\Theta](t):=\mathcal{E}_{0}[\Theta](t)+\mathcal{E}_{1}[\Theta](t),\quad
t\geq 0$

where $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ are defined as follows

(4.7) $\left\\{\begin{aligned}
&\mathcal{E}_{0}[\Theta](t):=\frac{m\kappa_{\textup{a}}}{2}(\|\Theta(t)\|^{2}_{L^{2}}+\|\Theta_{t}(t)\|^{2}_{L^{2}}+\|\Theta_{tt}(t)\|^{2}_{L^{2}}),\\\
&\mathcal{E}_{1}[\Theta](t):=\frac{m+\tau\ell}{2}\|\nabla\Theta(t)\|^{2}_{L^{2}}+\kappa_{\textup{a}}\|\nabla\Theta_{t}(t)\|^{2}_{L^{2}}+\kappa_{\textup{a}}\|\Delta\Theta(t)\|^{2}_{L^{2}}.\end{aligned}\right.$

The associated dissipation rate to $\mathcal{E}[\Theta](t)$ is

(4.8)
$\mathcal{D}[\Theta](t):=\mathcal{D}_{0}[\Theta](t)+\mathcal{D}_{1}[\Theta](t),$

with

(4.9) $\left\\{\begin{aligned}
&\mathcal{D}_{0}[\Theta](t):=\ell\kappa_{\textup{a}}(\|\Theta(t)\|^{2}_{L^{2}}+\|\Theta_{t}(t)\|^{2}_{L^{2}}+\|\Theta_{tt}(t)\|^{2}_{L^{2}}),\\\
&\mathcal{D}_{1}[\Theta](t):=\ell\|\nabla\Theta(t)\|^{2}_{L^{2}}+\kappa_{\textup{a}}\|\nabla\Theta_{t}(t)\|^{2}_{L^{2}}+\kappa_{\textup{a}}\|\Delta\Theta(t)\|^{2}_{L^{2}}.\end{aligned}\right.$

The definition of $\mathcal{E}[\Theta](t)$ and $\mathcal{D}[\Theta](t)$ are
inspired from a damped wave equation in $\Theta$ that can be obtained by
combining the two equations in system (4.1). See equation (4.20) below.

The following lemma will allow us to estimate $\mathcal{E}[\Theta]$ in terms
of $E^{\tau}[\Theta,q]$ uniformly with respect to $\tau$.

######  Lemma 4.1.

Let $\bar{\tau}>0$ be a fixed small number. Then for all $t\geq 0$, the
estimate

(4.10) $\displaystyle\mathcal{E}[\Theta](t)\lesssim(1+\bar{\tau}+$
$\displaystyle\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](t)+\|f\|^{2}_{H^{1}L^{2}}\big{)}.$

holds uniformly in $\tau\in(0,\bar{\tau}]$. The hidden constant in (4.10) does
not depend on $\tau$.

###### Proof.

It is clear that we have for all $t\geq 0$,

(4.11) $\mathcal{E}_{0}[\Theta](t)\leq E^{\tau}[\Theta,q](t).$

Now, we want to show that $\mathcal{E}_{1}[\Theta](t)$ is also bounded by
$E^{\tau}[\Theta,q](t)$, we can simply make use of the second equation in
(4.1) to get

(4.12) $\kappa_{\textup{a}}^{2}\|\nabla\Theta\|_{L^{2}}^{2}\leq
2\tau^{2}\|q_{t}\|_{L^{2}}^{2}+2\|q\|_{L^{2}}^{2}\leq
C(\tau)E^{\tau}[\Theta,q].$

However the constant $C(\tau)\rightarrow\infty$ as $\tau\rightarrow 0$. To
avoid this, we take the time derivative $\partial_{t}^{k},k=0,1$ of the system
(4.1) to obtain

(4.13) $\left\\{\begin{aligned}
&m\partial_{t}^{k}\Theta_{t}+\nabla\cdot(\partial_{t}^{k}q)+\ell\partial_{t}^{k}\Theta=\partial_{t}^{k}f,\qquad&\text{in}\
\Omega\times(0,T),\\\
&\tau\partial_{t}^{k}q_{t}+\partial_{t}^{k}q+\kappa_{\textup{a}}\nabla(\partial_{t}^{k}\Theta)=0,\qquad&\text{in}\
\Omega\times(0,T).\end{aligned}\right.$

We multiply the second equation in (4.13) by
$\nabla(\partial_{t}^{k}\Theta),k=0,1$ and integrate over $\Omega$. Using
integration by parts, we obtain

(4.14)
$\kappa_{\textup{a}}\|\nabla(\partial_{t}^{k}\Theta)\|_{L^{2}}^{2}=-\tau\int_{\Omega}\partial_{t}^{k}q_{t}\cdot\nabla(\partial_{t}^{k}\Theta)\,\textup{d}x+\int_{\Omega}\nabla\cdot(\partial_{t}^{k}q)\partial_{t}^{k}\Theta\,\textup{d}x.$

Note that we can recover the last term on the right from the first equation in
(4.13) by testing by $\partial_{t}^{k}\Theta$ and integrating over $\Omega$.
Hence, we get

(4.15)
$\int_{\Omega}\nabla\cdot(\partial_{t}^{k}q)\partial_{t}^{k}\Theta\,\textup{d}x=-m\int_{\Omega}\partial_{t}^{k}\Theta_{t}\partial_{t}^{k}\Theta\,\textup{d}x-\ell\|\partial_{t}^{k}\Theta\|^{2}_{L^{2}}+\int_{\Omega}\partial_{t}^{k}f\partial_{t}^{k}\Theta\,\textup{d}x.$

Collecting (4.14) and (4.15), we have

(4.16)
$\displaystyle\kappa_{\textup{a}}\|\nabla(\partial_{t}^{k}\Theta)\|_{L^{2}}^{2}=-\tau$
$\displaystyle\int_{\Omega}\partial_{t}^{k}q_{t}\cdot\nabla(\partial_{t}^{k}\Theta)\,\textup{d}x-m\int_{\Omega}\partial_{t}^{k}\Theta_{t}\partial_{t}^{k}\Theta\,\textup{d}x$
$\displaystyle-\ell\|\partial_{t}^{k}\Theta\|^{2}_{L^{2}}+\int_{\Omega}\partial_{t}^{k}f\partial_{t}^{k}\Theta\,\textup{d}x.$

Thus using Cauchy-Schwarz and Young inequalities, it results that

(4.17)
$\displaystyle\frac{\kappa_{\textup{a}}}{2}\|\nabla\partial_{t}^{k}\Theta\|_{L^{2}}^{2}+\frac{\ell}{2}\|\partial_{t}^{k}\Theta\|_{L^{2}}^{2}$
$\displaystyle\leq\frac{\tau^{2}}{2\kappa_{\textup{a}}}\|\partial_{t}^{k}q_{t}\|^{2}_{L^{2}}+\frac{m^{2}}{\ell}\|\partial_{t}^{k}\Theta_{t}\|^{2}_{L^{2}}+\frac{1}{\ell}\|\partial_{t}^{k}f\|^{2}_{L^{2}}$
$\displaystyle\leq\Big{(}\frac{2m}{\ell\kappa_{\textup{a}}}+\frac{\tau}{\kappa_{\textup{a}}}\Big{)}E^{\tau}[\Theta,q]+\frac{1}{\ell}\|\partial_{t}^{k}f\|^{2}_{L^{2}},\quad
k=0,1.$

The above estimate also implies

(4.18)
$\frac{\tau\ell}{2}\|\nabla\Theta\|_{L^{2}}^{2}\lesssim(\tau+\tau^{2})E^{\tau}[\Theta,q]+\tau\|f\|^{2}_{L^{2}}.$

Next, we focus on estimating the term $\|\Delta\Theta\|_{L^{2}}$ in
$\mathcal{E}_{1}[\Theta](t)$. First, we take the time derivative of the first
equation in (4.1) and apply the divergence operator to the second equation, so
that we have

(4.19) $\left\\{\begin{aligned} &m\Theta_{tt}+\nabla\cdot
q_{t}+\ell\Theta_{t}=f_{t},\\\ &\tau\nabla\cdot q_{t}+\nabla\cdot
q+\kappa_{\textup{a}}\Delta\Theta=0.\\\ \end{aligned}\right.$

Combining these two equations, and using the first equation in (4.1), we
obtain

(4.20) $\tau
m\Theta_{tt}+(m+\tau\ell)\Theta_{t}+\ell\Theta-\kappa_{\textup{a}}\Delta\Theta=f+\tau
f_{t}.$

From here, we infer that

(4.21) $\displaystyle\kappa_{\textup{a}}^{2}\|\Delta\Theta\|^{2}_{L^{2}}$
$\displaystyle\lesssim\ell^{2}\|\Theta\|^{2}_{L^{2}}+(m+\tau\ell)^{2}\|\Theta_{t}\|^{2}_{L^{2}}+\tau^{2}m^{2}\|\Theta_{tt}\|^{2}_{L^{2}}+\|f\|^{2}_{L^{2}}+\tau^{2}\|f_{t}\|^{2}_{L^{2}}$
$\displaystyle\lesssim(1+\tau^{2})\big{(}E^{\tau}[\Theta,q]+\|f\|^{2}_{H^{1}L^{2}}\big{)}.$

Putting together the estimates (4.11), (4.17), (4.18), (4.21) and noting that
$\tau\in(0,\bar{\tau}]$, we get the estimate (4.10). This completes the proof
of Lemma 4.1. ∎

######  Proposition 4.1.

Given $T>0$, and $f\in H^{2}(0,T;L^{2}(\Omega))$. Assume that the initial data
satisfy

$(\Theta_{0},q_{0})\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\times(H^{1}(\Omega))^{d}.$

Then, the system (4.1) has a unique solution $(\Theta,q)\in X_{\Theta}\times
X_{q}$. Furthermore, the solution satisfies

(4.22)
$\displaystyle\mathcal{E}[\Theta](t)+\int_{0}^{t}\mathcal{D}[\Theta](s)\,\textup{d}s\lesssim(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}$

and

(4.23)
$\displaystyle\|q(t)\|_{H^{1}}^{2}+\sum_{k=0}^{2}\int^{t}_{0}\|\partial_{t}^{k}q(s)\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\|q_{0}\|_{H^{1}}^{2}+(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}$

for all $0\leq t\leq T$, where the hidden constants in (4.22) and (4.23) are
uniform with respect to $\tau\in(0,\bar{\tau}]$.

We rely on the Faedo–Galerkin approach to prove the result above. In the first
step, we show some uniform a priori estimates to hold for the approximations
of the solution $(\Theta,q)$. Precisely, we take the eigenfunctions of the
Dirichlet–Laplacian as approximations for the temperature $\Theta$. As for the
heat flux $q$, we exploit the fact that $(H^{1}(\Omega))^{d}$ is separable,
yielding the existence of a dense sequence that can be used to approximate $q$
(see [17]). In the second step, due to the fact that the estimates are
uniform, we can pass to the limit as in [6, chapter 7] (see also [17, Chapter
1]) to obtain the existence of solutions. Further, uniqueness follows by
noting that the only solution to the homogeneous problem ($f=0,\Theta_{0}=0$
and $q_{0}=0$) is $(\Theta,q)=(0,0)$. Lastly, from the weak and weak-$\star$
semi-continuity of norms, we get that the estimate (4.22) also holds for the
solution $(\Theta,q)$. Since the approach is quite classical, our emphasis in
the following is on the energy analysis, and we refer to [6, 17] for the
procedure of passing to the limit. To start with, we prove the following
intermediate estimates.

######  Lemma 4.2.

Let $\tau\in(0,\bar{\tau}]$. Assume that $\partial_{t}^{k}f\in
L^{2}(\Omega),k=0,1,2$. Then for all $t\geq 0$, we have the following
estimates

(4.24)
$\frac{\textup{d}}{\textup{d}t}E_{k}[\Theta,q](t)+D_{k}[\Theta,q](t)\lesssim\|\partial_{t}^{k}f\|^{2}_{L^{2}},\quad
k=0,1,2.$

where the hidden constant is independent of $\tau$.

###### Proof.

Applying $\partial_{t}^{k},\ k=0,1,2$ to system (4.1), we get

(4.25) $\left\\{\begin{aligned}
&m\partial_{t}^{k}\Theta_{t}+\nabla\cdot\partial_{t}^{k}q+\ell\partial_{t}^{k}\Theta=\partial_{t}^{k}f,\qquad&\text{in}\
\Omega\times(0,T),\\\
&\tau\partial_{t}^{k}q_{t}+\partial_{t}^{k}q+\kappa_{\textup{a}}\nabla\partial_{t}^{k}\Theta=0,\qquad&\text{in}\
\Omega\times(0,T).\end{aligned}\right.$

Multiplying the first equation in (4.25) by
$\kappa_{\textup{a}}\partial_{t}^{k}\Theta$ and integrating over $\Omega$, it
follows

(4.26)
$\frac{m\kappa_{\textup{a}}}{2}\frac{\textup{d}}{\textup{d}t}\|\partial_{t}^{k}\Theta\|^{2}_{L^{2}}+\kappa_{\textup{a}}\int_{\Omega}\partial_{t}^{k}\Theta\nabla\cdot\partial_{t}^{k}q\,\textup{d}x+\ell\kappa_{\textup{a}}\|\partial_{t}^{k}\Theta\|^{2}_{L^{2}}=\kappa_{\textup{a}}\int_{\Omega}\partial_{t}^{k}f\partial_{t}^{k}\Theta\,\textup{d}x.$

Integrating by parts the second term on the left and using the fact that
$\partial_{t}^{k}\Theta|_{\partial\Omega}=0$ (due to (1.12b)), we obtain

(4.27)
$\frac{m\kappa_{\textup{a}}}{2}\frac{\textup{d}}{\textup{d}t}\|\partial_{t}^{k}\Theta\|^{2}_{L^{2}}-\kappa_{\textup{a}}\int_{\Omega}\partial_{t}^{k}q\cdot\nabla\partial_{t}^{k}\Theta\,\textup{d}x+\ell\kappa_{\textup{a}}\|\partial_{t}^{k}\Theta\|^{2}_{L^{2}}=\kappa_{\textup{a}}\int_{\Omega}\partial_{t}^{k}f\partial_{t}^{k}\Theta\,\textup{d}x.$

Next, multiplying the second equation in (4.25) by $\partial_{t}^{k}q$ and
integrating over $\Omega$, we get

(4.28)
$\frac{\tau}{2}\frac{\textup{d}}{\textup{d}t}\|\partial_{t}^{k}q\|^{2}_{L^{2}}+\|\partial_{t}^{k}q\|^{2}_{L^{2}}+\kappa_{\textup{a}}\int_{\Omega}\partial_{t}^{k}q\cdot\nabla\Theta\,\textup{d}x=0.$

Collecting (4.27) and (4.28) yields

(4.29)
$\frac{\textup{d}}{\textup{d}t}E_{k}[\Theta,q]+D_{k}[\Theta,q]=\kappa_{\textup{a}}\int_{\Omega}\partial_{t}^{k}f\partial_{t}^{k}\Theta\,\textup{d}x,$

where $E_{k}[\Theta,q]$ and $D_{k}[\Theta,q]$ are given by (4.3), (4.5),
respectively. Using Cauchy–Schwarz inequality together with Young’s inequality
on the right-hand side gives the desired estimate (4.24). ∎

######  Remark 1.

Recalling (4.2) and using (4.24), we obtain

(4.30)
$\frac{\textup{d}}{\textup{d}t}E^{\tau}[\Theta,q]+cE^{\tau}[\Theta,q]\lesssim\|f\|^{2}_{L^{2}}+\|f_{t}\|^{2}_{L^{2}}+\|f_{tt}\|^{2}_{L^{2}}$

with

(4.31) $c=\min\left\\{\frac{\ell}{m},\frac{2}{\tau}\right\\}.$

Multiplying (4.30) by $e^{ct}$, we obtain

(4.32)
$\frac{\textup{d}}{\textup{d}t}\left(e^{ct}E^{\tau}[\Theta,q]\right)(t)\lesssim
e^{ct}(\|f\|^{2}_{L^{2}}+\|f_{t}\|^{2}_{L^{2}}+\|f_{tt}\|^{2}_{L^{2}}),\quad
t\geq 0$

Hence, integrating in time, we have

(4.33) $\displaystyle E^{\tau}[\Theta,q](t)$ $\displaystyle\lesssim
e^{-ct}E^{\tau}[\Theta,q](0)+\int_{0}^{t}e^{-c(t-s)}(\|f(s)\|^{2}_{L^{2}}+\|f_{t}(s)\|^{2}_{L^{2}}+\|f_{tt}(s)\|^{2}_{L^{2}})\,\textup{d}s$
$\displaystyle\lesssim
e^{-ct}E^{\tau}[\Theta,q](0)+\frac{1-e^{-ct}}{c}\|f\|_{H^{2}L^{2}}^{2}.$

It is clear from (4.33) that if $f\in H^{2}(0,t,L^{2}(\Omega))$, then
$E^{\tau}[\Theta,q](t)$ has an exponential decay.

###  4.1. Proof of Proposition 4.1

First, we integrate over time the estimate (4.24), sum up over $k=0,1,2$ and
recall (4.2) and (4.4) to find

(4.34) $E^{\tau}[\Theta,q](t)+\int_{0}^{t}D[\Theta,q](s)\,\textup{d}s\lesssim
E^{\tau}[\Theta,q](0)+\|f\|_{H^{2}L^{2}}^{2},$

which also implies, by recalling (4.7) and (4.9) that

(4.35)
$\mathcal{E}_{0}[\Theta](t)+\int_{0}^{t}\mathcal{D}_{0}[\Theta](s)\,\textup{d}s\lesssim
E^{\tau}[\Theta,q](0)+\|f\|_{H^{2}L^{2}}^{2}.$

On the other hand, applying the divergence operator to the second equation in
(4.1) and testing the resulting equation by $\Delta\Theta$, we obtain for all
$t\geq 0$

(4.36) $\tau\int_{\Omega}\nabla\cdot
q_{t}\Delta\Theta\,\textup{d}x+\int_{\Omega}\nabla\cdot
q\Delta\Theta\,\textup{d}x+\kappa_{\textup{a}}\|\Delta\Theta\|^{2}_{L^{2}}=0.$

Next, we multiply the first equation in (4.1) by $\Delta\Theta$ and integrate
by parts, to get

(4.37) $\displaystyle\int_{\Omega}\nabla\cdot q\Delta\Theta\,\textup{d}x$
$\displaystyle=\frac{m}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla\Theta\|^{2}_{L^{2}}+\ell\|\nabla\Theta\|^{2}_{L^{2}}+\int_{\Omega}f\Delta\Theta\,\textup{d}x.$

Further, differentiating the first equation in (4.1) with respect to $t$, we
find

(4.38) $m\Theta_{tt}+\nabla\cdot q_{t}+\ell\Theta_{t}=f_{t}.$

Testing the above equation by $\tau\Delta\Theta$ and integrating over
$\Omega$, it follows

(4.39) $\displaystyle\tau\int_{\Omega}\nabla\cdot
q_{t}\Delta\Theta\,\textup{d}x$ $\displaystyle=-\tau
m\int_{\Omega}\Theta_{tt}\Delta\Theta\,\textup{d}x+\frac{\tau\ell}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla\Theta\|^{2}_{L^{2}}+\tau\int_{\Omega}f_{t}\Delta\Theta\,\textup{d}x.$

Plugging (4.37) and (4.39) into (4.36), we deduce

(4.40)
$\displaystyle\frac{1}{2}(m+\tau\ell)\frac{\textup{d}}{\textup{d}t}\|\nabla\Theta\|^{2}_{L^{2}}+\ell\|\nabla\Theta\|^{2}_{L^{2}}+\kappa_{\textup{a}}\|\Delta\Theta\|^{2}_{L^{2}}$
$\displaystyle=\tau$ $\displaystyle
m\int_{\Omega}\Theta_{tt}\Delta\Theta\,\textup{d}x-\int_{\Omega}f\Delta\Theta\,\textup{d}x-\tau\int_{\Omega}f_{t}\Delta\Theta\,\textup{d}x.$

Therefore, by applying Cauchy-Schwarz and Young inequalities, we have

(4.41)
$\frac{1}{2}(m+\tau\ell)\frac{\textup{d}}{\textup{d}t}\|\nabla\Theta\|^{2}_{L^{2}}+\ell\|\nabla\Theta\|^{2}_{L^{2}}+\frac{\kappa_{\textup{a}}}{4}\|\Delta\Theta\|^{2}_{L^{2}}\lesssim\tau^{2}\|\Theta_{tt}\|^{2}_{L^{2}}+\|f\|^{2}_{L^{2}}+\tau^{2}\|f_{t}\|^{2}_{L^{2}}.$

Integrating with respect to $t$, to get

(4.42)
$\displaystyle\frac{1}{2}(m+\tau\ell)\|\nabla\Theta(t)\|^{2}_{L^{2}}+\ell\int_{0}^{t}\|\nabla\Theta\|^{2}_{L^{2}}+\frac{\kappa_{\textup{a}}}{4}\int_{0}^{t}\|\Delta\Theta\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\frac{1}{2}(m+\tau\ell)\|\nabla\Theta_{0}\|^{2}_{L^{2}}+\tau^{2}\int_{0}^{t}\|\Theta_{tt}\|^{2}_{L^{2}}+\|f\|^{2}_{L^{2}L^{2}}+\tau^{2}\|f_{t}\|^{2}_{L^{2}L^{2}}.$

Using inequality (4.10), we have for all $\tau\in(0,\bar{\tau}]$

(4.43) $\displaystyle(1+\tau)\|\nabla\Theta(t)\|_{L^{2}}^{2}$
$\displaystyle\lesssim(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](t)+\|f(t)\|^{2}_{H^{1}L^{2}}\big{)}.$

Further, since

(4.44) $\|\partial_{t}^{k}f(0)\|^{2}_{L^{2}}\leq
C_{T}\big{(}\|\partial_{t}^{k}f\|^{2}_{L^{2}L^{2}}+\|\partial_{t}^{k}f_{t}\|^{2}_{L^{2}L^{2}}\big{)},\quad
k=0,1,$

it follows that

(4.45)
$(1+\tau)\|\nabla\Theta_{0}\|_{L^{2}}^{2}\lesssim(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}.$

Moreover, from estimate (4.34), we have

(4.46)
$\tau^{2}\int_{0}^{t}\|\Theta_{tt}\|^{2}_{L^{2}}\lesssim\bar{\tau}^{2}\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}.$

Combining the estimates (4.42), (4.45) and (4.46), we obtain

(4.47) $\displaystyle\frac{1}{2}(m+$
$\displaystyle\tau\ell)\|\nabla\Theta(t)\|^{2}_{L^{2}}+\ell\int_{0}^{t}\|\nabla\Theta\|^{2}_{L^{2}}\,\textup{d}s+\frac{\kappa_{\textup{a}}}{4}\int_{0}^{t}\|\Delta\Theta\|^{2}_{L^{2}}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}.$

Next, according to the first inequality in (4.17) for $k=1$ and the estimate
(4.34), we have for all $t\geq 0$

(4.48)
$\displaystyle\kappa_{\textup{a}}\int_{0}^{t}\|\nabla\Theta_{t}(s)\|^{2}_{L^{2}}\,\textup{d}s$
$\displaystyle\lesssim\tau^{2}\int_{0}^{t}\|q_{tt}(s)\|^{2}_{L^{2}}\,\textup{d}s+\int_{0}^{t}\|\Theta_{tt}(s)\|^{2}_{L^{2}}\,\textup{d}s+\|f_{t}\|^{2}_{L^{2}L^{2}}$
$\displaystyle\lesssim(1+\bar{\tau}^{2})(E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}).$

On the other hand, thanks to the 1D embedding $H^{1}(0,T)\hookrightarrow
L^{\infty}(0,T)$, we have $f,f_{t}\in L^{\infty}(0,T;L^{2}(\Omega))$. Then,
the estimate (4.10) together with (4.34) gives for all $t\geq 0$

(4.49)
$\displaystyle\kappa_{\textup{a}}\|\nabla\Theta_{t}(t)\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}.$

Similarly, we make use of (4.10), (4.34) and the fact that $f,f_{t}\in
L^{\infty}(0,T;L^{2}(\Omega))$ to get

(4.50)
$\displaystyle\kappa_{\textup{a}}\|\Delta\Theta(t)\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}.$

Collecting the estimates (4.35), (4.47), (4.48), (4.49) and (4.50), we arrive
at the estimate (4.22).

To establish estimate (4.23), first observe that from (4.34), we immediately
have for $t\geq 0$

(4.51)
$\int_{0}^{t}(\|q(s)\|_{L^{2}}^{2}+\|q_{t}(s)\|_{L^{2}}^{2}+\|q_{tt}(s)\|_{L^{2}}^{2})\,\textup{d}s\lesssim
E^{\tau}[\Theta,q](0)+\|f\|_{H^{2}L^{2}}^{2}.$

Thus, it remains to prove that $q$ is in $(H^{1}(\Omega))^{d}$. From the
second equation in (4.1), we obtain

(4.52)
$q(x,t)=q_{0}(x)-\frac{\kappa_{\textup{a}}}{\tau}\int_{0}^{t}\nabla\Theta
e^{-(t-s)/\tau}\,\textup{d}s,$

which implies that for all $t\geq 0$

(4.53)
$\|q(t)\|_{H^{1}}\leq\|q_{0}\|_{H^{1}}+\frac{\kappa_{\textup{a}}}{\tau}\int_{0}^{t}e^{-(t-s)/\tau}\|\nabla\Theta(s)\|_{H^{1}}\,\textup{d}s.$

Using elliptic regularity, one finds

(4.54) $\displaystyle\|q(t)\|_{H^{1}}$
$\displaystyle\leq\|q_{0}\|_{H^{1}}+\kappa_{\textup{a}}(1-e^{-t/\tau})\|\Delta\Theta\|_{L^{\infty}L^{2}}$
$\displaystyle\lesssim\|q_{0}\|_{H^{1}}+\|\Delta\Theta\|_{L^{\infty}L^{2}}.$

This last estimate along with (4.50) gives

(4.55) $\displaystyle\|q(t)\|_{H^{1}}^{2}\lesssim\|$ $\displaystyle
q_{0}\|_{H^{1}}^{2}+(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E^{\bar{\tau}}[\Theta,q](0)+\|f\|^{2}_{H^{2}L^{2}}\big{)}$

Summing up the estimates (4.51) and (4.55), we obtain (4.23). Further, from
(4.22), (4.23), we deduce that $(\Theta,q)\in X_{\Theta}\times X_{q}$. This
concludes the proof of Proposition 4.1.

##  5\. The linearized Westervelt equation

In this section, we consider the linearization of the Westervelt equation
(1.1):

(5.1) $\alpha(x,t)p_{tt}-r(x,t)\Delta p-b\Delta p_{t}=g(x,t),\qquad
x\in\Omega,\quad t\geq 0,$

supplemented by initial conditions (1.12c) and boundary conditions (1.12b).

We define the energies associated to (5.1) as follows

(5.2) $\displaystyle E_{1}[p](t)$
$\displaystyle:=\frac{1}{2}\big{(}\|\sqrt{\alpha(t)}p_{t}(t)\|^{2}_{L^{2}}+\|\sqrt{r(t)}\nabla
p(t)\|^{2}_{L^{2}}\big{)},$ $\displaystyle E_{2}[p](t)$
$\displaystyle:=\frac{1}{2}\big{(}\|\sqrt{\alpha(t)}p_{tt}(t)\|^{2}_{L^{2}}+\|\sqrt{r(t)}\nabla
p_{t}(t)\|^{2}_{L^{2}}+\|\sqrt{b}\Delta p(t)\|^{2}_{L^{2}}\big{)},$
$\displaystyle E_{3}[p](t)$ $\displaystyle:=\frac{1}{2}\big{(}\|\sqrt{b}\nabla
p_{tt}(t)\|_{L^{2}}^{2}+\|\sqrt{b}\nabla\Delta p(t)\|^{2}_{L^{2}}\big{)}.$

The total acoustic energy is given by

(5.3) $\mathfrak{E}[p](t):=\sum_{k=1}^{3}E_{k}[p](t),\qquad t\geq 0.$

We denote by $\mathfrak{D}[p]$ its associated dissipation rate given by

(5.4) $\displaystyle\mathfrak{D}[p](t):=\mathfrak{D}_{0}[p](t)+b\|\nabla\Delta
p_{t}\|^{2}_{L^{2}}+b\|\Delta p_{tt}(s)\|^{2}_{L^{2}},$

where $\mathfrak{D}_{0}[p]$ is given by

(5.5) $\displaystyle\mathfrak{D}_{0}[p](t):=b\|\nabla p_{t}(t)$
$\displaystyle\|^{2}_{L^{2}}+b\|\nabla
p_{tt}(t)\|^{2}_{L^{2}}+\|\sqrt{r}\Delta p(t)\|_{L^{2}}^{2}+b\|\Delta
p_{t}(t)\|^{2}_{L^{2}}$ $\displaystyle+\|\sqrt{r}\nabla\Delta
p(t)\|^{2}_{L^{2}}+\|\sqrt{\alpha}p_{ttt}(t)\|^{2}_{L^{2}}.$

We make the following regularity and non-degeneracy assumptions on the
coefficients $\alpha$ and $r$.

######  Assumption 1.

Assume that

  1. (A)

     * •

$\alpha\in L^{\infty}(0,T;L^{\infty}(\Omega))\cap L^{2}(0,T;W^{1,3}(\Omega))$,

     * •

$\alpha_{t}\in L^{2}(0,T;L^{3}(\Omega))\cap
L^{\frac{4}{4-d}}(0,T;L^{2}(\Omega))$,

     * •

There exist $0<\alpha_{0}\leq\alpha_{1}$ such that

(5.6) $\alpha_{0}\leq\alpha(x,t)\leq\alpha_{1}\quad\text{a.e.
in}\quad\Omega\times(0,T).$

  2. (R)

     * •

$r\in L^{\infty}(0,T;L^{\infty}(\Omega))\cap L^{2}(0,T;W^{1,3}(\Omega))$,

     * •

$r_{t}\in L^{2}(0,T;L^{3}(\Omega))\cap L^{\frac{4}{4-d}}(0,T;L^{2}(\Omega))$,

     * •

There exist $0<r_{0}\leq r_{1}$

(5.7) $r_{0}\leq r(x,t)\leq r_{1}\quad\text{a.e. in}\quad\Omega\times(0,T).$

The main result of this section reads as follows.

######  Proposition 5.1.

Given $T>0$ and $g\in H^{1}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;H^{1}_{0}(\Omega))$. Let

(5.8) $(p_{0},p_{1})\in H^{3}(\Omega)\cap H^{1}_{0}(\Omega)\times
H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$

Then under Assumption 1, the linearized pressure equation (5.1) has a unique
solution $p\in X_{p}$, which satisfies for all $0\leq t\leq T$

(5.9) $\displaystyle\mathfrak{E}[p](t)+b\|\Delta
p_{t}(t)\|^{2}_{L^{2}}+\int_{0}^{t}\mathfrak{D}[p](s)\,\textup{d}s\lesssim$
$\displaystyle\,\mathfrak{E}[p](0)\exp\Big{(}\int_{0}^{t}(1+\Lambda(s))\,\textup{d}s\Big{)}$
$\displaystyle+\int_{0}^{t}\mathfrak{F}(s)\exp\Big{(}\int_{s}^{t}(1+\Lambda(\sigma))\textup{d}\sigma\Big{)}\,\textup{d}s$

where

(5.10) $\displaystyle\Lambda(t)=\|\alpha_{t}(t)\|_{L^{2}}^{2}$
$\displaystyle+\|\alpha_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\nabla
r(t)\|_{L^{2}}^{2}+\|r_{t}(t)\|_{L^{3}}^{2}$
$\displaystyle+\|\alpha_{t}(t)\|_{L^{3}}^{2}+\|\nabla
r(t)\|_{L^{3}}^{2}+\|\nabla\alpha(t)\|_{L^{3}}^{2},$

and

(5.11) $\mathfrak{F}(t)=\|\nabla g(t)\|_{L^{2}}^{2}+\|g_{t}(t)\|^{2}_{L^{2}}.$

We carry out the proof of Proposition 5.1 by means of the Faedo-Galerkin
method using the smooth eigenfunctions of the Dirichlet-Laplacian as
approximations of the solution in space, see [6]. Below, we present the energy
analysis for the mentioned approximation, which leads to uniform a priori
estimates, and we direct the reader interested in the process of taking the
limit towards [6, chapter 7].
Before embarking on the proof of Proposition 5.1, we first prove two lemmas.

######  Lemma 5.1.

Given $g\in L^{2}(\Omega)$ such that $g_{t}\in L^{2}(\Omega)$. Let Assumption
1 hold. Then for all $t\geq 0$, we have the following energy estimate

(5.12)
$\displaystyle\frac{\textup{d}}{\textup{d}t}\Big{(}E_{1}[p](t)+E_{2}[p](t)\Big{)}+b\|\nabla
p_{t}(t)\|^{2}_{L^{2}}$ $\displaystyle+b\|\nabla
p_{tt}(t)\|^{2}_{L^{2}}+\|\sqrt{r}\Delta p(t)\|_{L^{2}}^{2}+b\|\Delta
p_{t}(t)\|^{2}_{L^{2}}$ $\displaystyle\lesssim$
$\displaystyle\,(1+\Lambda_{1}(t))(E_{1}[p](t)+E_{2}[p](t))+\|g\|_{L^{2}}^{2}+\|g_{t}\|^{2}_{H^{-1}}.$

where

(5.13) $\displaystyle\Lambda_{1}(t)=$
$\displaystyle\,\|r_{t}(t)\|_{L^{3}}^{2}+\|\nabla
r(t)\|_{L^{3}}^{2}+\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\nabla
r(t)\|_{L^{2}}^{2}$
$\displaystyle+\|\alpha_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\alpha_{t}(t)\|_{L^{2}}^{2}.$

###### Proof.

The proof is based on ideas that are drawn from [21]. Multiplying the equation
(5.1) by $p_{t}$, integrating over $\Omega$ and using integration by parts, we
find

(5.14)
$\displaystyle\frac{\textup{d}}{\textup{d}t}E_{1}[p](t)+b\int_{\Omega}|\nabla
p_{t}|^{2}\,\textup{d}x$ $\displaystyle=$
$\displaystyle\,\int_{\Omega}gp_{t}\,\textup{d}x+\frac{1}{2}\int_{\Omega}\alpha_{t}p_{t}^{2}\,\textup{d}x-\int_{\Omega}\nabla
r\cdot\nabla pp_{t}\,\textup{d}x+\frac{1}{2}\int_{\Omega}r_{t}|\nabla
p|^{2}\,\textup{d}x.$

Our goal now is to estimate the terms on the right-hand side of (5.14).
Applying Young and Poincaré inequalities, we obtain

(5.15) $\int_{\Omega}gp_{t}\,\textup{d}x\leq
C(\varepsilon)\|g\|^{2}_{L^{2}}+\varepsilon\|\nabla p_{t}\|^{2}_{L^{2}}.$

The next two terms on the right-hand side of (5.14) can be estimated using
Hölder, Young and Poincaré’s inequalities as well as the embedding
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$. Hence, we have

(5.16) $\displaystyle\int_{\Omega}\alpha_{t}p_{t}^{2}\,\textup{d}x$
$\displaystyle\leq\|\alpha_{t}\|_{L^{2}}\|p_{t}\|_{L^{4}}^{2}$
$\displaystyle\leq C(\varepsilon)\|\alpha_{t}\|_{L^{2}}^{2}\|\sqrt{r}\nabla
p_{t}\|_{L^{2}}^{2}+\varepsilon\Big{\|}\frac{1}{\sqrt{r}}\Big{\|}_{L^{\infty}}^{2}\|\nabla
p_{t}\|_{L^{2}}^{2}.$

Considering $\varepsilon$ suitably small and using Assumption 1, we can absorb
the last terms on the above estimate by the dissipation terms on the left-hand
side of $\eqref{identity_1}$.
Likewise, we obtain

(5.17) $\displaystyle\int_{\Omega}\nabla r\cdot\nabla pp_{t}\,\textup{d}x$
$\displaystyle\leq\|\nabla r\|_{L^{2}}\|\nabla p\|_{L^{4}}\|p_{t}\|_{L^{4}}$
$\displaystyle\leq C(\varepsilon)\|\nabla r\|^{2}_{L^{2}}\|\sqrt{r}\nabla
p_{t}\|^{2}_{L^{2}}+\varepsilon\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{2}\|\sqrt{r}\Delta
p\|^{2}_{L^{2}}$

where we have also taken into account the elliptic estimate

(5.18) $\|\nabla p\|_{H^{1}}\leq\|p\|_{H^{2}}\leq C\|\Delta p\|_{L^{2}}.$

To estimate the last term on the right-hand side of (5.14), we employ the
Ladyzhenskaya inequality (3.4) together with (5.18) to find

(5.19) $\displaystyle\int_{\Omega}r_{t}|\nabla p|^{2}\,\textup{d}x$
$\displaystyle\leq\|r_{t}\|_{L^{2}}\|\nabla p\|^{2}_{L^{4}}$
$\displaystyle\lesssim\|r_{t}\|_{L^{2}}\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}\|\sqrt{r}\nabla
p\|_{L^{2}}^{2(1-\frac{d}{4})}\|\sqrt{r}\Delta p\|_{L^{2}}^{\frac{d}{2}}$
$\displaystyle\leq
C(\varepsilon)\|r_{t}\|_{L^{2}}^{\frac{4}{4-d}}\|\sqrt{r}\nabla
p\|_{L^{2}}^{2}+\varepsilon\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{\frac{4}{d}}\|\sqrt{r}\Delta
p\|^{2}_{L^{2}}.$

Altogether, the estimates above yield for all $t\geq 0$

(5.20) $\displaystyle\frac{\textup{d}}{\textup{d}t}E_{1}[p](t)+b\|\nabla
p_{t}(t)\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}E_{1}[p](t)$
$\displaystyle+(\|\alpha_{t}(t)\|_{L^{2}}^{2}+\|\nabla
r(t)\|^{2}_{L^{2}})E_{2}[p](t)$
$\displaystyle+\|g(t)\|_{L^{2}}^{2}+\varepsilon\|\sqrt{r(t)}\Delta
p(t)\|^{2}_{L^{2}}.$

In order to get an estimate for the energy $E_{2}[p]$, we differentiate (5.1)
with respect to $t$, to obtain

(5.21) $\alpha p_{ttt}-r\Delta p_{t}-b\Delta
p_{tt}=-\alpha_{t}p_{tt}+r_{t}\Delta p+g_{t}.$

Next, we multiply (5.21) by $p_{tt}$ and integrate over $\Omega$, using
integration by parts, we find

(5.22) $\displaystyle\frac{1}{2}$
$\displaystyle\,\frac{\textup{d}}{\textup{d}t}\Big{(}\|\sqrt{\alpha}p_{tt}\|^{2}_{L^{2}}+\|\sqrt{r}\nabla
p_{t}\|^{2}_{L^{2}}\Big{)}+b\|\nabla p_{tt}\|^{2}_{L^{2}}$
$\displaystyle=-\frac{1}{2}$
$\displaystyle\int_{\Omega}\alpha_{t}p_{tt}^{2}\,\textup{d}x+\frac{1}{2}\int_{\Omega}r_{t}|\nabla
p_{t}|^{2}\,\textup{d}x-\int_{\Omega}\nabla r\cdot\nabla
p_{t}p_{tt}\,\textup{d}x$ $\displaystyle+\int_{\Omega}r_{t}\Delta
pp_{tt}\,\textup{d}x+\int_{\Omega}g_{t}p_{tt}\,\textup{d}x.$

First, Ladyzhenskaya and Young inequalities along with the embedding
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$ allow us to get the following
upper bound for the first term on the right-hand side of (5.22). That is

(5.23)
$\displaystyle-\frac{1}{2}\int_{\Omega}\alpha_{t}p_{tt}^{2}\,\textup{d}x$
$\displaystyle\leq\frac{1}{2}\|\alpha_{t}\|_{L^{2}}\|p_{tt}\|_{L^{4}}^{2}$
$\displaystyle\lesssim\Big{\|}\frac{1}{\alpha}\Big{\|}_{L^{\infty}}^{1-\frac{d}{4}}\|\alpha_{t}\|_{L^{2}}\|\sqrt{\alpha}p_{tt}\|_{L^{2}}^{2(1-\frac{d}{4})}\|\nabla
p_{tt}\|_{L^{2}}^{\frac{d}{2}}$ $\displaystyle\leq
C(\varepsilon)\|\alpha_{t}\|_{L^{2}}^{\frac{4}{4-d}}\|\sqrt{\alpha}p_{tt}\|_{L^{2}}^{2}+\varepsilon\Big{\|}\frac{1}{\alpha}\Big{\|}_{L^{\infty}}^{\frac{4}{d}-1}\|\nabla
p_{tt}\|_{L^{2}}^{2}.$

We can take $\varepsilon$ as small as needed in order to absorb the last term
in the estimate above into the dissipation term on the left-hand side of
(5.22).
Similarly, we can derive the following estimate for the second term on the
right-hand side of (5.22)

(5.24) $\displaystyle\int_{\Omega}r_{t}|\nabla p_{t}|^{2}\,\textup{d}x$
$\displaystyle\leq\|r_{t}\|_{L^{2}}\|\nabla p_{t}\|^{2}_{L^{4}}$
$\displaystyle\lesssim\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{1-\frac{d}{4}}\|r_{t}\|_{L^{2}}\|\sqrt{r}\nabla
p_{t}\|_{L^{2}}^{2(1-\frac{d}{4})}\|\Delta p_{t}\|_{L^{2}}^{\frac{d}{2}}$
$\displaystyle\leq
C(\varepsilon)\|r_{t}\|_{L^{2}}^{\frac{4}{4-d}}\|\sqrt{r}\nabla
p_{t}\|^{2}_{L^{2}}+\varepsilon\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{\frac{4}{d}-1}\|\Delta
p_{t}\|^{2}_{L^{2}}.$

Moreover, making use of the embedding $H^{1}(\Omega)\hookrightarrow
L^{6}(\Omega)$ and Poincaré’s inequality, it follows

(5.25) $\displaystyle\int_{\Omega}\nabla r\cdot\nabla
p_{t}p_{tt}\,\textup{d}x$ $\displaystyle\leq\|\nabla r\|_{L^{3}}\|\nabla
p_{t}\|_{L^{2}}\|p_{tt}\|_{L^{6}}$ $\displaystyle\leq C(\varepsilon)\|\nabla
r\|^{2}_{L^{3}}\|\sqrt{r}\nabla
p_{t}\|_{L^{2}}^{2}+\varepsilon\Big{\|}\frac{1}{\sqrt{r}}\Big{\|}_{L^{\infty}}^{2}\|\nabla
p_{tt}\|^{2}_{L^{2}}.$

Again, we call on the same tools to estimate the two remaining terms on the
right of (5.22). So we can show that

(5.26) $\displaystyle\int_{\Omega}r_{t}\Delta pp_{tt}\,\textup{d}x$
$\displaystyle\leq\Big{\|}\frac{r_{t}}{b}\Big{\|}_{L^{3}}\|p_{tt}\|_{L^{6}}\|\sqrt{b}\Delta
p\|_{L^{2}}$ $\displaystyle\leq
C(\varepsilon)\|r_{t}\|_{L^{3}}^{2}\|\sqrt{b}\Delta
p\|_{L^{2}}^{2}+\varepsilon\|\nabla p_{tt}\|_{L^{2}}^{2}.$

In addition, we have

(5.27) $\int_{\Omega}g_{t}p_{tt}\,\textup{d}x\leq
C(\varepsilon)\|g_{t}\|^{2}_{L^{2}}+\varepsilon\|\nabla p_{tt}\|_{L^{2}}^{2}.$

Collecting the above estimates and selecting $\varepsilon$ small enough we
obtain for all $t\geq 0$

(5.28)
$\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\Big{(}\|\sqrt{\alpha(t)}p_{tt}(t)\|^{2}_{L^{2}}+\|\sqrt{r(t)}\nabla
p_{t}(t)\|^{2}_{L^{2}}\Big{)}+\frac{b}{2}\|\nabla p_{tt}(t)\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\Big{(}\|r_{t}(t)\|_{L^{3}}^{2}+\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\alpha_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\nabla
r(t)\|^{2}_{L^{3}}\Big{)}E_{2}[p](t)$
$\displaystyle+\varepsilon\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{\frac{4}{d}-1}\|\Delta
p_{t}(t)\|^{2}_{L^{2}}+\|g_{t}(t)\|^{2}_{L^{2}}.$

Now, we focus on establishing estimates for $\|\Delta p\|_{L^{2}}$ and
$\|\Delta p_{t}\|_{L^{2}}$. Testing the equation (5.1) by $-\Delta p$ and
integrating over $\Omega$ gives

(5.29) $\frac{b}{2}\frac{\textup{d}}{\textup{d}t}\|\Delta
p\|^{2}_{L^{2}}+\int_{\Omega}r|\Delta
p|^{2}\,\textup{d}x=-\int_{\Omega}g(\Delta p)\,\textup{d}x+\int_{\Omega}\alpha
p_{tt}\Delta p\,\textup{d}x.$

Applying Young and Poincaré inequalities, we get the following bound

(5.30) $\displaystyle\frac{b}{2}\frac{\textup{d}}{\textup{d}t}\|\Delta
p\|^{2}_{L^{2}}+\|\sqrt{r}\Delta p\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\,2\|g\|^{2}_{L^{2}}+\frac{2}{b}\|\sqrt{b}\Delta
p\|_{L^{2}}^{2}+C(\varepsilon)\|\sqrt{b}\Delta
p\|_{L^{2}}^{2}+\varepsilon\|\alpha\|_{L^{\infty}}^{2}\|\nabla
p_{tt}\|_{L^{2}}^{2}.$

Adding up (5.30) and (5.28), now the last term on the right of (5.30) can be
absorbed by the left-hand side of (5.28). Hence, we get

(5.31)
$\displaystyle\frac{\textup{d}}{\textup{d}t}E_{2}[p](t)+\frac{b}{3}\|\nabla
p_{tt}(t)\|^{2}_{L^{2}}+\|\sqrt{r(t)}\Delta p(t)\|_{L^{2}}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,(1+\|r_{t}(t)\|_{L^{3}}^{2}+\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\alpha_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\nabla
r(t)\|^{2}_{L^{3}})E_{2}[p](t)$
$\displaystyle+\varepsilon\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{\frac{4}{d}-1}\|\Delta
p_{t}(t)\|^{2}_{L^{2}}+\|g\|^{2}_{L^{2}}+\|g_{t}\|^{2}_{L^{2}}.$

Next, we multiply the equation (5.1) by $-\Delta p_{t}$ and we intergrate over
$\Omega$ to get

(5.32) $b\|\Delta p_{t}\|^{2}_{L^{2}}=\int_{\Omega}\alpha p_{tt}\Delta
p_{t}\,\textup{d}x-\int_{\Omega}r\Delta p\Delta
p_{t}\,\textup{d}x+\int_{\Omega}g(-\Delta p_{t})\,\textup{d}x.$

Using Young’s inequality and recalling Assumption 1, we obtain

(5.33) $\displaystyle b\|\Delta p_{t}\|^{2}_{L^{2}}\leq$
$\displaystyle\,C(\varepsilon)\Big{(}\|\sqrt{\alpha}p_{tt}\|^{2}_{L^{2}}+\|\sqrt{b}\Delta
p\|^{2}_{L^{2}}+\|g\|^{2}_{L^{2}}\Big{)}+\varepsilon\|\Delta
p_{t}\|^{2}_{L^{2}}.$

Taking $\varepsilon$ small enough, we find

(5.34) $b\|\Delta p_{t}(t)\|^{2}_{L^{2}}\lesssim
E_{2}[p](t)+\|g(t)\|^{2}_{L^{2}},\quad t\geq 0.$

Thus, putting (5.31) and (5.34) together, we have

(5.35)
$\displaystyle\frac{\textup{d}}{\textup{d}t}E_{2}[p](t)+\frac{b}{4}\|\nabla
p_{tt}(t)\|^{2}_{L^{2}}+\|\sqrt{r(t)}\Delta
p(t)\|_{L^{2}}^{2}+\frac{b}{2}\|\Delta p_{t}(t)\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\Big{(}1+\|r_{t}(t)\|_{L^{3}}^{2}+\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\alpha_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\nabla
r(t)\|_{L^{3}}^{2}\Big{)}E_{2}[p](t)$
$\displaystyle+\|g(t)\|^{2}_{L^{2}}+\|g_{t}(t)\|^{2}_{L^{2}}.$

Finally, summing up the estimates (5.20), (5.35) and selecting $\varepsilon$
small enough in the last term on the right-hand side of (5.20), we obtain
(5.12). This completes the proof of Lemma 5.1. ∎

######  Lemma 5.2.

Let $g\in H^{1}_{0}(\Omega)$ and $g_{t}\in L^{2}(\Omega)$. Then, under
Assumption 1, the energy $E_{3}$ satisfies for $t\geq 0$ the estimate

(5.36)
$\displaystyle\frac{\textup{d}}{\textup{d}t}E_{3}[p](t)+\|\sqrt{r}\nabla\Delta
p(t)\|^{2}_{L^{2}}+\|\sqrt{\alpha}p_{ttt}(t)\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,(1+\Lambda_{2}(t))E_{3}[p](t)+\|\sqrt{b}\Delta
p_{t}(t)\|_{L^{2}}^{2}+\varepsilon\|\nabla p_{tt}(t)\|_{L^{2}}^{2}$
$\displaystyle+\|\nabla g(t)\|^{2}_{L^{2}}+\|g_{t}(t)\|_{L^{2}}^{2}$

where

(5.37)
$\Lambda_{2}(t)=\|r_{t}(t)\|_{L^{3}}^{2}+\|\alpha_{t}(t)\|_{L^{3}}^{2}+\|\nabla\alpha(t)\|_{L^{3}}^{2}+\|\nabla
r(t)\|_{L^{3}}^{2}.$

###### Proof.

We test the linearized Westervelt equation (5.1) by $\Delta^{2}p$ and we
integrate over $\Omega$, using integration by parts, we obtain

(5.38) $\displaystyle\int_{\Omega}(r|\nabla\Delta p|^{2}+b\nabla\Delta
p_{t}\cdot\nabla\Delta p)\,\textup{d}x$ $\displaystyle=$
$\displaystyle\,\int_{\Omega}(-\alpha\nabla p_{tt}-p_{tt}\nabla\alpha-\Delta
p\nabla r-\nabla g)\cdot\nabla\Delta p\,\textup{d}x$

Using Hölder’s inequality, we find

(5.39)
$\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{b}\nabla\Delta
p\|_{L^{2}}^{2}+\|\sqrt{r}\nabla\Delta p\|^{2}_{L^{2}}$
$\displaystyle\leq\|\alpha$ $\displaystyle\|_{L^{\infty}}\|\nabla
p_{tt}\|_{L^{2}}\|\nabla\Delta
p\|_{L^{2}}+\|\nabla\alpha\|_{L^{3}}\|p_{tt}\|_{L^{6}}\|\nabla\Delta
p\|_{L^{2}}$ $\displaystyle+\|\nabla r\|_{L^{3}}\|\Delta
p\|_{L^{6}}\|\nabla\Delta p\|_{L^{2}}+\|\nabla g\|_{L^{2}}\|\nabla\Delta
p\|_{L^{2}}.$

Moreover, taking advantage of Assumption 1, the embedding
$H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$ and applying Young’s inequality,
we find

(5.40)
$\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{b}\nabla\Delta
p\|_{L^{2}}^{2}+\|\sqrt{r}\nabla\Delta p\|^{2}_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\,C(\varepsilon)\Big{(}1+\|\nabla\alpha\|_{L^{3}}^{2}\Big{)}\|\sqrt{b}\nabla\Delta
p\|^{2}_{L^{2}}+\varepsilon\|\nabla p_{tt}\|_{L^{2}}^{2}$
$\displaystyle+C(\varepsilon)\Big{(}\|\nabla
r\|_{L^{3}}^{2}\|\sqrt{b}\nabla\Delta p\|^{2}_{L^{2}}+\|\nabla
g\|^{2}_{L^{2}}\Big{)}+\varepsilon\Big{\|}\frac{1}{r}\Big{\|}_{L^{\infty}}^{2}\|\sqrt{r}\nabla\Delta
p\|_{L^{2}}^{2}.$

We can fix $\varepsilon$ as small as needed to absorb the last term on the
right-hand side into the dissipative term on the left-hand side. Hence, we
have

(5.41)
$\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{b}\nabla\Delta
p\|_{L^{2}}^{2}+\|\sqrt{r}\nabla\Delta p\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,(1+\|\nabla\alpha\|_{L^{3}}^{2}+\|\nabla
r\|_{L^{3}}^{2})\|\sqrt{b}\nabla\Delta p\|^{2}_{L^{2}}+\|\nabla
g\|^{2}_{L^{2}}+\varepsilon\|\nabla p_{tt}\|_{L^{2}}^{2}.$

Next, we multiply the time-differentiated equation (5.21) by $p_{ttt}$ and we
integrate over $\Omega$ to find

(5.42) $\frac{b}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla
p_{tt}\|^{2}_{L^{2}}+\|\sqrt{\alpha}p_{ttt}\|^{2}_{L^{2}}=\int_{\Omega}(-\alpha_{t}p_{tt}+r_{t}\Delta
p+r\Delta p_{t}+g_{t})p_{ttt}\,\textup{d}x.$

Our goal now is to estimate the terms of the right-hand side of (5.42).
First, we have by using Hölder’s inequality

(5.43) $\displaystyle\frac{b}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla
p_{tt}\|^{2}_{L^{2}}+\|\sqrt{\alpha}p_{ttt}\|^{2}_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\,\|\alpha_{t}\|_{L^{3}}\|p_{tt}\|_{L^{6}}\|p_{ttt}\|_{L^{2}}+\|r_{t}\|_{L^{3}}\|\Delta
p\|_{L^{6}}\|p_{ttt}\|_{L^{2}}$ $\displaystyle+\|r\|_{L^{\infty}}\|\Delta
p_{t}\|_{L^{2}}\|p_{ttt}\|_{L^{2}}+\|g_{t}\|_{L^{2}}\|p_{ttt}\|_{L^{2}}.$

Furthermore, making use of Young’s inequality together with the embedding
$H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, we obtain

(5.44) $\displaystyle\frac{b}{2}\frac{\textup{d}}{\textup{d}t}\|\nabla
p_{tt}\|^{2}_{L^{2}}+\|\sqrt{\alpha}p_{ttt}\|^{2}_{L^{2}}$ $\displaystyle\leq
C$
$\displaystyle(\varepsilon)\Big{(}\|\alpha_{t}\|_{L^{3}}^{2}\|\sqrt{b}\nabla
p_{tt}\|_{L^{2}}^{2}+\|r_{t}\|_{L^{3}}^{2}\|\sqrt{b}\nabla\Delta
p\|_{L^{2}}^{2}+\|\sqrt{b}\Delta
p_{t}\|_{L^{2}}^{2}+\|g_{t}\|_{L^{2}}^{2}\Big{)}$
$\displaystyle+\varepsilon\Big{\|}\frac{1}{\sqrt{\alpha}}\Big{\|}_{L^{\infty}}^{2}\|\sqrt{\alpha}p_{ttt}\|_{L^{2}}^{2}.$

By selecting $\varepsilon$ small enough, we find

(5.45) $\displaystyle\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\|\sqrt{b}\nabla
p_{tt}\|^{2}_{L^{2}}+\|\sqrt{\alpha}p_{ttt}\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\|\alpha_{t}\|_{L^{3}}^{2}\|\sqrt{b}\nabla
p_{tt}\|_{L^{2}}^{2}+\|r_{t}\|_{L^{3}}^{2}\|\sqrt{b}\nabla\Delta
p\|_{L^{2}}^{2}+\|\sqrt{b}\Delta p_{t}\|_{L^{2}}^{2}+\|g_{t}\|_{L^{2}}^{2}.$

Consequently, collecting all the estimates (5.41) and (5.45), we arrive at
(5.36). ∎

###### Proof of the Proposition 5.1.

Collecting the estimates (5.12) and (5.36) and employing Poincaré’s inequality
to find for all $t\geq 0$

(5.46)
$\displaystyle\frac{\textup{d}}{\textup{d}t}\mathfrak{E}[p](t)+\mathfrak{D}_{0}[p](t)$
$\displaystyle\lesssim$
$\displaystyle\,(1+\Lambda_{1}(t)+\Lambda_{2}(t))\mathfrak{E}[p](t)+\|\sqrt{b}\Delta
p_{t}(t)\|^{2}_{L^{2}}+\varepsilon\|\nabla p_{tt}(t)\|_{L^{2}}^{2}$
$\displaystyle+\|\nabla g(t)\|^{2}_{L^{2}}+\|g_{t}(t)\|^{2}_{L^{2}},$

where $\mathfrak{E}[p]$ and $\mathfrak{D}_{0}[p]$ are defined in (5.3) and
(5.5) respectively. So now the terms depending on $\varepsilon$ can be
absorbed into $\mathfrak{D}_{0}[p]$ for small values of $\varepsilon$. In
addition, recalling (5.34) and using Poincaré’s inequality, we obtain the
following estimate for the total acoustic energy

(5.47)
$\displaystyle\frac{\textup{d}}{\textup{d}t}\mathfrak{E}[p](t)+\mathfrak{D}_{0}[p](t)\lesssim(1+\Lambda(t))\mathfrak{E}[p](t)+\mathfrak{F}(t),$

where the functions $\Lambda$ and $\mathfrak{F}$ are given in (5.10) and
(5.11), respectively.

Next, to estimate the second term on the left-hand side of (5.9), we multiply
the equation (5.21) by $-\Delta p_{tt}$ and integrate over $\Omega$, we have

(5.48) $\displaystyle b\|\Delta p_{tt}\|^{2}_{L^{2}}=$
$\displaystyle\int_{\Omega}(\alpha p_{ttt}+\alpha_{t}p_{tt}-r_{t}\Delta
p-r\Delta p_{t}-g_{t})\Delta p_{tt}\,\textup{d}x.$

Using Hölder’s inequality, it follows that

(5.49) $\displaystyle b\|\Delta
p_{tt}\|^{2}_{L^{2}}\leq\|\sqrt{\alpha}\|_{L^{\infty}}$
$\displaystyle\|\sqrt{\alpha}p_{ttt}\|_{L^{2}}\|\Delta
p_{tt}\|_{L^{2}}+\|\alpha_{t}\|_{L^{3}}\|p_{tt}\|_{L^{6}}\|\Delta
p_{tt}\|_{L^{2}}$ $\displaystyle+\|r_{t}\|_{L^{3}}$ $\displaystyle\|\Delta
p\|_{L^{6}}\|\Delta p_{tt}\|_{L^{2}}+\|r\|_{L^{\infty}}\|\Delta
p_{t}\|_{L^{2}}\|\Delta p_{tt}\|_{L^{2}}$
$\displaystyle+\|g_{t}\|_{L^{2}}\|\Delta p_{tt}\|_{L^{2}}.$

Applying Young’s inequality, making use of the continuous embedding
$H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$, and keeping in mind Assumption
1, we find

(5.50) $\displaystyle b\|\Delta p_{tt}\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,\|\sqrt{\alpha}p_{ttt}\|_{L^{2}}^{2}+\|\alpha_{t}\|_{L^{3}}^{2}\|\sqrt{b}\nabla
p_{tt}\|_{L^{2}}^{2}+\|r_{t}\|_{L^{3}}^{2}\|\sqrt{b}\nabla\Delta
p\|_{L^{2}}^{2}$ $\displaystyle+\|\sqrt{b}\Delta
p_{t}\|_{L^{2}}^{2}+\|g_{t}\|_{L^{2}}^{2}+\varepsilon\|\Delta
p_{tt}\|_{L^{2}}^{2}.$

Fixing $\varepsilon>0$ small enough and keeping in mind (5.34), we get for all
$t\geq 0$

(5.51) $\displaystyle\frac{b}{2}\|\Delta p_{tt}(t)\|^{2}\lesssim$
$\displaystyle\,\|\sqrt{\alpha}p_{ttt}(t)\|_{L^{2}}^{2}+(\|\alpha_{t}(t)\|_{L^{3}}^{2}+\|r_{t}(t)\|_{L^{3}}^{2})E_{3}[p](t)$
$\displaystyle+E_{2}[p](t)+\|g(t)\|_{L^{2}}^{2}+\|g_{t}(t)\|_{L^{2}}^{2}.$

We sum up $\gamma\times\eqref{Delta_p_tt}$ and the estimate (5.47), then we
take $\gamma>0$ suitably small in order to absorb the first term on the right
of (5.51) by the dissipation $\mathfrak{D}_{0}[p]$ in (5.47), then, it follows
that

(5.52)
$\displaystyle\frac{\textup{d}}{\textup{d}t}\mathfrak{E}[p](t)+\mathfrak{D}_{0}[p](t)+b\|\Delta
p_{tt}(t)\|^{2}\lesssim(1+\Lambda(t))\mathfrak{E}[p](t)+\mathfrak{F}(t).$

On the other hand, by multiplying the equation (5.1) by $\Delta^{2}p_{t}$ and
integrating over $\Omega$, we obtain

(5.53) $\displaystyle b\|\nabla\Delta
p_{t}\|^{2}_{L^{2}}=\int_{\Omega}(\alpha\nabla
p_{tt}+p_{tt}\nabla\alpha-\Delta p\nabla r-r\nabla\Delta p-\nabla
g)\cdot\nabla\Delta p_{t}\,\textup{d}x.$

Applying Hölder’s inequality, we find

(5.54) $\displaystyle b\|\nabla\Delta p_{t}\|^{2}_{L^{2}}\leq\|$
$\displaystyle\alpha\|_{L^{\infty}}\|\nabla p_{tt}\|_{L^{2}}\|\nabla\Delta
p_{t}\|_{L^{2}}+\|\nabla\alpha\|_{L^{3}}\|p_{tt}\|_{L^{6}}\|\nabla\Delta
p_{t}\|_{L^{2}}$ $\displaystyle+\|r\|_{L^{\infty}}\|\nabla\Delta
p\|_{L^{2}}\|\nabla\Delta p_{t}\|_{L^{2}}+\|\nabla r\|_{L^{3}}\|\Delta
p\|_{L^{6}}\|\nabla\Delta p_{t}\|_{L^{2}}$ $\displaystyle+\|\nabla
g\|_{L^{2}}\|\nabla\Delta p_{t}\|_{L^{2}}.$

Furthermore, the embedding $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$
together with Young’s inequality and Assumption 1 yield

(5.55) $\displaystyle b\|\nabla\Delta p_{t}\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,\|\sqrt{b}\nabla
p_{tt}\|_{L^{2}}^{2}+\|\nabla\alpha\|_{L^{3}}^{2}\|\sqrt{b}\nabla
p_{tt}\|_{L^{2}}^{2}+\|\sqrt{b}\nabla\Delta p\|^{2}_{L^{2}}$
$\displaystyle+\|\nabla r\|_{L^{3}}^{2}\|\sqrt{b}\nabla\Delta
p\|^{2}_{L^{2}}+\|\nabla g\|^{2}_{L^{2}}+\varepsilon\|\nabla\Delta
p_{t}\|^{2}_{L^{2}}.$

Hence, selectiong $\varepsilon$ as small as needed, we infer that

(5.56) $\displaystyle\frac{b}{2}\|\nabla\Delta p_{t}(t)\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,(1+\|\nabla\alpha\|_{L^{3}}^{2}+\|\nabla
r\|_{L^{3}}^{2})E_{3}[p](t)+\|\nabla g(t)\|^{2}_{L^{2}},\quad t\geq 0.$

Adding up the estimates (5.52) and (5.56) gives

(5.57)
$\displaystyle\frac{\textup{d}}{\textup{d}t}\mathfrak{E}[p](t)+\mathfrak{D}[p](t)\lesssim(1+\Lambda(t))\mathfrak{E}[p](t)+\mathfrak{F}(t).$

Consequently, applying Gronwall’s inequality cf. Lemma 3.1, we infer that

(5.58)
$\displaystyle\mathfrak{E}[p](t)+\int_{0}^{t}\mathfrak{D}[p](s)\,\textup{d}s\lesssim$
$\displaystyle\,\mathfrak{E}[p](0)\exp\Big{(}\int_{0}^{t}(1+\Lambda(s))\,\textup{d}s\Big{)}$
$\displaystyle+\int_{0}^{t}\mathfrak{F}(s)\exp\Big{(}\int_{s}^{t}(1+\Lambda(\sigma))\textup{d}\sigma\Big{)}\,\textup{d}s.$

Furthermore, the estimates (5.58) and (5.34) yield

(5.59) $\displaystyle b\|\Delta p_{t}(t)\|^{2}_{L^{2}}\lesssim$
$\displaystyle\,\mathfrak{E}[p](t)+\|g(t)\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\mathfrak{E}[p](0)\exp\Big{(}\int_{0}^{t}(1+\Lambda(s))\,\textup{d}s\Big{)}$
$\displaystyle+\int_{0}^{t}\mathfrak{F}(s)\exp\Big{(}\int_{s}^{t}(1+\Lambda(\sigma))\textup{d}\sigma\Big{)}\,\textup{d}s.$

where we employed the inequality (3.2) to bound from above the term
$\|g(t)\|^{2}_{L^{2}}$. Then, it suffices to sum up (5.58) and (5.59) in order
to reach the estimate (5.9) for the smooth eigenfunctions of the Dirichlet-
Laplacian. Standard compactness method allows passing to the limit; thus
proving the existence of a solution $p\in X_{p}$ to (5.1). From the weak and
weak-$\star$ lower semi-continuity of norms, we get that $p$ satisfies the
same energy bound (5.9). The uniqueness of the solution $p$ is guaranteed,
since the only solution to the homogeneous problem

(5.60) $\alpha(x,t)p_{tt}-r(x,t)\Delta p-b\Delta p_{t}=0,\quad
p(x,0)=p_{t}(x,0)=0,\quad p|_{\partial\Omega}=0$

is zero. Indeed, from (5.9), we have $\mathfrak{E}[p](t)=0$ which immediately
gives $p=0$. We conclude the proof by noting that $p$ belonging to $X_{p}$
implies that (see [17, Chapter 1, Lemma 2.1] and [6, Chapter 5, Theorem 2])

$p\in C(0,T;H^{3}(\Omega)\cap H^{1}_{0}(\Omega)),\qquad p_{t}\in
C(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)).$

This completes the proof of Proposition 5.1. ∎

##  6\. Uniform local well-posedness: Proof of Theorem 2.1

In this section we prove Theorem 2.1. The proof is accomplished using Banach’s
fixed point theorem. For this purpose, we recall (2.2) and define the ball $B$
as

(6.1) $\displaystyle B=\Big{\\{}$
$\displaystyle(p^{\ast},\Theta^{\ast},q^{\ast})\in\mathcal{X}:(p^{\ast}(0),p^{\ast}_{t}(0),\Theta^{\ast}(0),q^{\ast}(0))=(p_{0},p_{1},\Theta_{0},q_{0})$
$\displaystyle\quad\|p^{\ast}\|_{L^{\infty}L^{\infty}}\leq\gamma<\frac{1}{2k_{1}},\quad\|p^{\ast}\|_{X_{p}}\leq
R_{1},\quad\|(\Theta^{\ast},q^{\ast})\|_{X_{\Theta}\times X_{q}}\leq
R_{2}\Big{\\}},$

where $\mathcal{X}:=X_{p}\times X_{\Theta}\times X_{q}$. Note that we have two
different radii. The reason behind this will be made clear in the proof below,
as we will need to impose a smallness condition on $R_{1}>0$ but not on
$R_{2}>0$.

To check that the ball $B$ is a non-empty subset of $\mathcal{X}$, one can
take for instance (see [21]) $\alpha=r=1$ and $f=g=0$ in (5.1) and (4.1). Then
the solution $(p,\Theta,q)\in\mathcal{X}$ lies in $B$ if we choose
$R_{1},R_{2}$ and $\gamma$ such that

(6.2) $C_{T}\mathfrak{E}[p](0)\leq
R_{1}^{2}\leq\gamma^{2}<\frac{1}{(2k_{1})^{2}},\quad
C_{T}\big{(}\|q_{0}\|_{H^{1}}^{2}+(1+\bar{\tau}+\bar{\tau}^{2})E^{\bar{\tau}}[\Theta,q](0)\big{)}\leq
R_{2}^{2}.$

The solution spaces $X_{p}$, $X_{\Theta}$ and $X_{q}$ (defined in (2.2)) are
endowed with the norms

(6.3) $\displaystyle\|p\|_{X_{p}}:=$
$\displaystyle\,\|p\|_{L^{\infty}H^{3}}+\|p_{t}\|_{L^{\infty}H^{2}}+\|\nabla\Delta
p_{t}\|_{L^{2}L^{2}}+\|\nabla p_{tt}\|_{L^{\infty}L^{2}}$
$\displaystyle+\|\Delta p_{tt}\|_{L^{2}L^{2}}+\|p_{ttt}\|_{L^{2}L^{2}},$
$\displaystyle\|\Theta\|_{X_{\Theta}}:=$
$\displaystyle\,\|\Theta\|_{L^{\infty}H^{2}}+\|\Theta_{t}\|_{L^{\infty}H^{1}}+\|\Theta_{tt}\|_{L^{\infty}L^{2}},$
$\displaystyle\|q\|_{X_{q}}:=$
$\displaystyle\,\|q\|_{L^{\infty}H^{1}}+\sum_{k=1}^{2}\|\partial_{t}^{k}q\|_{L^{2}L^{2}}.$

Then, the product space $\mathcal{X}$ is equipped with the norm

$\|(p,\Theta,q)\|_{\mathcal{X}}^{2}=\|p\|_{X_{p}}^{2}+\|\Theta\|_{X_{\Theta}}^{2}+\|q\|_{X_{q}}^{2}.$

These norms are clearly equivalent to the energies $\mathcal{E}[\Theta]$ and
$\mathfrak{E}[p]$. We have for all $t\geq 0$

(6.4)
$\sup_{t\in(0,T)}\mathcal{E}[\Theta](t)+\|q\|_{L^{\infty}H^{1}}^{2}+\sum_{k=1}^{2}\int_{0}^{T}\|\partial_{t}^{k}q\|_{L^{2}}^{2}\lesssim\|(\Theta,q)\|^{2}_{X_{\Theta}\times
X_{q}}$

and

(6.5) $\displaystyle\|(\Theta,q)\|^{2}_{X_{\Theta}\times
X_{q}}\lesssim\sup_{t\in(0,T)}\mathcal{E}[\Theta](t)+\|q\|_{L^{\infty}H^{1}}^{2}+\sum_{k=1}^{2}\int_{0}^{T}\|\partial_{t}^{k}q\|_{L^{2}}^{2}.$

where $\|(\Theta,q)\|^{2}_{X_{\Theta}\times
X_{q}}=\|\Theta\|_{X_{\Theta}}^{2}+\|q\|_{X_{q}}^{2}$. Furthermore, taking
into account Assumption 1, especially the boundedness of the functions
$\alpha$ and $r$, we obtain

(6.6) $\displaystyle\|p\|_{X_{p}}^{2}\lesssim$
$\displaystyle\,\sup_{t\in(0,T)}\mathfrak{E}[p](t)+\sup_{t\in(0,T)}b\|\Delta
p_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{T}\|\sqrt{\alpha}p_{ttt}\|^{2}_{L^{2}}\,\textup{d}s$
$\displaystyle+\int_{0}^{T}b\|\nabla\Delta
p_{t}\|^{2}_{L^{2}}\,\textup{d}s+\int_{0}^{T}b\|\Delta
p_{tt}\|^{2}_{L^{2}}\,\textup{d}s$ $\displaystyle\lesssim$
$\displaystyle\,\sup_{t\in(0,T)}\mathfrak{E}[p](t)+\sup_{t\in(0,T)}b\|\Delta
p_{t}(t)\|_{L^{2}}+\int_{0}^{T}\mathfrak{D}[p](s)\,\textup{d}s.$

The inverse inequality also holds

(6.7)
$\displaystyle\sup_{t\in(0,T)}\mathfrak{E}[p](t)+\sup_{t\in(0,T)}b\|\Delta
p_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{T}\|\sqrt{\alpha}p_{ttt}\|^{2}_{L^{2}}$
$\displaystyle\,\,\textup{d}s$ $\displaystyle+\int_{0}^{T}b\|\nabla\Delta
p_{t}\|^{2}_{L^{2}}\,\textup{d}s+\int_{0}^{T}b\|\Delta
p_{tt}\|^{2}_{L^{2}}\,\textup{d}s$ $\displaystyle\,\lesssim\|p\|_{X_{p}}^{2}.$

Notice that the norm $\|(p,\Theta,q)\|_{\mathcal{X}}$ is independent of
$\tau$. This plays a crucial role in the study of the limit $\tau\rightarrow
0$.

We consider the operator $\mathcal{T}$ that maps
$(p^{\ast},\Theta^{\ast},q^{\ast})\in B\subset\mathcal{X}$ to
$(p,\Theta,q)\in\mathcal{X}$ the solution of the coupled problem

(6.8) $\left\\{\begin{aligned}
&(1-2k(\Theta^{\ast})p^{\ast})p_{tt}-h(\Theta^{\ast})\Delta p-b\Delta
p_{t}=2k(\Theta^{\ast})(p^{\ast}_{t})^{2},&\quad&\text{in}&\
&\Omega\times(0,T),\\\ &m\Theta_{t}+\nabla\cdot
q+\ell\Theta=\mathcal{Q}(p^{\ast}_{t}),&\quad&\text{in}&\
&\Omega\times(0,T),\\\ &\tau
q_{t}+q+\kappa_{\textup{a}}\nabla\Theta=0,&\quad&\text{in}&\
&\Omega\times(0,T).\end{aligned}\right.$

The existence of a unique solution of system (1.12) is equivalent to the
existence of a unique fixed point in $B$ to the mapping $\mathcal{T}$, which
will be guaranteed by Banach’s fixed-point theorem. Therefore, to ensure
applicability of the latter, we want to show that for $R_{2}$ large enough,
$R_{1}$ small enough and for $\delta$ small enough, we have:

(i):

The mapping $\mathcal{T}:B\rightarrow B$ is well defined.

(ii):

$\mathcal{T}$ is a contraction mapping.

This is achieved by following the lines of the analysis presented in [21,
Section 4]. The results are contained in the next two lemmas.

######  Lemma 6.1.

Given $\tau\in(0,\bar{\tau}]$. Then, for small enough $R_{1}$ and $\delta$,
the operator $\mathcal{T}$ is self-mapping; namely $\mathcal{T}(B)\subset B$.

###### Proof.

Given $(p^{\ast},\Theta^{\ast},q^{\ast})\in B$. We are looking to show that
$(p,\Theta,q)=\mathcal{T}(p^{\ast},\Theta^{\ast},q^{\ast})$ the solution of
(6.8) also lies in $B$. To do this, we write the system (6.8) in the context
of the Propositions 4.1 and 5.1. To this end, we set

(6.9) $\displaystyle\alpha(x,t)=1-2k(\Theta^{\ast})p^{\ast},\qquad
r(x,t)=h(\Theta^{\ast}),$ $\displaystyle
g(x,t)=2k(\Theta^{\ast})(p^{\ast}_{t})^{2},\qquad
f(x,t)=\mathcal{Q}(p^{\ast}_{t})=\frac{2b}{\rho_{\textup{a}}C_{\textup{a}}^{4}}(p_{t}^{\ast})^{2}$

and our goal is to prove that these functions satisfy Assumption 1. We begin
by checking the nondegeneracy condition. Indeed, we have

(6.10) $\|2k(\Theta^{\ast})p^{\ast}\|_{L^{\infty}L^{\infty}}\leq
2k_{1}\|p^{\ast}\|_{L^{\infty}L^{\infty}}\leq 2k_{1}\gamma,$

hence

(6.11) $0<\alpha_{0}=1-2k_{1}\gamma\leq\alpha(x,t)\leq
1+2k_{1}\gamma=\alpha_{1}.$

In addition, from (H1), we have

(6.12) $0<r_{0}=h_{1}\leq h(\Theta^{\ast}),$

which ensures that the functions $r,\alpha$ do not degenerate.

Next, we focus on the function $\Lambda(t)$ given in (5.10). Using the
properties of $k$ (see the assumption (K2)) and the embedding
$H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, we have

(6.13) $\displaystyle\|\alpha_{t}\|_{L^{2}L^{2}}$
$\displaystyle=\|\partial_{t}(2k(\Theta^{\ast})p^{\ast})\|_{L^{2}L^{2}}$
$\displaystyle\leq
2\|k(\Theta^{\ast})p_{t}^{\ast}\|_{L^{2}L^{2}}+2\|k^{\prime}(\Theta^{\ast})\Theta^{\ast}_{t}p^{\ast}\|_{L^{2}L^{2}}$
$\displaystyle\lesssim\|k(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|p_{t}^{\ast}\|_{L^{2}L^{2}}+\|k^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta^{\ast}_{t}\|_{L^{2}L^{2}}\|p^{\ast}\|_{L^{\infty}L^{\infty}}$
$\displaystyle\lesssim
k_{1}\|p_{t}^{\ast}\|_{L^{2}L^{2}}+(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}+1})\|\Theta^{\ast}_{t}\|_{L^{2}L^{2}}\|p^{\ast}\|_{L^{\infty}L^{\infty}}$
$\displaystyle\leq C_{T}(R_{1}+(1+R_{2}^{1+\gamma_{2}})R_{1}R_{2}).$

Let $\beta=\frac{4}{4-d}$. Then, we obtain

(6.14) $\displaystyle\|\alpha_{t}\|_{L^{\beta}L^{2}}$ $\displaystyle\leq
2\|k(\Theta^{\ast})p_{t}^{\ast}\|_{L^{\beta}L^{2}}+2\|k^{\prime}(\Theta^{\ast})\Theta^{\ast}_{t}p^{\ast}\|_{L^{\beta}L^{2}}$
$\displaystyle\lesssim\|k(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|p_{t}^{\ast}\|_{L^{\beta}L^{2}}+\|k^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta^{\ast}_{t}\|_{L^{\beta}L^{2}}\|p^{\ast}\|_{L^{\infty}L^{\infty}}$
$\displaystyle\lesssim
k_{1}\|p_{t}^{\ast}\|_{L^{\beta}L^{2}}+(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}+1})\|\Theta^{\ast}_{t}\|_{L^{\beta}L^{2}}\|p^{\ast}\|_{L^{\infty}H^{3}}$
$\displaystyle\leq C_{T}(R_{1}+(1+R_{2}^{1+\gamma_{2}})R_{1}R_{2}),$

where we used the embedding $L^{\infty}(0,T)\hookrightarrow L^{\beta}(0,T)$.
Further, using the embedding $H^{1}(\Omega)\hookrightarrow L^{3}(\Omega)$ we
can estimate $\|\alpha_{t}\|_{L^{2}L^{3}},\|\nabla\alpha\|_{L^{2}L^{3}}$ as
follows

(6.15) $\displaystyle\|\alpha_{t}\|_{L^{2}L^{3}}$ $\displaystyle\leq
2\|k(\Theta^{\ast})p_{t}^{\ast}\|_{L^{2}L^{3}}+2\|k^{\prime}(\Theta^{\ast})\Theta^{\ast}_{t}p^{\ast}\|_{L^{2}L^{3}}$
$\displaystyle\lesssim\|k(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|p_{t}^{\ast}\|_{L^{2}L^{3}}+\|k^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta^{\ast}_{t}\|_{L^{2}L^{3}}\|p^{\ast}\|_{L^{\infty}L^{\infty}}$
$\displaystyle\lesssim
k_{1}\|p_{t}^{\ast}\|_{L^{2}H^{1}}+(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}+1})\|\Theta^{\ast}_{t}\|_{L^{2}H^{1}}\|p^{\ast}\|_{L^{\infty}H^{3}},$

and

(6.16) $\displaystyle\|\nabla\alpha\|_{L^{2}L^{3}}$ $\displaystyle\leq
2\|k(\Theta^{\ast})\nabla
p^{\ast}\|_{L^{2}L^{3}}+2\|k^{\prime}(\Theta^{\ast})\nabla\Theta^{\ast}p^{\ast}\|_{L^{2}L^{3}}$
$\displaystyle\lesssim\|k(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\nabla
p^{\ast}\|_{L^{2}L^{3}}+\|k^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\nabla\Theta^{\ast}\|_{L^{2}L^{3}}\|p^{\ast}\|_{L^{\infty}L^{\infty}}$
$\displaystyle\lesssim
k_{1}\|p^{\ast}\|_{L^{2}H^{2}}+(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}+1})\|\Theta^{\ast}\|_{L^{2}H^{2}}\|p^{\ast}\|_{L^{\infty}H^{3}}.$

Hence, it results from (6.15) and (6.16) that

(6.17) $\|\alpha_{t}\|_{L^{2}L^{3}}+\|\nabla\alpha\|_{L^{2}L^{3}}\leq
C_{T}(R_{1}+(1+R_{2}^{1+\gamma_{2}})R_{1}R_{2}).$

Similarly, we can derive estimates for the terms of $\Lambda$ involving the
function $r$. On account of the properties of the function $h$ (see (H3)), we
find

(6.18) $\displaystyle\|r_{t}\|_{L^{\beta}L^{2}}$
$\displaystyle=\|h^{\prime}(\Theta^{\ast})\Theta^{\ast}_{t}\|_{L^{\beta}L^{2}}\leq\|h^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta^{\ast}_{t}\|_{L^{\beta}L^{2}}$
$\displaystyle\leq
C_{T}(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}+1})\|\Theta^{\ast}_{t}\|_{L^{\infty}L^{2}}\leq
C_{T}(1+R_{2}^{\gamma_{1}+1})R_{2},$

and

(6.19) $\displaystyle\|\nabla r\|_{L^{2}L^{2}}$
$\displaystyle=\|h^{\prime}(\Theta^{\ast})\nabla\Theta^{\ast}\|_{L^{2}L^{2}}\leq\|h^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\nabla\Theta^{\ast}\|_{L^{2}L^{2}}$
$\displaystyle\lesssim(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}+1})\|\nabla\Theta^{\ast}\|_{L^{2}L^{2}}\leq
C_{T}(1+R_{2}^{\gamma_{1}+1})R_{2}.$

Again, using the embedding $H^{1}(\Omega)\hookrightarrow L^{3}(\Omega)$, we
have

(6.20) $\displaystyle\|r_{t}\|_{L^{2}L^{3}}$
$\displaystyle=\|h^{\prime}(\Theta^{\ast})\Theta^{\ast}_{t}\|_{L^{2}L^{3}}\lesssim(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}+1})\|\Theta^{\ast}_{t}\|_{L^{2}L^{3}}$
$\displaystyle\lesssim(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}+1})\|\Theta^{\ast}_{t}\|_{L^{2}H^{1}}.$

Moreover, elliptic regularity allows one to get

(6.21) $\displaystyle\|\nabla r\|_{L^{2}L^{3}}$
$\displaystyle=\|h^{\prime}(\Theta^{\ast})\nabla\Theta^{\ast}\|_{L^{2}L^{3}}\leq\|h^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\nabla\Theta^{\ast}\|_{L^{2}L^{3}}$
$\displaystyle\lesssim(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}+1})\|\Theta^{\ast}\|_{L^{2}H^{2}}.$

Thus, it follows that

(6.22) $\|r_{t}\|_{L^{2}L^{3}}+\|\nabla r\|_{L^{2}L^{3}}\leq
C_{T}(1+R_{2}^{\gamma_{1}+1})R_{2}.$

For convenience, we recall the definition of $\Lambda$

(6.23) $\displaystyle\Lambda(t)=\|\alpha_{t}(t)\|_{L^{2}}^{2}$
$\displaystyle+\|\alpha_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|r_{t}(t)\|_{L^{2}}^{\frac{4}{4-d}}+\|\nabla
r(t)\|_{L^{2}}^{2}+\|r_{t}(t)\|_{L^{3}}^{2}$
$\displaystyle+\|\alpha_{t}(t)\|_{L^{3}}^{2}+\|\nabla
r(t)\|_{L^{3}}^{2}+\|\nabla\alpha(t)\|_{L^{3}}^{2},$

so altogether the above estimates imply that

(6.24) $\|\Lambda\|_{L^{1}(0,t)}\leq C_{1}(T,R_{1},R_{2}).$

Now, we turn our attention to the source term $g$. Using the embeddings
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$, $H^{1}(\Omega)\hookrightarrow
L^{6}(\Omega)$ and the fact that
$\|(p^{\ast}_{t})^{2}\|_{L^{3}}=\|p^{\ast}_{t}\|_{L^{6}}^{2}$, we have

(6.25) $\displaystyle\|g\|_{L^{2}H^{1}}+\|g_{t}\|_{L^{2}L^{2}}=$
$\displaystyle\|4k(\Theta^{\ast})\nabla
p^{\ast}_{t}p^{\ast}_{t}+2k^{\prime}(\Theta^{\ast})\nabla\Theta^{\ast}(p^{\ast}_{t})^{2}\|_{L^{2}L^{2}}$
$\displaystyle+\|4k(\Theta^{\ast})p^{\ast}_{t}p^{\ast}_{tt}+2k^{\prime}(\Theta^{\ast})\Theta^{\ast}_{t}(p^{\ast}_{t})^{2}\|_{L^{2}L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\|k(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\nabla
p^{\ast}_{t}\|_{L^{2}L^{4}}\|p^{\ast}_{t}\|_{L^{\infty}L^{4}}$
$\displaystyle+\|k^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\nabla\Theta^{\ast}\|_{L^{2}L^{6}}\|(p^{\ast}_{t})^{2}\|_{L^{\infty}L^{3}}$
$\displaystyle+\|k(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|p^{\ast}_{t}\|_{L^{\infty}L^{4}}\|p^{\ast}_{tt}\|_{L^{2}L^{4}}$
$\displaystyle+\|k^{\prime}(\Theta^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta^{\ast}_{t}\|_{L^{2}L^{6}}\|(p^{\ast}_{t})^{2}\|_{L^{\infty}L^{3}}$
$\displaystyle\lesssim$
$\displaystyle\,\|p^{\ast}_{t}\|_{L^{2}H^{2}}\|p^{\ast}_{t}\|_{L^{\infty}H^{1}}+(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}+1})\|\Theta^{\ast}\|_{L^{2}H^{2}}\|p^{\ast}_{t}\|_{L^{\infty}H^{1}}^{2}$
$\displaystyle+\|p^{\ast}_{t}\|_{L^{\infty}H^{1}}\|p^{\ast}_{tt}\|_{L^{2}H^{1}}+(1+\|\Theta^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}+1})\|\Theta^{\ast}_{t}\|_{L^{2}H^{1}}\|p^{\ast}_{t}\|_{L^{\infty}H^{1}}^{2},$

which gives

(6.26) $\|g\|_{L^{2}H^{1}}+\|g_{t}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}^{2}(1+R_{2}+R_{2}^{2+\gamma_{2}}).$

Thus, we can bound from above the function $\mathfrak{F}$ defined in (5.11) as

(6.27) $\displaystyle\|\mathfrak{F}\|_{L^{1}(0,t)}=\int_{0}^{t}(\|\nabla
g\|_{L^{2}}^{2}+\|g_{t}\|_{L^{2}}^{2})\,\textup{d}s\leq$
$\displaystyle\,\|g\|_{L^{2}H^{1}}^{2}+\|g_{t}\|_{L^{2}L^{2}}^{2}$
$\displaystyle\leq$
$\displaystyle\,C_{T}R_{1}^{4}(1+R_{2}^{2}(1+R_{2}^{2+2\gamma_{2}})).$

Hence, it results that

(6.28) $\|\mathfrak{F}\|_{L^{1}(0,t)}\leq R_{1}^{4}C_{2}(T,R_{2}).$

Consequently, from Proposition 5.1 we have the existence of a unique $p\in
X_{p}$ solution to the first equation in (6.8). Moreover, since
$(p^{\ast},\Theta^{\ast},q^{\ast})\in B$, we have $f=\mathcal{Q}(p^{\ast})\in
H^{2}(0,T;L^{2}(\Omega))$. Then, according to Proposition 4.1, there exists a
unique $(\Theta,q)\in X_{\Theta}\times X_{q}$ solution of the second and third
equations in (6.8). That is to say, the mapping $\mathcal{T}$ is well-defined.

On account of (5.9) and the fact that $\mathfrak{E}[p](0)\leq\delta$, we
obtain

(6.29) $\displaystyle\|p\|_{X_{p}}^{2}\lesssim$
$\displaystyle\,\sup_{t\in(0,T)}\mathfrak{E}[p](t)+\sup_{t\in(0,T)}b\|\Delta
p_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{T}\mathfrak{D}[p](s)\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\delta\exp(T(1+C_{1}(T,R_{1},R_{2})))+T^{2}R_{1}^{4}\exp(T(1+C_{1}(T,R_{1},R_{2})))C_{2}(T,R_{2}).$

Thus, by choosing $\delta$ and $R_{1}$ small enough, we get

(6.30) $\|p\|_{X_{p}}\leq R_{1}.$

Also, observing that

(6.31) $\|p\|_{L^{\infty}L^{\infty}}\lesssim\|\Delta
p\|_{L^{\infty}L^{2}}\lesssim\|p\|_{X_{p}},$

we obtain an upper bound $\gamma<\dfrac{1}{2k_{1}}$ for
$\|p\|_{L^{\infty}L^{\infty}}$ by possibly reducing $R_{1}$.
Therefore, it remains to verify that $\|(\Theta,q)\|_{X_{\Theta}\times
X_{q}}\leq R_{2}$. First, we have

(6.32) $\displaystyle\|f\|_{H^{2}L^{2}}^{2}\lesssim$
$\displaystyle\,\|(p_{t}^{\ast})^{2}\|_{L^{2}L^{2}}^{2}+\|2p_{t}^{\ast}p_{tt}^{\ast}\|_{L^{2}L^{2}}^{2}+\|2(p_{tt}^{\ast})^{2}+2p_{t}^{\ast}p_{ttt}^{\ast}\|_{L^{2}L^{2}}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\|p_{t}^{\ast}\|_{L^{\infty}L^{4}}^{2}\|p_{t}^{\ast}\|_{L^{2}L^{4}}^{2}+\|p_{t}^{\ast}\|_{L^{\infty}L^{4}}^{2}\|p_{tt}^{\ast}\|_{L^{2}L^{4}}^{2}$
$\displaystyle+\|p_{tt}^{\ast}\|_{L^{\infty}L^{4}}^{2}\|p_{tt}^{\ast}\|_{L^{2}L^{4}}^{2}+\|p_{t}^{\ast}\|_{L^{\infty}L^{\infty}}^{2}\|p_{ttt}^{\ast}\|_{L^{2}L^{2}}^{2}.$

Thanks to the embedding $H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$, we find

(6.33) $\|f\|_{H^{2}L^{2}}^{2}\leq C_{T}\|p\|_{X_{p}}^{4}\leq C_{T}R_{1}^{4}.$

Then, according to Proposition 4.1 and particularly the estimates (4.22) and
(4.23), we get

(6.34) $\displaystyle\|(\Theta,q)\|_{X_{\Theta}\times X_{q}}^{2}\lesssim
C_{T}\Big{(}\|q_{0}\|^{2}_{H^{1}}+(1+\bar{\tau}+\bar{\tau}^{2})(E^{\bar{\tau}}[\Theta,q](0)+\|f\|_{H^{2}L^{2}}^{2})\Big{)}.$

We emphasize that the constant $C_{T}>0$ in this inequality does not depend on
the parameter $\tau$. So it suffices to take $R_{2}$ large enough such that

(6.35)
$C_{T}\Big{(}\|q_{0}\|^{2}_{H^{1}}+(1+\bar{\tau}+\bar{\tau}^{2})(E^{\bar{\tau}}[\Theta,q](0)+R_{1}^{4})\Big{)}\leq
R_{2}^{2},$

in order to conclude that the solution $(p,\Theta,q)$ of the system (6.8)
remains in $B$. ∎

######  Lemma 6.2.

Let $\tau\in(0,\bar{\tau}]$. If $R_{1}$ and $\delta$ are sufficiently small,
then the mapping $\mathcal{T}$ is a contraction on $B$.

###### Proof.

Let
$(p^{\ast}_{1},\Theta^{\ast}_{1},q_{1}^{\ast}),(p^{\ast}_{2},\Theta^{\ast}_{2},q_{2}^{\ast})\in
B$ and let $(p_{1},\Theta_{1},q_{1}),(p_{2},\Theta_{2},q_{2})$ stand for their
corresponding images by the operator $\mathcal{T}$; that is

(6.36)
$\mathcal{T}(p^{\ast}_{1},\Theta^{\ast}_{1},q_{1}^{\ast})=(p_{1},\Theta_{1},q_{1})\quad\text{and}\quad\mathcal{T}(p^{\ast}_{2},\Theta^{\ast}_{2},q_{2}^{\ast})=(p_{2},\Theta_{2},q_{2}).$

Since $(p_{1},\Theta_{1},q_{1}),(p_{2},\Theta_{2},q_{2})$ are both solutions
to system (6.8), clearly the differences

(6.37)
$\displaystyle\hat{p}=p_{1}-p_{2},\quad\hat{\Theta}=\Theta_{1}-\Theta_{2},\quad\hat{q}=q_{1}-q_{2},$
$\displaystyle\hat{p}^{\ast}=p_{1}^{\ast}-p_{2}^{\ast},\quad\hat{\Theta}^{\ast}=\Theta_{1}^{\ast}-\Theta_{2}^{\ast},\quad\hat{q}^{\ast}=q_{1}^{\ast}-q_{2}^{\ast}$

solve the following system

(6.38) $\left\\{\begin{aligned}
&(1-2k(\Theta^{\ast}_{1})p_{1}^{\ast})\hat{p}_{tt}-h(\Theta_{1}^{\ast})\Delta\hat{p}-b\Delta\hat{p}_{t}=g_{1},&\qquad&\\\
&m\hat{\Theta}_{t}+\nabla\cdot\hat{q}+\ell\hat{\Theta}=f_{1},\\\
&\tau\hat{q}_{t}+\hat{q}+\kappa_{\textup{a}}\nabla\hat{\Theta}=0,\\\
\end{aligned}\text{in}\quad\Omega\times(0,T),\right.$

with the initial and boundary conditions

(6.39) $\displaystyle\hat{p}(x,0)=\hat{p}_{t}(x,0)=\hat{\Theta}(x,0)=0,\quad$
$\displaystyle\hat{q}(x,0)=0,$ $\displaystyle\quad\text{in}\quad\ \Omega.$
$\displaystyle\hat{p}=\hat{\Theta}=0,$
$\displaystyle\text{on}\quad\partial\Omega\times(0,T).$

The forcing terms $f_{1}$ and $g_{1}$ are given by

(6.40) $\displaystyle f_{1}=$
$\displaystyle\,\mathcal{Q}(p_{1t}^{\ast})-\mathcal{Q}(p_{2t}^{\ast}),$
$\displaystyle g_{1}=$
$\displaystyle\,2(k(\Theta_{1}^{\ast})p_{1}^{\ast}-k(\Theta_{2}^{\ast})p_{2}^{\ast})p_{2tt}+(h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\Delta
p_{2}+2k(\Theta_{1}^{\ast})(p^{\ast}_{1t})^{2}-2k(\Theta_{2}^{\ast})(p^{\ast}_{2t})^{2}$
$\displaystyle=$
$\displaystyle\,2(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2})+(h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\Delta
p_{2}$
$\displaystyle+2k(\Theta_{1}^{\ast})\big{(}(p^{\ast}_{1t}+p^{\ast}_{2t})\hat{p}^{\ast}_{t}+\hat{p}^{\ast}p_{2tt}\big{)}$
$\displaystyle:=$ $\displaystyle\,g_{11}+g_{12}+g_{13}.$

We start by recalling some estimates derived in [21] on the functions $k,h$
and their derivatives. We have

(6.41)
$k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast})=(\Theta_{1}^{\ast}-\Theta_{2}^{\ast})\int_{0}^{1}k^{\prime}(\Theta_{2}^{\ast}+\sigma(\Theta_{1}^{\ast}-\Theta_{2}^{\ast}))\textup{d}\sigma;$

hence using (K2), together with the Sobolev embedding $H^{2}\hookrightarrow
L^{\infty}$, we get

(6.42a)
$\displaystyle\|k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}$
$\displaystyle=\Big{\|}(\Theta_{1}^{\ast}-\Theta_{2}^{\ast})\int_{0}^{1}k^{\prime}(\Theta_{2}^{\ast}+\sigma(\Theta_{1}^{\ast}-\Theta_{2}^{\ast}))\textup{d}\sigma\Big{\|}_{L^{\infty}L^{\infty}}$
$\displaystyle\lesssim\|\Theta_{1}^{\ast}-\Theta_{2}^{\ast}\|_{L^{\infty}L^{\infty}}\Big{(}1+\|\Theta_{2}^{\ast}+\sigma(\Theta_{1}^{\ast}-\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}^{1+\gamma_{2}}\Big{)}$
$\displaystyle\lesssim\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}\Big{(}1+\|\Theta_{1}^{\ast}\|_{L^{\infty}L^{\infty}}^{1+\gamma_{2}}+\|\Theta_{2}^{\ast}\|_{L^{\infty}L^{\infty}}^{1+\gamma_{2}}\Big{)}.$
In a similar fashion, on account of (H2), (H3) and (K2), we can show that
(6.42b)
$\|k^{\prime}(\Theta_{1}^{\ast})-k^{\prime}(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\lesssim\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}\Big{(}1+\|\Theta_{1}^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}}+\|\Theta_{2}^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{2}}\Big{)};$
(6.43a)
$\displaystyle\|h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\lesssim\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}\Big{(}1+\|\Theta_{1}^{\ast}\|_{L^{\infty}L^{\infty}}^{1+\gamma_{1}}+\|\Theta_{2}^{\ast}\|_{L^{\infty}L^{\infty}}^{1+\gamma_{1}}\Big{)};$
(6.43b)
$\displaystyle\|h^{\prime}(\Theta_{1}^{\ast})-h^{\prime}(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\lesssim\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}\Big{(}1+\|\Theta_{1}^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}}+\|\Theta_{2}^{\ast}\|_{L^{\infty}L^{\infty}}^{\gamma_{1}}\Big{)}.$

These estimates will be repeatedly used in the course of this proof.
First, we focus on estimating the $L^{2}$-norm of the gradient of $g_{1}$.
Recall that

(6.44)
$g_{11}=2(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2}).$

This entails that

(6.45) $\displaystyle\nabla g_{11}=$
$\displaystyle\,2\nabla(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2})$
$\displaystyle+2(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))\nabla(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2}).$

Since the heat flux $q$ is not present in the expression of the source term
$g_{1}$; we can proceed as in [21, Section 4] to estimate $\|\nabla
g_{11}\|_{L^{2}L^{2}}$. The fact that the operator $\mathcal{T}$ is self-
mapping along with the estimate

(6.46)
$\|\Theta_{1}^{\ast}-\Theta_{2}^{\ast}\|_{X_{\Theta}}^{2}+\|p_{1}^{\ast}-p_{2}^{\ast}\|_{X_{p}}^{2}\leq\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}^{2},$

allows us to obtain the following bound

(6.47) $\displaystyle\|\nabla g_{11}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}^{2}(1+R_{2}^{\gamma_{2}}+R_{2}^{1+\gamma_{2}})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Analogously, we can follow the arguments in [21] in order to deal with the
gradient of the second contribution

(6.48) $\nabla g_{12}=\nabla(h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\Delta
p_{2}+h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\nabla\Delta p_{2}.$

In fact, using the assumption (H3) on the function $h$ together with (6.43a)
and (6.43b), we find

(6.49) $\|\nabla g_{12}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}(1+R_{2}^{\gamma_{1}}+R_{2}^{1+\gamma_{1}})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Thus, it remains to bound the gradient of the last component of $g_{1}$

(6.50) $\nabla
g_{13}=2\big{(}\nabla(k(\Theta_{1}^{\ast}))\big{(}(p^{\ast}_{1t}+p^{\ast}_{2t})\hat{p}^{\ast}_{t}+\hat{p}^{\ast}p_{2tt}\big{)}+k(\Theta_{1}^{\ast})\nabla\big{(}(p^{\ast}_{1t}+p^{\ast}_{2t})\hat{p}^{\ast}_{t}+\hat{p}^{\ast}p_{2tt}\big{)}\big{)}.$

Again, we can do as in [21] to get

(6.51) $\|\nabla g_{13}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}(1+R_{2}+R_{2}^{\gamma_{2}+2})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Collecting the estimates of the components of $g_{1}$, it follows that

(6.52) $\|\nabla g_{1}\|_{L^{2}L^{2}}\leq
R_{1}C(T,R_{1},R_{2})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Next, we seek a similar estimate for the derivative in time of $g_{1}$.
Differentiating $g_{11}$ with respect to $t$, we have

(6.53) $\displaystyle\partial_{t}g_{11}=$ $\displaystyle
2\Big{(}\partial_{t}(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2})+(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))\partial_{t}(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2})\Big{)}$
$\displaystyle=$ $\displaystyle
2\Big{(}\big{(}k^{\prime}(\Theta_{1}^{\ast})\hat{\Theta}^{\ast}_{t}+(k^{\prime}(\Theta_{1}^{\ast})-k^{\prime}(\Theta_{2}^{\ast}))\Theta_{2t}^{\ast}\big{)}(p_{2}^{\ast}p_{2tt}+(p^{\ast}_{2t})^{2})$
$\displaystyle+(k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast}))(p_{2t}^{\ast}p_{2tt}+p_{2}^{\ast}p_{2ttt}+2p^{\ast}_{2t}p^{\ast}_{2tt})\Big{)}.$

Using Hölder’s inequality, we obtain

(6.54) $\displaystyle\|\partial_{t}g_{11}\|_{L^{2}L^{2}}\lesssim$
$\displaystyle\,\big{(}\|k^{\prime}(\Theta_{1}^{\ast})\|_{L^{\infty}L^{\infty}}\|\hat{\Theta}^{\ast}_{t}\|_{L^{2}L^{4}}+\|k^{\prime}(\Theta_{1}^{\ast})-k^{\prime}(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta_{2t}^{\ast}\|_{L^{2}L^{4}}\big{)}$
$\displaystyle\times\big{(}\|p_{2}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{2tt}\|_{L^{\infty}L^{4}}+\|p_{2t}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{2t}^{\ast}\|_{L^{\infty}L^{4}}\big{)}$
$\displaystyle+\|k(\Theta_{1}^{\ast})-k(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\big{(}\|p_{2t}^{\ast}\|_{L^{\infty}L^{4}}\|p_{2tt}\|_{L^{2}L^{4}}+\|p_{2}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{2ttt}\|_{L^{2}L^{2}}$
$\displaystyle+\|p^{\ast}_{2t}\|_{L^{\infty}L^{4}}\|p^{\ast}_{2tt}\|_{L^{2}L^{4}}\big{)}$

Then taking into account (K2), (6.42b) and using the embedding
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$, it follows

(6.55) $\|\partial_{t}g_{11}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}^{2}(1+R_{2}^{\gamma_{2}}+R_{2}^{1+\gamma_{2}})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

The time derivative of $g_{12}$ is given by

(6.56) $\displaystyle\partial_{t}g_{12}=$
$\displaystyle\partial_{t}(h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\Delta
p_{2}+(h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\Delta p_{2t},$
$\displaystyle=$
$\displaystyle\big{(}h^{\prime}(\Theta_{1}^{\ast})\hat{\Theta}^{\ast}_{t}+(h^{\prime}(\Theta_{1}^{\ast})-h^{\prime}(\Theta_{2}^{\ast}))\Theta_{2t}^{\ast}\big{)}\Delta
p_{2}+(h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast}))\Delta p_{2t}.$

Hence, it holds that

(6.57) $\displaystyle\|\partial_{t}g_{11}\|_{L^{2}L^{2}}\lesssim$
$\displaystyle\,\big{(}\|h^{\prime}(\Theta_{1}^{\ast})\|_{L^{\infty}L^{\infty}}\|\hat{\Theta}^{\ast}_{t}\|_{L^{2}L^{4}}+\|h^{\prime}(\Theta_{1}^{\ast})-h^{\prime}(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta_{2t}^{\ast}\|_{L^{2}L^{4}}\big{)}\|\Delta
p_{2}\|_{L^{\infty}L^{4}}$
$\displaystyle+\|h(\Theta_{1}^{\ast})-h(\Theta_{2}^{\ast})\|_{L^{\infty}L^{\infty}}\|\Delta
p_{2t}\|_{L^{2}L^{2}},$

which, thanks to (H3), (6.43b) implies

(6.58) $\|\partial_{t}g_{12}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}(1+R_{2}^{\gamma_{1}}+R_{2}^{1+\gamma_{1}})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Differentiating in time the last contribution $g_{13}$, we find

(6.59) $\displaystyle\partial_{t}g_{13}=$ $\displaystyle
2\big{(}k^{\prime}(\Theta_{1}^{\ast})\Theta_{1t}^{\ast}\big{(}(p^{\ast}_{1t}+p^{\ast}_{2t})\hat{p}^{\ast}_{t}+\hat{p}^{\ast}p_{2tt}\big{)}$
$\displaystyle+k(\Theta_{1}^{\ast})\big{(}(p^{\ast}_{1tt}+p^{\ast}_{2tt})\hat{p}^{\ast}_{t}+(p^{\ast}_{1t}+p^{\ast}_{2t})\hat{p}^{\ast}_{tt}+\hat{p}^{\ast}_{t}p_{2tt}+\hat{p}^{\ast}p_{2ttt}\big{)}\big{)}.$

Applying Hölder’s inequality yields

(6.60) $\displaystyle\|\partial_{t}g_{13}\|_{L^{2}L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\|k^{\prime}(\Theta_{1}^{\ast})\|_{L^{\infty}L^{\infty}}\|\Theta_{1t}^{\ast}\|_{L^{2}L^{4}}\big{(}\|p^{\ast}_{1t}+p^{\ast}_{2t}\|_{L^{\infty}L^{4}}\|\hat{p}^{\ast}_{t}\|_{L^{\infty}L^{\infty}}+\|\hat{p}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{2tt}\|_{L^{\infty}L^{4}}\big{)}$
$\displaystyle+\|k(\Theta_{1}^{\ast})\|_{L^{\infty}L^{\infty}}\big{(}\|p^{\ast}_{1tt}+p^{\ast}_{2tt}\|_{L^{2}L^{4}}\|\hat{p}^{\ast}_{t}\|_{L^{\infty}L^{4}}+\|p^{\ast}_{1t}+p^{\ast}_{2t}\|_{L^{\infty}L^{4}}\|\hat{p}^{\ast}_{tt}\|_{L^{2}L^{4}}$
$\displaystyle+\|\hat{p}^{\ast}_{t}\|_{L^{\infty}L^{4}}\|p_{2tt}\|_{L^{2}L^{4}}+\|\hat{p}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{2ttt}\|_{L^{2}L^{2}}\big{)}.$

Then, from the assumption (K2) and the fact that the mapping $\mathcal{T}$
being self-mapping, we arrive at

(6.61) $\|\partial_{t}g_{13}\|_{L^{2}L^{2}}\leq
C_{T}R_{1}(1+R_{2}+R_{2}^{2+\gamma_{2}})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Putting together the above estimates results in

(6.62) $\|\partial_{t}g_{1}\|_{L^{2}L^{2}}\leq
R_{1}C(T,R_{1},R_{2})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

As for the source term $f_{1}$ in the temperature equation, we can handle it
as follows

(6.63) $\displaystyle\|f_{1}\|_{H^{2}L^{2}}=$
$\displaystyle\,\|f_{1}\|_{L^{2}L^{2}}+\|f_{1t}\|_{L^{2}L^{2}}+\|f_{1tt}\|_{L^{2}L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\|(p_{1t}^{\ast})^{2}-(p_{2t}^{\ast})^{2}\|_{L^{2}L^{2}}+\|(p_{1tt}^{\ast}-p_{2tt}^{\ast})(p_{1t}^{\ast})\|_{L^{2}L^{2}}+\|(p_{1t}^{\ast}-p_{2t}^{\ast})(p_{2tt}^{\ast})\|_{L^{2}L^{2}}$
$\displaystyle+\|(p_{1ttt}^{\ast}-p_{2ttt}^{\ast})p_{1t}^{\ast}\|_{L^{2}L^{2}}+\|(p_{1tt}^{\ast}-p_{2tt}^{\ast})p_{1tt}^{\ast}\|_{L^{2}L^{2}}$
$\displaystyle+\|(p_{1tt}^{\ast}-p_{2tt}^{\ast})p_{2tt}^{\ast}\|_{L^{2}L^{2}}+\|(p_{1t}^{\ast}-p_{2t}^{\ast})p_{2ttt}^{\ast}\|_{L^{2}L^{2}}.$

Thus, we get

(6.64) $\displaystyle\|f_{1}\|_{H^{2}L^{2}}\lesssim$
$\displaystyle\,\|p_{1t}^{\ast}-p_{2t}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{1t}^{\ast}+p_{2t}^{\ast}\|_{L^{2}L^{2}}+\|p_{1tt}^{\ast}-p_{2tt}^{\ast}\|_{L^{2}L^{4}}\|p_{1t}^{\ast}\|_{L^{\infty}L^{4}}$
$\displaystyle+\|p_{1t}^{\ast}-p_{2t}^{\ast}\|_{L^{\infty}L^{4}}\|p_{2tt}^{\ast}\|_{L^{2}L^{4}}+\|p_{1ttt}^{\ast}-p_{2ttt}^{\ast}\|_{L^{2}L^{2}}\|p_{1t}^{\ast}\|_{L^{\infty}L^{\infty}}$
$\displaystyle+\|p_{1tt}^{\ast}-p_{2tt}^{\ast}\|_{L^{\infty}L^{4}}\|p_{1tt}^{\ast}\|_{L^{2}L^{4}}+\|p_{1tt}^{\ast}-p_{2tt}^{\ast}\|_{L^{\infty}L^{4}}\|p_{2tt}^{\ast}\|_{L^{2}L^{4}}$
$\displaystyle+\|p_{1t}^{\ast}-p_{2t}^{\ast}\|_{L^{\infty}L^{\infty}}\|p_{2ttt}^{\ast}\|_{L^{2}L^{2}}.$

Using the embeddings $H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$,
$H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ and elliptic regularity, we
deduce that

(6.65) $\|f_{1}\|_{H^{2}L^{2}}\leq
R_{1}C_{T}\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

From the previous Lemma 6.1, we know that the coefficients of the first
equation in the system (6.38)

$\alpha_{1}=1-2k(\Theta_{1}^{\ast})p_{1}^{\ast},\qquad
r_{1}=h(\Theta_{1}^{\ast})$

satisfy the requirements of Proposition 5.1. Further, setting $g=g_{1}$ and
$f=f_{1}$ and taking into account the above estimates, we deduce that the
results of the Propositions 4.1, 5.1 hold. In particular, the energy estimates
(4.22), (4.23) and (5.9) hold for the solution
$(\hat{p},\hat{\Theta},\hat{q})$; that is

(6.66) $\displaystyle\|(\hat{p},\hat{\Theta},\hat{q})\|_{\mathcal{X}}^{2}=$
$\displaystyle\|\mathcal{T}(p^{\ast}_{1},\Theta^{\ast}_{1},q^{\ast}_{1})-\mathcal{T}(p^{\ast}_{2},\Theta^{\ast}_{2},q^{\ast}_{2})\|_{\mathcal{X}}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\sup_{t\in(0,T)}\mathfrak{E}[\hat{p}](t)+\sup_{t\in(0,T)}b\|\Delta\hat{p}_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{T}\mathfrak{D}[\hat{p}](s)\,\textup{d}s$
$\displaystyle+\sup_{t\in(0,T)}\mathcal{E}[\hat{\Theta}](t)+\|\hat{q}\|_{L^{\infty}H^{1}}^{2}+\sum_{k=1}^{2}\int_{0}^{T}\|\partial_{t}^{k}\hat{q}(s)\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\exp\big{(}\int_{0}^{T}(1+\Lambda(s))\,\textup{d}s\big{)}\int_{0}^{T}(\|\nabla
g_{1}\|_{L^{2}}^{2}+\|g_{1t}\|_{L^{2}}^{2})\,\textup{d}s$
$\displaystyle+(1+\bar{\tau}+\bar{\tau}^{2})\|f_{1}\|_{H^{2}L^{2}}^{2}.$

Hence, due to estimates (6.24), (6.52), (6.62) and (6.65), we find

(6.67)
$\displaystyle\|(\hat{p},\hat{\Theta},\hat{q})\|_{\mathcal{X}}^{2}\lesssim$
$\displaystyle\,R_{1}^{2}C(T,R_{1},R_{2})\exp(T+TC_{1}(T,R_{1},R_{2}))\big{\|}(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\big{\|}_{\mathcal{X}}^{2}$
$\displaystyle+(1+\bar{\tau}+\bar{\tau}^{2})R_{1}^{2}C_{T}^{2}\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}^{2},$

where $C_{1}(T,R_{1},R_{2})$ is the constant in (6.24). From here, we infer
that

(6.68)
$\|\mathcal{T}(p^{\ast}_{1},\Theta^{\ast}_{1},q^{\ast}_{1})-\mathcal{T}(p^{\ast}_{2},\Theta^{\ast}_{2},q^{\ast}_{2})\|_{\mathcal{X}}\lesssim
R_{1}C(T,R_{1},R_{2},\bar{\tau})\|(\hat{p}^{\ast},\hat{\Theta}^{\ast},\hat{q}^{\ast})\|_{\mathcal{X}}.$

Consequently, it suffices to select $R_{1}$ as small as needed to ensure that
the mapping $\mathcal{T}$ is a strict contraction. ∎

##  7\. Limiting behaviour as $\tau\searrow 0$: Proof of Theorem 2.2

We aim in this section to prove Theorem 2.2. More precisely, we prove that the
solution of the Westervelt–Pennes–Cattaneo system (1.12) converges as $\tau$
goes to zero, to the solution of the limiting system Westervelt–Pennes–Fourier
system corresponding to $\tau=0$. For the sake of simplicity and to be able to
relate it to the Westervelt–Pennes–Fourier problem analyzed in [21], we
replace the system (4.1) by the equivalent telegraph equation derived in
(4.20). Let $(p^{\tau},\Theta^{\tau})$ be the solution of the following
$\tau$-dependent problem

(7.1) $\left\\{\begin{aligned}
&(1-2k(\Theta^{\tau})p^{\tau})p_{tt}^{\tau}-h(\Theta^{\tau})\Delta
p^{\tau}-b\Delta p_{t}^{\tau}=2k(\Theta^{\tau})(p_{t}^{\tau})^{2},&\
&\text{in}\ \Omega\times(0,T),\\\ &\tau
m\Theta_{tt}^{\tau}+(m+\tau\ell)\Theta_{t}^{\tau}+\ell\Theta^{\tau}-\kappa_{\textup{a}}\Delta\Theta^{\tau}=\mathcal{Q}(p_{t}^{\tau})+\tau\partial_{t}\big{(}\mathcal{Q}(p_{t}^{\tau})\big{)},&\
&\text{in}\ \Omega\times(0,T),\end{aligned}\right.$

subject to homogeneous Dirichlet boundary conditions (1.12b) and initial
conditions

(7.2)
$(p^{\tau},p_{t}^{\tau})|_{t=0}=(p_{0}^{\tau},p_{1}^{\tau}),\quad(\Theta^{\tau},\Theta_{t}^{\tau})|_{t=0}=(\Theta_{0}^{\tau},\Theta_{1}^{\tau})$

with $\Theta_{1}^{\tau}=\frac{1}{m}\big{(}-\nabla\cdot
q^{\tau}_{0}-\ell\Theta_{0}^{\tau}+\mathcal{Q}(p_{1}^{\tau})\big{)}$.

The proof presented below is based on the method employed in [14, 20] to
examine the weak limit of the Jordan–Moore–Gibson–Thompson-type equations.

###  7.1. Proof of Theorem 2.2

According to Theorem 2.1, we know that if the initial data
$p_{0}^{\tau},p_{1}^{\tau}$, $\Theta_{0}^{\tau},\Theta_{1}^{\tau}$ satisfies

(7.3) $\displaystyle\mathfrak{E}[p](0)\leq\delta,$
$\displaystyle\mathcal{E}[\Theta](0)\leq
C\Big{(}\|q_{0}\|^{2}_{H^{1}}+(1+\bar{\tau}+\bar{\tau}^{2})\big{(}E[\Theta,q](0)+R_{1}^{2}\big{)}\Big{)}\leq
R_{2}^{2},$

then the system (7.1) admits a unique solution verifying

(7.4) $\|p^{\tau}\|_{X_{p}}\leq R_{1},\quad\|\Theta^{\tau}\|_{X_{\Theta}}\leq
R_{2}.$

Since the radii $R_{1},R_{2}$ and $\delta$ are independent of $\tau$, we can
find subsequences such that

(7.5) $\displaystyle(p_{0}^{\tau},p_{1}^{\tau})\rightharpoonup(p_{0},p_{1})$
weakly in $\displaystyle H^{3}(\Omega)$ $\displaystyle\cap
H^{1}_{0}(\Omega)\times H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$
$\displaystyle(\Theta_{0}^{\tau},\Theta_{1}^{\tau})\rightharpoonup(\Theta_{0},\Theta_{1})$
weakly in $\displaystyle H^{2}(\Omega)$ $\displaystyle\cap
H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega).$

as $\tau$ tends to zero. Further, owing to the compactness of the embedding
$H^{s+1}\hookrightarrow H^{s},s\in\mathbb{N}$, it follows that there exists
strongly convergent subsequences still denoted by
$(p_{0}^{\tau},p_{1}^{\tau}),(\Theta_{0}^{\tau},\Theta_{1}^{\tau})$, such that

(7.6) $\displaystyle(p_{0}^{\tau},p_{1}^{\tau})\rightarrow(p_{0},p_{1})$ in
$\displaystyle H^{2}($ $\displaystyle\Omega)\cap H^{1}_{0}(\Omega)\times
H^{1}_{0}(\Omega),$
$\displaystyle(\Theta_{0}^{\tau},\Theta_{1}^{\tau})\rightarrow(\Theta_{0},\Theta_{1})$
in $\displaystyle H^{1}_{0}($ $\displaystyle\Omega)\times L^{2}(\Omega).$

Similarly, since the solution $(p^{\tau},\Theta^{\tau})$ is bounded uniformly
in $\tau$, there exists subsequences (which we keep on denoting by the index
$\tau$) converging in the weak-$\star$ topology

(7.7) $\displaystyle p^{\tau}\rightharpoonup p$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;H^{3}(\Omega)\cap H^{1}_{0}(\Omega));$
$\displaystyle p^{\tau}_{t}\rightharpoonup p_{t}$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega));$
$\displaystyle p^{\tau}_{tt}\rightharpoonup p_{tt}$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;H^{1}_{0}(\Omega));$ $\displaystyle
p^{\tau}_{ttt}\rightharpoonup p_{ttt}$ weakly in $\displaystyle
L^{2}(0,T;L^{2}(\Omega));$ $\displaystyle\Theta^{\tau}\rightharpoonup\Theta$
weakly-$\star$ in $\displaystyle L^{\infty}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega));$ $\displaystyle\Theta^{\tau}_{t}\rightharpoonup\Theta_{t}$
weakly-$\star$ in $\displaystyle L^{\infty}(0,T;H^{1}_{0}(\Omega));$
$\displaystyle\Theta^{\tau}_{tt}\rightharpoonup\Theta_{tt}$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;L^{2}(\Omega)).$

Next, we seek to connect the limit of the initial data sequence found in (7.6)
to the initial state of the limit functions in (7.7). For this purpose, we
invoke Aubin–Lions lemma [17, Lemma 1.2], which we can apply here thanks to
the compactness of the embeddings $H^{s+1}\hookrightarrow
H^{s},s\in\mathbb{N}$. Then, we get (up to subsequences)

(7.8) $\displaystyle p^{\tau}\rightarrow p$ strongly in $\displaystyle
C(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega));$ $\displaystyle
p^{\tau}_{t}\rightarrow p_{t}$ strongly in $\displaystyle
C(0,T;H^{1}_{0}(\Omega));$ $\displaystyle p^{\tau}_{tt}\rightarrow p_{tt}$
strongly in $\displaystyle L^{2}(0,T;L^{2}(\Omega));$
$\displaystyle\Theta^{\tau}\rightarrow\Theta$ strongly in $\displaystyle
C(0,T;H^{1}_{0}(\Omega));$
$\displaystyle\Theta^{\tau}_{t}\rightarrow\Theta_{t}$ strongly in
$\displaystyle C(0,T;L^{2}(\Omega)).$

This also leads to strong convergence of the initial data as follows

(7.9) $\displaystyle(p^{\tau},p_{t}^{\tau})(0)\rightarrow(p,p_{t})(0)$ in
$\displaystyle H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega);$
$\displaystyle(\Theta^{\tau},\Theta_{t}^{\tau})(0)\rightarrow(\Theta,\Theta_{t})(0)$
in $\displaystyle H^{1}_{0}(\Omega)\times L^{2}(\Omega).$

Hence, the uniqueness of the limit along with (7.6), (7.9) and (7.5) gives

(7.10) $\displaystyle(p,p_{t})(0)=(p_{0},p_{1})\in\,H^{3}(\Omega)\cap
H^{1}_{0}(\Omega)\times H^{2}(\Omega)\cap H^{1}_{0}(\Omega),$
$\displaystyle(\Theta,\Theta_{t})(0)=(\Theta_{0},\Theta_{1})\in\,H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega).$

This means that the initial states of the limit functions in (7.7) are well-
defined and have the necessary smoothness. Therefore, we can now focus on the
task of passing to the limit in system (7.1) when $\tau$ goes to zero. Let
$v\in L^{1}(0,T;L^{2}(\Omega))$. We denote by $(\hat{p},\hat{\Theta})$ the
differences $\hat{p}=p-p^{\tau},\hat{\Theta}=\Theta-\Theta^{\tau}$. Then, we
have

(7.11)
$\displaystyle\int^{T}_{0}\int_{\Omega}\big{(}m\Theta_{t}-\kappa_{\textup{a}}\Delta\Theta+\ell\Theta-\mathcal{Q}(p_{t})\big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle=$
$\displaystyle\int^{T}_{0}\int_{\Omega}\big{(}m\hat{\Theta}_{t}-\kappa_{\textup{a}}\Delta\hat{\Theta}+\ell\hat{\Theta}-\tau
m\Theta_{tt}^{\tau}-\tau\ell\Theta^{\tau}_{t}+\mathcal{Q}(p_{t}^{\tau})-\mathcal{Q}(p_{t})+\tau\partial_{t}\mathcal{Q}(p_{t}^{\tau})\big{)}v\,\textup{d}x\,\textup{d}t.$

From the weak-$\star$ convergence (7.7), we get

(7.12)
$\int^{T}_{0}\int_{\Omega}\big{(}m\hat{\Theta}_{t}-\kappa_{\textup{a}}\Delta\hat{\Theta}+\ell\hat{\Theta}\big{)}v\,\textup{d}x\,\textup{d}t\longrightarrow
0\quad\text{as}\ \tau\rightarrow 0.$

Moreover, recalling that
$\mathcal{Q}(p_{t})=\frac{2b}{\rho_{\textup{a}}C_{\textup{a}}^{4}}(p_{t})^{2}$
and keeping in mind the regularity provided by the fact that
$(p^{\tau},\Theta^{\tau})\in X_{p}\times X_{\Theta}$, we obtain

(7.13) $\displaystyle\int^{T}_{0}\int_{\Omega}\big{(}-\tau
m\Theta_{tt}^{\tau}-\tau\ell\Theta^{\tau}_{t}+\frac{4b\tau}{\rho_{\textup{a}}C_{\textup{a}}^{4}}p_{t}^{\tau}p_{tt}^{\tau}\big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle\lesssim$
$\displaystyle\,\tau\big{(}\|\Theta_{tt}^{\tau}\|_{L^{\infty}L^{2}}+\|\Theta_{t}^{\tau}\|_{L^{\infty}L^{2}}+\|p_{t}^{\tau}\|_{L^{\infty}L^{4}}\|p_{tt}^{\tau}\|_{L^{\infty}L^{4}}\big{)}\|v\|_{L^{1}L^{2}}\longrightarrow
0$

as $\tau$ tends to zero. The two remaining terms on the right-hand side of
(7.11) can be treated as follows

(7.14)
$\displaystyle\int^{T}_{0}\int_{\Omega}\big{(}\mathcal{Q}(p_{t}^{\tau})-\mathcal{Q}(p_{t})\big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle=-\int^{T}_{0}\int_{\Omega}\frac{2b}{\rho_{\textup{a}}C_{\textup{a}}^{4}}(p_{t}^{\tau}+p_{t})\hat{p}_{t}v\,\textup{d}x\,\textup{d}t$
$\displaystyle\lesssim\|(p_{t}^{\tau}+p_{t})\|_{L^{\infty}L^{4}}\|\hat{p}_{t}\|_{L^{\infty}L^{4}}\|v\|_{L^{1}L^{2}}.$

Then, the embedding $H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$ together with
the boundedness of $p^{\tau}_{t},p_{t}$ and the convergence (7.8) yields

(7.15)
$\int^{T}_{0}\int_{\Omega}\big{(}\mathcal{Q}(p_{t}^{\tau})-\mathcal{Q}(p_{t})\big{)}v\,\textup{d}x\,\textup{d}t\longrightarrow
0\quad\text{as}\ \tau\rightarrow 0.$

Consequently, we infer that $(p,\Theta)$ satisfies in
$L^{\infty}(0,T;L^{2}(\Omega))$ the equation

(7.16)
$m\Theta_{t}-\kappa_{\textup{a}}\Delta\Theta+\ell\Theta=\mathcal{Q}(p_{t}).$

Next, we claim that for all $v\in L^{1}(0,T;L^{2}(\Omega))$, it holds that

(7.17) $\int^{T}_{0}\int_{\Omega}\Big{(}(1-2k(\Theta)p)p_{tt}-h(\Theta)\Delta
p-b\Delta
p_{t}-2k(\Theta)(p_{t})^{2}\Big{)}v\,\textup{d}x\,\textup{d}t\longrightarrow
0\quad\text{as}\ \tau\rightarrow 0.$

To see this, we first observe that

(7.18)
$\displaystyle\int^{T}_{0}\int_{\Omega}\Big{(}(1-2k(\Theta)p)p_{tt}-h(\Theta)\Delta
p-b\Delta p_{t}-2k(\Theta)(p_{t})^{2}\Big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle=$
$\displaystyle\int^{T}_{0}\int_{\Omega}\Big{(}(1-2k(\Theta)p)\hat{p}_{tt}+(1-2k(\Theta)p)p_{tt}^{\tau}-h(\Theta)\Delta\hat{p}-h(\Theta)\Delta
p^{\tau}$ $\displaystyle-b\Delta\hat{p}_{t}-b\Delta
p_{t}^{\tau}-2k(\Theta)((p_{t})^{2}-(p_{t}^{\tau})^{2})-2k(\Theta)(p_{t}^{\tau})^{2}\Big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle=$
$\displaystyle\int^{T}_{0}\int_{\Omega}\Big{(}(1-2k(\Theta)p)\hat{p}_{tt}-h(\Theta)\Delta\hat{p}-b\Delta\hat{p}_{t}+(1-2k(\Theta^{\tau})p^{\tau})p_{tt}^{\tau}+2\big{(}k(\Theta^{\tau})p^{\tau}-k(\Theta)p\big{)}p_{tt}^{\tau}$
$\displaystyle-(h(\Theta)-h(\Theta^{\tau}))\Delta
p^{\tau}-h(\Theta^{\tau})\Delta p^{\tau}-b\Delta
p_{t}^{\tau}-2k(\Theta)(p_{t}+p_{t}^{\tau})\hat{p}_{t}-2k(\Theta)(p_{t}^{\tau})^{2}\Big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle=$
$\displaystyle\int^{T}_{0}\int_{\Omega}\Big{(}(1-2k(\Theta)p)\hat{p}_{tt}-h(\Theta)\Delta\hat{p}-b\Delta\hat{p}_{t}-2k(\Theta^{\tau})\hat{p}p_{tt}^{\tau}+2\big{(}k(\Theta^{\tau})-k(\Theta)\big{)}pp_{tt}^{\tau}$
$\displaystyle-(h(\Theta)-h(\Theta^{\tau}))\Delta
p^{\tau}-2k(\Theta)(p_{t}+p_{t}^{\tau})\hat{p}_{t}+2\big{(}k(\Theta^{\tau})-k(\Theta)\big{)}(p_{t}^{\tau})^{2}\Big{)}v\,\textup{d}x\,\textup{d}t.$

We want to prove that the integral on right-hand side goes to zero as the
relaxation time vanishes. Again, the weak-$\star$ convergence of $p^{\tau}$ to
$p$ in (7.7) along with the boundedness of $(p,\Theta)\in X_{p}\times
X_{\Theta}$ implies that

(7.19) $\displaystyle h(\Theta)\Delta p^{\tau}\rightharpoonup h(\Theta)\Delta
p$ weakly-$\star$ in $\displaystyle L^{\infty}(0,T;H^{1}_{0}(\Omega));$
$\displaystyle((1-2k(\Theta)p)p^{\tau}_{tt}\rightharpoonup((1-2k(\Theta)p)p_{tt}$
weakly-$\star$ in $\displaystyle L^{\infty}(0,T;H^{1}_{0}(\Omega)).$

Then, we can conclude that

(7.20)
$\int^{T}_{0}\int_{\Omega}\Big{(}(1-2k(\Theta)p)\hat{p}_{tt}-h(\Theta)\Delta\hat{p}-b\Delta\hat{p}_{t}\Big{)}v\,\textup{d}x\,\textup{d}t\longrightarrow
0\quad\text{as}\ \tau\rightarrow 0.$

Further, relying on (7.8), we have

(7.21)
$\displaystyle\int^{T}_{0}\int_{\Omega}\Big{(}-2k(\Theta^{\tau})\hat{p}p_{tt}^{\tau}-2k(\Theta)(p_{t}+p_{t}^{\tau})\hat{p}_{t}\Big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle\lesssim$
$\displaystyle\,\Big{(}\|k(\Theta^{\tau})\|_{L^{\infty}L^{\infty}}\|\hat{p}\|_{L^{\infty}L^{4}}\|p_{tt}^{\tau}\|_{L^{\infty}L^{4}}+\|k(\Theta)\|_{L^{\infty}L^{\infty}}\|p_{t}+p_{t}^{\tau}\|_{L^{\infty}L^{4}}\|\hat{p}_{t}\|_{L^{\infty}L^{4}}\Big{)}\|v\|_{L^{1}L^{2}},$

which yields that

(7.22)
$\int^{T}_{0}\int_{\Omega}\Big{(}-2k(\Theta^{\tau})\hat{p}p_{tt}^{\tau}-2k(\Theta)(p_{t}+p_{t}^{\tau})\hat{p}_{t}\Big{)}v\,\textup{d}x\,\textup{d}t\longrightarrow
0\quad\text{as}\ \tau\rightarrow 0.$

In order to handle the terms involving the difference
$k(\Theta^{\tau})-k(\Theta)$ or $h(\Theta^{\tau})-h(\Theta)$, we call upon
analogous estimates to (6.42a), (6.43a). Since we can show as in (6.42a) that

(7.23)
$\|k(\Theta^{\tau})-k(\Theta)\|_{L^{\infty}L^{4}}\lesssim\|\Theta^{\tau}-\Theta\|_{L^{\infty}L^{4}}\Big{(}1+\|\Theta^{\tau}\|^{1+\gamma_{2}}_{L^{\infty}L^{4}}+\|\Theta\|^{1+\gamma_{2}}_{L^{\infty}L^{4}}\Big{)},$

the boundedness of $\Theta,\Theta^{\tau}$ in $X_{\Theta}$ allows to obtain
that

(7.24)
$\displaystyle\int^{T}_{0}\int_{\Omega}2\Big{(}\big{(}k(\Theta^{\tau})-k(\Theta)\big{)}pp_{tt}^{\tau}+\big{(}k(\Theta^{\tau})-k(\Theta)\big{)}(p_{t}^{\tau})^{2}\Big{)}v\,\textup{d}x\,\textup{d}t$
$\displaystyle\leq$
$\displaystyle\,2\Big{(}\|p\|_{L^{\infty}L^{\infty}}\|p_{tt}^{\tau}\|_{L^{\infty}L^{4}}+\|p^{\tau}_{t}\|_{L^{\infty}L^{4}}^{2}\Big{)}\|k(\Theta^{\tau})-k(\Theta)\|_{L^{\infty}L^{4}}\|v\|_{L^{1}L^{2}}\rightarrow
0\quad\text{as}\ \tau\rightarrow 0.$

Likewise, the following estimate

(7.25)
$\|h(\Theta^{\tau})-h(\Theta)\|_{L^{\infty}L^{4}}\lesssim\|\Theta^{\tau}-\Theta\|_{L^{\infty}L^{4}}\Big{(}1+\|\Theta^{\tau}\|^{1+\gamma_{1}}_{L^{\infty}L^{4}}+\|\Theta\|^{1+\gamma_{1}}_{L^{\infty}L^{4}}\Big{)},$

leads to

(7.26)
$\int^{T}_{0}\int_{\Omega}\big{(}h(\Theta^{\tau})-h(\Theta)\big{)}\Delta
p^{\tau}v\,\textup{d}x\,\textup{d}t\lesssim\|h(\Theta^{\tau})-h(\Theta)\|_{L^{\infty}L^{4}}\|\Delta
p^{\tau}\|_{L^{\infty}L^{4}}\|v\|_{L^{1}L^{2}}$

converging to zero as $\tau$ goes to zero. Thus, the terms on the right-hand
side of (7.18) all vanish as $\tau$ tends to zero, then we infer that the
limit $p$ satisfies the equation

(7.27) $(1-2k(\Theta)p)p_{tt}-h(\Theta)\Delta p-b\Delta
p_{t}=2k(\Theta)(p_{t})^{2},$

which is understood in $L^{\infty}(0,T;L^{2}(\Omega))$.

We conclude that there exists a subsequence converging in the sense of (7.5),
(7.7) whose limit $(p,\Theta)$ is a solution to the limiting system (2.6). In
addition, since we have slightly more regularity than that provided by the
well-posedness result in [21] especially in terms of the derivatives in time
$p_{t},p_{tt},\Theta_{t},\Theta_{tt}$, the uniqueness of the solution obtained
in [21] extends to the functional setting at hand. Thus, we can confirm that
the whole sequence $(p^{\tau},\Theta^{\tau})_{\tau\in(0,\bar{\tau}]}$
converges to the solution of the parabolic system (2.6) based on [30,
Proposition 10.13].

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