# Well-posedness and exponential stability of a thermoelastic system with
internal delay

Smain Moulay Khatir  UR Analysis and Control of PDEs, UR 13ES64, Department of
Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia
Laboratory of Analysis and Control of Partial Differential Equations, Djillaly
Liabes University, Sidi Bel Abbes, Algeria<EMAIL_ADDRESS>and
Farhat Shel  UR Analysis and Control of PDEs, UR 13ES64, Department of
Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia
<EMAIL_ADDRESS>
###### Abstract.

The presence of a delay in a thermoelastic system destroys the well-posedness
and the stabilizing effect of the heat conduction [17]. To avoid this problem
we add to the system, at the delayed equation, a Kelvin-Voigt damping. At
first, we prove the well-posedness of the system by the semigroup theory.
Next, under appropriate assumptions, we prove the exponential stability of the
system by introducing a suitable Lyapunov functional.

###### Key words and phrases:

Thermoelastic system, delay, Kelvin-voigt damping, well posedness, exponential
stability

###### 2010 Mathematics Subject Classification:

35B35, 35B40, 93D05 93D20

###### Contents

  1. 1 Introduction
  2. 2 Well-posedness of the problem
  3. 3 Exponential stability

##  1\. Introduction

Let us consider the following thermoelastic system with delay

(1.1) $\left\\{\begin{tabular}[]{l}$u_{tt}(x,t)-\alpha
u_{xx}(x,t-\tau)+\gamma\theta_{x}(x,t)=0,\;\;\;\;\;\;\text{ in
}(0,\ell)\times(0,\infty),$\\\ $\theta_{t}(x,t)-\kappa\theta_{xx}(x,t)+\gamma
u_{xt}(x,t)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\text{ in
}(0,\ell)\times(0,\infty),$\\\
$u(0,t)=u(\ell,t)=\theta_{x}(0,t)=\theta_{x}(\ell,t)=0,\;\;\;\;\;\;\;\;\;t\geq
0$\end{tabular}\right.$

where $\alpha,\gamma,\kappa$ and $\ell$ are some positive constants. The
functions $u=u(x,t)$ and $\theta=\theta(x,t)$ describe respectively the
displacement and the temperature difference, with $x\in(0,\ell)$ and $t\geq
0.$ Moreover, $\tau>0$ is the time delay. Racke proved in [17] that, under
some initial and boundary conditions, the system (1.1) is not well posed and
unstable even if $\tau$ is relatively small. However, it is well known that,
in the absence of delay, the damping through the heat conduction is strong
enough to produce an exponential stable system (see for example [16, 9, 18]),
and specially,[16] and [10] where various types of boundary conditions are
associated to the one dimensional thermoelastic systems.

In recent years, the PDEs with time delays effects become an active area of
research. In fact, time delays so often arise in many applications since, most
physical phenomena not only depend on the present state but also on some past
occurrences, see for instance [19] and references therein, but as for the
classical thermoelastic system, an arbitrary small delay may destroy the well-
posedness of the problem or may destroy the stability, see also [15, 3, 6, 5].

In order to solve the problem, additional conditions or control terms have
been used, we refer to [7, 5, 13, 4, 1, 14], see also [11] and references
therein . In this paper we add to the delayed equation, a Kelvin-Voigt damping
of the form $-\beta u_{xxt}(x,t)$ for some real positive number $\beta,$ which
eventually depends on $\alpha,\gamma,\kappa$ and $\tau.$ Then our system takes
the form

(1.2) $\left\\{\begin{tabular}[]{l}$u_{tt}(x,t)-\alpha u_{xx}(x,t-\tau)-\beta
u_{xxt}(x,t)+\gamma\theta_{x}(x,t)=0,\;\;\text{ in
}\Omega\times(0,\infty),$\\\ $\theta_{t}(x,t)-\kappa\theta_{xx}(x,t)+\gamma
u_{xt}(x,t)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in }\Omega\times(0,\infty),$\\\
$u(0,t)=u(\ell,t)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in }(0,\infty),$\\\
$\theta_{x}(0,t)=\theta_{x}(\ell,t)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in }(0,\infty),$\\\
$u_{x}(x,t-\tau)=f_{0}(x,t-\tau),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in }\Omega\times(0,\tau),$\\\
$u(x,0)=u_{0}(x),u_{t}(x,0)=u_{1}(x),\theta(x,0)=\theta_{0}(x),\;\;\;\;\;\;\;\;\;\;\;\;\text{
in }\Omega$\end{tabular}\right.$

where the initial data $(u_{0},u_{1},f_{0},\theta_{0})$ belongs to a suitable
space and with $\Omega=(0,\ell).$ We meanly investigate well-posedness and
exponential stability of such initial-boundary value problem.

This idea arises from [2] where the authors added a Kelvin-Voigt damping term
to the abstract equation. More precisely, they considered the following system

(1.3)
$\left\\{\begin{tabular}[]{l}$u^{\prime\prime}(t)+aBB^{*}u^{\prime}(t)+BB^{*}u(t-\tau)=0,\;\;\text{
in }(0,\infty),$\\\ $u(0)=u_{0},\;\;u^{\prime}(0)=u_{1},$\\\
$B^{*}u(t-\tau)=f_{0}(t-\tau),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in }(0,\tau),$\end{tabular}\right.$

where a ”prime” denotes a one-dimensional derivative with respect to ”t” and
where $B:\mathcal{D}(B)\subset H_{1}\rightarrow H$ is a linear unbounded
operator from a Hilbert space $H_{1}$ to a Hilbert space $H,$ such that
$B^{*},$ the adjoint of $B,$ satisfies some properties of coercivity and
compact embedding. They obtained an exponential decay result under the
assumption $\tau\leq a.$

In [12], the authors dropped the time delay in the harmonic term of the
elastic equation in (1.1) and added a delay term of the form
$\int_{\tau_{1}}^{\tau_{2}}\mu(s)\theta_{xx}(x,t-s\tau)ds$ in the heat
equation, where $\tau_{1}$ and $\tau_{2}$ are non-negative constants such that
$\tau_{1}<\tau_{2}$ and $\mu:[\tau_{1},\tau_{2}]\rightarrow\mathbb{R}$ is a
bounded function. They proved an exponential decay result under the condition
$\int_{\tau_{1}}^{\tau_{2}}|\mu(s)|ds<\kappa.$

We define the energy of a solution of problem (1.2) as

$E(t):=\frac{1}{2}\int_{\Omega}\left(u_{t}^{2}(x,t)+\alpha
u_{x}^{2}(x,t)+\theta^{2}(x,t)\right)dx+\xi\int_{\Omega}\int_{0}^{1}u_{x}^{2}(x,t-\tau\rho)d\rho
dx$

where $\xi>0$ is a parameter fixed later on.

The paper is organized as follow. In section 2 we first formulate the problem
(1.2) into an appropriate Hilbert space, and then we study the well-posedness
of the system using semigroup theory. In section 3, we prove, using Lyapunov’s
method, a result of exponential stability of system (1.2).

##  2\. Well-posedness of the problem

We introduce, as in [2], the new variable

(2.1) $z(x,\rho,t)=u_{x}(x,t-\tau\rho),\;\;\;\;\;\;\text{ in
}\Omega\times(0,1)\times(0,\infty),$

Clearly, $z(x,\rho,t)$ satisfies

(2.2) $\displaystyle\tau z_{t}(x,\rho,t)+z_{\rho}(x,\rho,t)$ $\displaystyle=$
$\displaystyle
0,\;\;\;\;\;\;\;\;\;\;\;\;x\in\Omega,\;\;\rho\in(0,1),\;\;t\in(0,+\infty),$
(2.3) $\displaystyle z(x,0,t)$ $\displaystyle=$ $\displaystyle
u_{x}(x,t),\;\;\;x\in\Omega,\;\;t\in(0,+\infty).$

Then, problem (1.2) takes the form

(2.4) $\displaystyle u_{tt}(x,t)-\alpha z_{x}(x,1,t)-\beta
u_{xxt}(x,t)+\gamma\theta_{x}(x,t)=0,$ $\displaystyle\text{ in
}\Omega\times(0,\infty),$ (2.5) $\displaystyle\tau
z_{t}(x,\rho,t)+z_{\rho}(x,\rho,t)=0,$ $\displaystyle\text{ in
}\Omega\times(0,1)\times(0,+\infty),$ (2.6)
$\displaystyle\theta_{t}(x,t)-\kappa\theta_{xx}(x,t)+\gamma u_{xt}(x,t)=0,$
$\displaystyle\text{ in }\Omega\times(0,\infty),$ (2.7) $\displaystyle
u(0,t)=u(\ell,t)=0,$ $\displaystyle\text{ in }(0,\infty),$ (2.8)
$\displaystyle\theta_{x}(0,t)=\theta_{x}(\ell,t)=0,$ $\displaystyle\text{ in
}(0,\infty),$ (2.9) $\displaystyle z(x,0,t)=u_{x}(x,t),$ $\displaystyle\text{
in }\Omega\times(0,\infty),$ (2.10) $\displaystyle
u(x,0)=u_{0}(x),u_{t}(x,0)=u_{1}(x),\theta(x,0)=\theta_{0}(x),$
$\displaystyle\text{ in }\Omega,$ (2.11) $\displaystyle
z(x,\rho,0)=f_{0}(x,-\tau\rho),$ $\displaystyle\text{ in }\Omega\times(0,1).$

Observe that it follows from (2.6)-(2.8) that
$\int_{\Omega}\theta_{t}(x,t)dx=0$ that is, $\int_{\Omega}\theta(x,t)dx$ is
conservative all the time. Without loss of generality, we assume that
$\int_{\Omega}\theta(x,t)dx=0$. Otherwise, we can make the substitution
$\tilde{\theta}(x,t)=\theta(x,t)-\frac{1}{\ell}\int_{\Omega}\theta_{0}(x)dx,$
in fact $(u,v,z,\theta)$ and $(u,v,z,\tilde{\theta})$ satisfy the same system
(2.4)-(2.11).

Let

$\mathcal{H}=\left\\{(f,g,p,h)\in H^{1}_{0}(\Omega)\times L^{2}(\Omega)\times
L^{2}(\Omega\times(0,1))\times
L^{2}(\Omega)\mid\int_{\Omega}h(x)dx=0\right\\}.$

Equipped with the following inner product: for any
$U_{k}=(f_{k},g_{k},p_{k},h_{k})\in\mathcal{H},\;\;k=1,2,$

$\left\langle U_{1},U_{2}\right\rangle_{\mathcal{H}}=\int_{\Omega}\left(\alpha
f_{1x}(x)f_{2x}(x)+g_{1}(x)g_{2}(x)+h_{1}(x)h_{2}(x)\right)dx+\xi\int_{\Omega}\int_{0}^{1}p_{1}(x,\rho)p_{2}(x,\rho)d\rho
dx,$

$\mathcal{H}$ is a Hilbert space.

Define

$U:=(u,u_{t},z,\theta)$

then, problem (1.2) can be formulated as a first order system of the form

(2.12) $\left\\{\begin{array}[]{c}U^{\prime}=\mathcal{A}U,\\\
U(0)=(u_{0},u_{1},f_{0}(.,-.\tau),\theta_{0})\end{array}\right.$

where the operator $\mathcal{A}$ is defined by

$\mathcal{A}\left(\begin{array}[]{c}u\\\ v\\\ z\\\
\theta\end{array}\right)=\left(\begin{array}[]{c}v\\\ (\alpha z(.,1)+\beta
v_{x})_{x}-\gamma\theta_{x}\\\ -\frac{1}{\tau}z_{\rho}\\\ -\gamma
v_{x}+\kappa\theta_{xx}\end{array}\right)$

with domain

$\mathcal{D}(\mathcal{A})=\left\\{\begin{array}[]{c}U=(u,v,z,\theta)\in\mathcal{H}\cap\left[H^{1}_{0}(\Omega)\times
H^{1}_{0}(\Omega)\times L^{2}(\Omega;H^{1}(0,1))\times
H^{2}(\Omega)\right]\mid\\\ \;\;z(.,0)=u_{x}\;\;\;\text{and}\;\;(\alpha
z(.,1)+\beta v_{x})\in H^{1}(\Omega)\end{array}\right\\}$

in the Hilbert space $\mathcal{H}.$

For to establish the existence of solution, we will prove that the operator
$\mathcal{A}$ generates a $\mathcal{C}_{0}$-semigroup, and to do this, we will
prove that $\mathcal{A}-mId$ generates $\mathcal{C}_{0}$-semigroup (of
contractions), for an appropriate real number $m,$ function of
$\xi,\alpha,\beta$ and $\tau.$ Then we apply the bounded perturbation theorem
(Sect. III.1 of [8]). In fact, we begin by the following result

######  Lemma 2.1.

If $\xi>\frac{2\tau\alpha^{2}}{\beta},$ then there exists $m\in\mathbb{R}$
such that $\mathcal{A}-mId$ is dissipative maximal.

###### Proof.

Take $U=(u,v,z,h)\in\mathcal{D}(\mathcal{A}).$

(2.13) $\displaystyle\left\langle\mathcal{A}U,U\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\alpha\int_{\Omega}v_{x}(x)u_{x}(x)dx+\int_{\Omega}\left((\alpha
z(.,1)+\beta v_{x})_{x}(x)-\gamma\theta_{x}\right)v(x)dx$ $\displaystyle+$
$\displaystyle\int_{\Omega}\left(-\gamma
v_{x}+\kappa\theta_{xx}\right)(x)\theta(x)dx-\frac{\xi}{\tau}\int_{\Omega}\int_{0}^{1}z_{\rho}(x,\rho)z(x,\rho)d\rho
dx.$

Integrating by parts, using boundary conditions of $u,v$ and $\theta$ to get

$\displaystyle\int_{\Omega}\left((\alpha z(.,1)+\beta
v_{x})_{x}-\gamma\theta_{x}\right)(x)v(x)dx+\int_{\Omega}\left(-\gamma
v_{x}+\kappa\theta_{xx}\right)(x)\theta(x)dx$
$\displaystyle=-\alpha\int_{\Omega}z(x,1)v(x)dx-\beta\int_{\Omega}v^{2}_{x}(x)dx-\kappa\int_{\Omega}\theta_{x}^{2}(x)dx.$

Integrating by parts in $\rho$, we get

$\int_{\Omega}\int_{0}^{1}z_{\rho}(x,\rho)z(x,\rho)d\rho
dx=\frac{1}{2}\int_{\Omega}\left(z^{2}(x,1)-z^{2}(x,0)\right)dx.$

Then (2.13) become

$\displaystyle\left\langle\mathcal{A}U,U\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\alpha\int_{\Omega}v_{x}(x)u_{x}(x)dx-\alpha\int_{\Omega}z(x,1)v_{x}(x)dx$
$\displaystyle-$
$\displaystyle\beta\int_{\Omega}v^{2}_{x}(x)dx-\kappa\int_{\Omega}\theta_{x}^{2}(x)dx-\frac{\xi}{2\tau}\int_{\Omega}\left(z^{2}(x,1)-z^{2}(x,0)\right)dx,$

from which follows, using the Young’s inequality and that $z(x,0)=u_{x}(x),$

$\displaystyle\left\langle\mathcal{A}U,U\right\rangle_{\mathcal{H}}$
$\displaystyle\leq$
$\displaystyle\left(\alpha\varepsilon-\beta\right)\int_{\Omega}v_{x}^{2}(x)dx+\left(\frac{\alpha}{2\varepsilon}-\frac{\xi}{2\tau}\right)\int_{\Omega}z^{2}(x,1)dx+\left(\frac{\alpha}{2\varepsilon}+\frac{\xi}{2\tau}\right)\int_{\Omega}u^{2}_{x}(x)dx$
$\displaystyle-$ $\displaystyle\kappa\int_{\Omega}\theta_{x}^{2}(x)dx.$

Choosing $\alpha\varepsilon=\frac{\beta}{2}$, or equivalently,
$\varepsilon=\frac{\beta}{2\alpha},$ we get

$\left\langle\mathcal{A}U,U\right\rangle_{\mathcal{H}}\leq-\frac{\beta}{2}\int_{\Omega}v_{x}^{2}(x)dx+\left(\frac{\alpha^{2}}{\beta}-\frac{\xi}{2\tau}\right)\int_{\Omega}z^{2}(x,1)dx+\left(\frac{\alpha^{2}}{\beta}+\frac{\xi}{2\tau}\right)\int_{\Omega}u^{2}_{x}(x)dx-\kappa\int_{\Omega}\theta_{x}^{2}(x)dx.$

Then we choose $\xi>0$ such that
$\frac{\alpha^{2}}{\beta}-\frac{\xi}{2\tau}<0,$ that is,
$\xi>\frac{2\tau\alpha^{2}}{\beta}.$ Furthermore we take
$m=\frac{\alpha^{2}}{\beta}+\frac{\xi}{2\tau}>\frac{2\alpha^{2}}{\beta},$ to
get

$\left\langle\left(\mathcal{A}-mId\right)U,U\right\rangle_{\mathcal{H}}\leq-\frac{\beta}{2}\int_{\Omega}v_{x}^{2}(x)dx+\left(\frac{\alpha^{2}}{\beta}-\frac{\xi}{2\tau}\right)\int_{\Omega}z^{2}(x,1)dx-\kappa\int_{\Omega}\theta_{x}^{2}(x)dx\leq
0$

which means that the operator $\mathcal{A}-mId\;$ is dissipative.

Now, we will prove the maximality of $\mathcal{A}-mId.\;$ It suffices to show
that $\lambda Id-\mathcal{A}$ is surjective for a fixed $\lambda>m.$ Given
$(f,g,p,h)\in\mathcal{H},$ we look for
$U=(u,v,z,\theta)\in\mathcal{D}(\mathcal{A}),$ solution of

$\left(\lambda Id-\mathcal{A}\right)\left(\begin{array}[]{c}u\\\ v\\\ z\\\
\theta\end{array}\right)=\left(\begin{array}[]{c}f\\\ g\\\ p\\\
h\end{array}\right),$

that is verifying

(2.14) $\left\\{\begin{tabular}[]{l}$\lambda u-v=f,$\\\ $\lambda
v-\left(\alpha z(.,1)+\beta v_{x}\right)_{x}+\gamma\theta_{x}=g,$\\\ $\lambda
z-\frac{1}{\tau}z_{\rho}=p,$\\\ $\lambda\theta+\gamma
v_{x}-\kappa\theta_{xx}=h.$\end{tabular}\right.$

Suppose that we have found $u$ with the appropriate regularity. Then,

(2.15) $v=\lambda u-f.$

To determine $z,$ recall that $z(.,0)=u_{x},$ then, by $(\ref{max1})_{3}$, we
obtain

(2.16) $z(.,\rho)=e^{-\lambda\tau\rho}u_{x}+\tau
e^{-\lambda\tau\rho}\int_{0}^{\rho}p(s)e^{\lambda\tau s}ds,$

and, in particular

(2.17) $z(x,1)=e^{-\lambda\tau}u_{x}+z_{0},$

with $z_{0}\in L^{2}(\Omega)$ defined by

$z_{0}=\tau e^{-\lambda\tau}\int_{0}^{1}p(s)e^{\lambda\tau s}ds.$

Now, Multiplying $(\ref{max1})_{2}$ and $(\ref{max1})_{4}$ respectively by
$w\in H^{1}_{0}(\Omega)$ and $\varphi\in H^{2}(\Omega)$ such that
$\varphi_{x}(0)=\varphi_{x}(\ell)=0,$ we obtain after some integrations by
parts taking into account boundary conditions on $v,$ $\theta$ and $w,$

(2.18) $\lambda\int_{\Omega}vwdx+\int_{\Omega}\left(\alpha z(.,1)+\beta
v_{x}\right)w_{x}dx+\gamma\int_{\Omega}\theta_{x}wdx=\int_{\Omega}gwdx$

and

(2.19) $\lambda\int_{\Omega}\theta\varphi
dx-\gamma\int_{\Omega}v\varphi_{x}dx+\kappa\int_{\Omega}\theta_{x}\varphi_{x}dx=\int_{\Omega}h\varphi
dx.$

Substituting (2.15) and (2.17) into (2.18) and (2.19), we get

(2.20) $\lambda^{2}\int_{\Omega}uwdx+\left(\alpha
e^{-\lambda\tau}+\lambda\beta\right)\int_{\Omega}u_{x}w_{x}dx+\gamma\int_{\Omega}\theta_{x}wdx=\int_{\Omega}(g+\lambda
f)wdx+\int_{\Omega}(f_{x}-\alpha z_{0})w_{x}dx$

and

(2.21) $\lambda\int_{\Omega}\theta\varphi
dx-\lambda\gamma\int_{\Omega}u\varphi_{x}dx+\kappa\int_{\Omega}\theta_{x}\varphi_{x}dx=\int_{\Omega}(h-\gamma
f)\varphi dx.$

Summing (2.20), and (2.21) multiplied by $\frac{1}{\lambda},$ we get

(2.22) $b\left((u,\theta),(w,\varphi)\right)=F(w,\varphi)$

with

$b\left((u,\theta),(w,\varphi)\right)=\int_{\Omega}\left[\lambda^{2}uw+\left(\alpha
e^{-\lambda\tau}+\lambda\beta\right)u_{x}w_{x}\right]dx+\int_{\Omega}\left(\theta\varphi+\frac{\kappa}{\lambda}\theta_{x}\varphi_{x}\right)dx+\gamma\int_{\Omega}\left(\theta_{x}w-u\varphi_{x}\right)dx$

and

$F(w,\varphi)=\int_{\Omega}(g+\lambda f)wdx+\int_{\Omega}(f_{x}-\alpha
z_{0})w_{x}dx+\frac{1}{\lambda}\int_{\Omega}(h-\gamma f)\varphi dx.$

The space

$\mathcal{F}:=\left\\{(w,\varphi)\in H^{1}_{0}(\Omega)\times
H^{2}(\Omega)\mid\theta_{x}(0)=\theta_{x}(\ell)=0\right\\},$

equipped with the inner product

$\langle(w_{1},\varphi_{1}),(w_{2},\varphi_{2})\rangle_{\mathcal{F}}=\int_{\Omega}\left(w_{1}w_{2}+w_{1x}w_{2x}+\varphi_{1}\varphi_{2}+\varphi_{1x}\varphi_{2x}\right)dx,$

is a Hilbert space; the bilinear form $b$ on $\mathcal{F}\times\mathcal{F}$
and the linear form $F$ on $\mathcal{F}$ are continuous. Moreover, for every
$(w,\varphi)\in\mathcal{F},$

$|b\left((w,\varphi),(w,\varphi)\right)|\geq
c\|(w,\varphi)\|^{2}_{\mathcal{H}}$

with $c:=\min\left(\lambda^{2},\left(\alpha
e^{-\lambda\tau}+\lambda\beta\right),1,\frac{\kappa}{\lambda}\right)>0.$

By the Lax-Milgram lemma, equation (2.22) has a unique solution
$(u,\theta)\in\mathcal{F}.$ Immediately, from (2.15), we have that $v\in
H^{1}_{0}(\Omega).$ Now, if we consider
$(w,\varphi)\in\\{0\\}\times\mathcal{D}(\Omega)$ in (2.22) we deduce that
equation $(\ref{max1})_{4}$ holds true. The function $z,$ defined by (2.17),
belongs to $L^{1}(\Omega,H^{1}(0,1))$ and satisfies $(\ref{max1})_{3}$ and
$z(.,0)=u_{x}.$

The functions $z(.,1)$ and $v_{x}$ belong to $L^{2}(\Omega),$ then we take
$(w,\varphi)\in\mathcal{D}(\Omega)\times\\{0\\}$ in (2.22) to deduce that
$\alpha z(.,1)+\beta v_{x}$ belongs to $H^{1}_{0}(\Omega)$ and that equation
$(\ref{max1})_{1}$ holds true.

Let $\tilde{\theta}=\theta-\frac{1}{\ell}\int_{\Omega}\theta_{0}dx,$ then we
have that $U=(u,v,z,\tilde{\theta})$ belongs to $\mathcal{D}(\mathcal{A})$,
and $\mathcal{A}U=(f,g,p,h).$ Thus, $\lambda Id-\mathcal{A}$ is surjective for
every $\lambda>0$. ∎

In conclusion the operator $\mathcal{A}-mId$ generates a
$\mathcal{C}_{0}$-semigroup of contraction. By the bounded perturbation
theorem (Sect. III.1 of [8]), we have

######  Lemma 2.2.

The operator $\mathcal{A}$ generates a $\mathcal{C}_{0}$-semigroup on
$\mathcal{H}$.

Finally, the well-posedness result follows from semigroup theory.

######  Theorem 2.3.

For any initial datum $U_{0}\in\mathcal{H}$ there exists a unique solution
$U\in\mathcal{C}([0,+\infty),\mathcal{H})$ of problem (2.12). Moreover, if
$U_{0}\in\mathcal{D}(\mathcal{A}),$ then
$U\in\mathcal{C}([0,+\infty),\mathcal{D}(\mathcal{A}))\cap\mathcal{C}^{1}([0,+\infty),\mathcal{H}).$

##  3\. Exponential stability

Based on Lyapunov method, we prove that the system (1.2) is exponentially
stable for some $\beta>0.$ More precisely:

######  Theorem 3.1.

There exists $\beta_{0}>0$ such that for every $\beta\geq\beta_{0},$ the
system (1.2) is exponentially stable.

###### Proof.

We take as Lyapunov function

$V(t):=N_{1}V_{1}(t)+\alpha
N_{2}V_{2}(t)+N_{3}V_{3}(t)+N_{4}V_{4}(t)+N_{5}V_{5}(t)+N_{6}V_{6}(t)$

where

$\displaystyle V_{1}(t)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}\left\|u_{t}\right\|^{2}=\frac{1}{2}\int_{\Omega}u_{t}^{2}dx,\;\;V_{2}(t):=\;\frac{1}{2}\left\|u_{x}\right\|^{2}=\frac{1}{2}\int_{\Omega}u_{x}^{2}dx,\;\;V_{3}(t):=\;\frac{1}{2}\left\|\theta\right\|^{2}=\frac{1}{2}\int_{\Omega}\theta^{2}dx,$
$\displaystyle V_{4}(t)$ $\displaystyle:=$
$\displaystyle\int_{0}^{1}e^{-2\lambda\rho}\left\|z(.,\rho,t)\right\|^{2}d\rho=\int_{0}^{1}e^{-2\lambda\rho}\int_{\Omega}z^{2}(x,\rho,t)dxd\rho,$
$\displaystyle V_{5}(t)$ $\displaystyle:=$
$\displaystyle-\int_{0}^{1}e^{-\lambda\rho}f(\rho)\left\langle
z(.,\rho,t),u_{x}\right\rangle
d\rho=-\int_{0}^{1}e^{-\lambda\rho}f(\rho)\int_{\Omega}z(x,\rho,t)u_{x}(x,t)dxd\rho\;\;\text{and}$
$\displaystyle V_{6}(t)$ $\displaystyle:=$ $\displaystyle\langle
u,u_{t}\rangle=\int_{\Omega}uu_{t}dx.$

$f$ is a real function defined on $[0,1]$ and that will be determined later.
The constants $N_{1},N_{2},N_{3},N_{4},N_{5}$ and $N_{6}$ are positive numbers
to be fixed later too.

Denote by $\tilde{V}(t)$ the energy defined by

$\tilde{V}(t):=N_{1}V_{1}(t)+\alpha
N_{2}V_{2}(t)+N_{3}V_{3}(t)+N_{4}V_{4}(t).$

It is clear that $\tilde{V}(t)$ is equivalent to $E(t).$ Then for a suitable
choice of $f$ we will prove that we can find $\\{N_{1},...,N_{6}\\}$ and
$\beta>0$ such that the following two assumptions are satisfied:

  1. (A1)

$V(t)$ is equivalent to $\tilde{V}(t),$

  2. (A2)

$V^{\prime}(t)\leq-n_{0}\tilde{V}(t),$ for some positive number $n_{0}$.

The rest of the proof will be divided into three parts:

First part: it concerns the second assumption (A2). We start with the
following lemma

######  Lemma 3.2.

Let $V(t)$ be defined as before. By choosing a function $f$ satisfying

(3.1) $-e^{-2\lambda\rho}=(e^{-\lambda\rho}f(\rho))^{\prime},\;\;\lambda>0$

and by taking $N_{3}=N_{1},$ $N_{6}\beta=N_{2}\alpha$ and
$N_{6}\alpha=\frac{f(1)e^{-\lambda}}{\tau}N_{5}$ we have that for every
positive real numbers $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}$ and
$\varepsilon_{4},$

$\displaystyle V^{\prime}(t)$ $\displaystyle\leq$
$\displaystyle\left(-N_{4}\frac{e^{-2\lambda}}{\tau}+N_{1}\frac{\alpha\varepsilon_{1}}{2}\right)\left\|z(.,1,.)\right\|^{2}$
$\displaystyle+$
$\displaystyle\left(-2\frac{k_{1}}{\tau}N_{4}+\frac{N_{5}}{2\tau}\left(\varepsilon_{2}+\frac{\tau}{\varepsilon_{3}}\right)\right)V_{4}(t)$
$\displaystyle+$
$\displaystyle\left(\frac{N_{4}}{\tau}+\frac{N_{5}}{\tau}\left(\frac{\Gamma}{2\varepsilon_{2}}-\Lambda\right)+N_{5}\frac{\Psi\gamma\varepsilon_{4}c_{p}}{2\alpha\tau}\right)\left\|u_{x}\right\|^{2}$
$\displaystyle+$
$\displaystyle\left(N_{1}\left(\frac{\alpha}{2\varepsilon_{1}}-\beta\right)+N_{5}\left(\frac{\varepsilon_{3}}{2}\Phi+\frac{\Psi
c_{p}}{2\alpha\tau}\right)\right)\left\|u_{tx}\right\|^{2}$ $\displaystyle+$
$\displaystyle\left(-N_{1}\kappa+N_{5}\frac{\Psi\gamma}{2\alpha\tau\varepsilon_{4}}\right)\left\|\theta_{x}\right\|^{2}$

where $c_{p}>0$ is the Poincaré constant associated to $\Omega,$ (it can be
taken equal to $\frac{\ell^{2}}{2}$) and

$\Psi:=f(1)e^{-\lambda},\;\;\;\Lambda:=f(0)\;\;\;\text{and}\;\;\Phi:=\int_{0}^{1}f^{2}(\rho)d\rho.$

Furthermore,

$\Gamma:=\int_{0}^{1}e^{-2\lambda\rho}d\rho=\frac{1-e^{-2\lambda}}{2\lambda}.$

Notice that, in view of (3.1), we have

$\Lambda=\Psi+\Gamma.$

###### Proof.

Computing the derivatives of $V_{1},V_{2},V_{3},V_{4}$ and $V_{5}$ using
integration by parts, boundary conditions and Youg’s inequality, we have

$\displaystyle V^{\prime}_{1}(t)=\left\langle u_{tt},u_{t}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle(\alpha z(.,1,.)+\beta
u_{tx})_{x},u_{t}\right\rangle-\gamma\left\langle\theta_{x},u_{t}\right\rangle$
$\displaystyle=$ $\displaystyle-\left\langle\alpha z(.,1,.)+\beta
u_{tx},u_{tx}\right\rangle-\gamma\left\langle\theta_{x},u_{t}\right\rangle$
$\displaystyle=$ $\displaystyle-\left\langle\alpha
z(.,1,.),u_{tx}\right\rangle-\beta\left\|u_{tx}\right\|^{2}-\gamma\left\langle\theta_{x},u_{t}\right\rangle$
$\displaystyle\leq$
$\displaystyle\left(\frac{\alpha}{2\varepsilon_{1}}-\beta\right)\left\|u_{tx}\right\|^{2}+\frac{\alpha\varepsilon_{1}}{2}\left\|z(.,1,.)\right\|^{2}-\gamma\left\langle\theta_{x},u_{t}\right\rangle,$
$\displaystyle V^{\prime}_{2}(t)$ $\displaystyle=$ $\displaystyle\left\langle
u_{xt},u_{x}\right\rangle,$ $\displaystyle
V^{\prime}_{3}(t)=\left\langle\theta_{t},\theta\right\rangle$ $\displaystyle=$
$\displaystyle-\gamma\left\langle(u_{xt},\theta\right\rangle+\kappa\left\langle\theta_{xx},\theta\right\rangle$
$\displaystyle=$ $\displaystyle\gamma\left\langle
u_{t},\theta_{x}\right\rangle-\kappa\left\|\theta_{x}\right\|^{2}.$

The derivative of $V_{4}$ is

$\displaystyle V^{\prime}_{4}(t)$ $\displaystyle=$ $\displaystyle
2\int_{0}^{1}e^{-2\lambda\rho}\left\langle
z(.,\rho,.),z_{t}(.,\rho,.)\right\rangle d\rho$ $\displaystyle=$
$\displaystyle-\frac{2}{\tau}\int_{0}^{1}e^{-2\lambda\rho}\left\langle
z(.,\rho,.),z_{\rho}(.,\rho,.)\right\rangle d\rho$ $\displaystyle=$
$\displaystyle-\frac{e^{-2\lambda}}{\tau}\left\|z(.,1,.)\right\|^{2}+\frac{1}{\tau}\left\|u_{x}\right\|^{2}-\frac{2\lambda}{\tau}\int_{0}^{1}e^{-2\lambda\rho}\left\|z(.,\rho,.)\right\|^{2}d\rho$
$\displaystyle\leq$
$\displaystyle-\frac{e^{-2\lambda}}{\tau}\left\|z(.,1,.)\right\|^{2}+\frac{1}{\tau}\left\|u_{x}\right\|^{2}-\frac{2\lambda}{\tau}V_{4}(t).$

The derivative of $V_{5}$ is calculated as follows

$\displaystyle V^{\prime}_{5}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}\int_{0}^{1}e^{-\lambda\rho}f(\rho)\left\langle
z_{\rho}(.,\rho,.),u_{x}\right\rangle
d\rho-\int_{0}^{1}e^{-\lambda\rho}f(\rho)\left\langle
z(.,\rho,.),u_{xt}\right\rangle d\rho$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}e^{-\lambda}f(1)\left\langle
z(.,1,.),u_{x}\right\rangle-\frac{1}{\tau}f(0)\left\|u_{x}\right\|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{\tau}\int_{0}^{1}(e^{-\lambda\rho}f(\rho))^{\prime}\left\langle
z(.,\rho,.),u_{x}\right\rangle
d\rho-\int_{0}^{1}e^{-\lambda\rho}f(\rho)\left\langle
z(.,\rho,.),u_{xt}\right\rangle d\rho.$

Replacing $e^{-\lambda}f(1)$ by $\Psi,$ $f(0)$ by $\Lambda$ and
$(e^{-\lambda\rho}f(\rho))^{\prime}$ by $-e^{-2\lambda\rho},$ we obtain (using
Young’s inequality),

$\displaystyle V^{\prime}_{5}(t)$ $\displaystyle\leq$
$\displaystyle\frac{1}{\tau}\left(\frac{\Gamma}{2\varepsilon_{2}}-\Lambda\right)\left\|u_{x}\right\|^{2}+\frac{1}{2\tau}\left(\varepsilon_{2}+\frac{\tau}{\varepsilon_{3}}\right)V_{4}(t)+\frac{\varepsilon_{3}}{2}\Phi\left\|u_{tx}\right\|^{2}+\frac{\Psi}{\tau}\left\langle
z(.,1,.),u_{x}\right\rangle.$

Finally, the derivative of $V_{6}$ is

$\displaystyle V^{\prime}_{6}(t)$ $\displaystyle=$
$\displaystyle\left\|u_{t}\right\|^{2}-\alpha\left\langle
z(.,1,.),u_{x}\right\rangle-\beta\left\langle
u_{x},u_{xt}\right\rangle+\gamma\left\langle u,\theta_{x}\right\rangle.$

To conclude, it suffices to sum up $N_{1}V^{\prime}_{1}(t),$ $\alpha
N_{2}V^{\prime}_{2}(t),$ $N_{3}V^{\prime}_{3}(t),$ $N_{4}V^{\prime}_{4}(t),$
$N_{5}V^{\prime}_{5}(t),$ and $N_{6}V^{\prime}_{6}(t).$ ∎

In view of Lemma 3.2, for the assumption (A2) to be satisfied, it suffices
that

(3.2) $\displaystyle-
N_{4}\frac{e^{-2\lambda}}{\tau}+N_{1}\frac{\alpha\varepsilon_{1}}{2}$
$\displaystyle=$ $\displaystyle 0,$ (3.3) $\displaystyle
n_{1}:=-2\frac{\lambda}{\tau}N_{4}+\frac{N_{5}}{2\tau}\left(\varepsilon_{2}+\frac{\tau}{\varepsilon_{3}}\right)$
$\displaystyle<$ $\displaystyle 0,$ (3.4) $\displaystyle
n_{2}:=\frac{N_{4}}{\tau}+\frac{N_{5}}{\tau}\left(\frac{\Gamma}{2\varepsilon_{2}}-\Lambda\right)+N_{5}\frac{\Psi\gamma\varepsilon_{4}c_{p}}{2\alpha\tau}$
$\displaystyle<$ $\displaystyle 0,$ (3.5) $\displaystyle
n_{3}:=N_{1}\left(\frac{\alpha}{2\varepsilon_{1}}-\beta\right)+N_{5}\left(\frac{\varepsilon_{3}}{2}\Phi+\frac{\Psi
c_{p}}{\alpha\tau}\right)$ $\displaystyle<$ $\displaystyle 0,$ (3.6)
$\displaystyle
n_{4}:=-N_{1}\kappa+N_{5}\frac{\Psi\gamma}{2\alpha\tau\varepsilon_{4}}$
$\displaystyle<$ $\displaystyle 0.$

The first condition (3.2) is equivalent to

$N_{4}=aN_{1}$

with $a:=\frac{1}{2}\alpha\varepsilon_{1}\tau e^{2\lambda}.$

The second condition (3.3) means that there exists $0<k<1$ such that

$N_{5}=bN_{4}=abN_{1}$

with $b:=\frac{4\lambda k}{\varepsilon_{2}+\frac{\tau}{\varepsilon_{3}}}.$

Note that, we have then

$N_{6}=\frac{\Psi}{\alpha\tau}N_{5}=\frac{ab\Psi}{\alpha\tau}N_{1}\;\;\;\text{and}\;\;\;N_{2}=\frac{\beta}{\alpha}N_{6}=\frac{ab\Psi\beta}{\alpha^{2}\tau}N_{1}.$

Replacing $N_{5}$ by $abN_{1}$ and $a$ by
$\frac{1}{2}\alpha\varepsilon_{1}\tau e^{2\lambda}$ in (3.5), then multiplying
the inequality by $\frac{\alpha}{\varepsilon_{1}}$, we obtain

(3.7) $\frac{1}{2}b\Psi\alpha e^{2\lambda}c_{p}+\frac{1}{4}\tau
b\Phi\varepsilon_{3}\alpha^{2}e^{2\lambda}<\frac{\alpha}{\varepsilon_{1}}\left(\beta-\frac{\alpha}{2\varepsilon_{1}}\right).$

We take $\varepsilon_{1}=\frac{\alpha}{\beta},$ then (3.7) turns into

(3.8) $b\Psi\alpha e^{2\lambda}c_{p}+\frac{1}{2}\tau
b\Phi\varepsilon_{3}\alpha^{2}e^{2\lambda}<\beta^{2}.$

Return back to (3.6), replacing $N_{5}$ by $abN_{1}$ to obtain

(3.9) $\frac{ab\Psi\gamma}{2\alpha\tau\varepsilon_{4}}<\kappa.$

Also inequality (3.4) becomes

(3.10)
$\frac{b\Psi\gamma\varepsilon_{4}c_{p}}{2\alpha}<\left(\left(\Lambda-\frac{\Gamma}{2\varepsilon_{2}}\right)b-1\right).$

Already, it is necessary that $\Lambda-\frac{\Gamma}{2\varepsilon_{2}}>0,$
that is $\Lambda>\frac{\Gamma}{2\varepsilon_{2}},$ and
$\left(\Lambda-\frac{\Gamma}{2\varepsilon_{2}}\right)b-1>0,$ that is,

(3.11) $b=\frac{A}{\Lambda-\frac{\Gamma}{2\varepsilon_{2}}},\;\;\;A>1,$

hence, (3.10) turns into

(3.12) $\frac{b\Psi\gamma\varepsilon_{4}c_{p}}{2\alpha}<\left(A-1\right).$

Combining (3.9) and (3.12) to obtain

(3.13)
$\frac{ab\Psi\gamma}{2\alpha\tau\kappa}<\varepsilon_{4}<\frac{2\alpha}{\gamma
b\Psi c_{p}}(A-1).$

Replacing $a$ by $\frac{1}{2\beta}\alpha^{2}\tau e^{2\lambda}$ in (3.13) to
get

(3.14) $\frac{\alpha\gamma
b\Psi}{4\beta\kappa}e^{2\lambda}<\varepsilon_{4}<\frac{2\alpha}{\gamma b\Psi
c_{p}}(A-1).$

Now, going back with more detail on assumption (3.11). To do this, replacing
$b$ by $\frac{4\lambda k}{\varepsilon_{2}+\frac{\tau}{\varepsilon_{3}}},$ we
obtain

(3.15) $\frac{A}{\Lambda-\frac{\Gamma}{2\varepsilon_{2}}}=\frac{4\lambda
k}{\varepsilon_{2}+\frac{\tau}{\varepsilon_{3}}},$

or equivalently,

(3.16) $\frac{A}{4\lambda
k}\varepsilon_{2}^{2}-\left(\Lambda-\frac{A}{4\lambda
k}\frac{\tau}{\varepsilon_{3}}\right)\varepsilon_{2}+\frac{\Gamma}{2}=0$

the discriminant of such equation in $\varepsilon_{2}$ is

(3.17) $\Delta:=\left(\Lambda-\frac{A}{4\lambda
k}\frac{\tau}{\varepsilon_{3}}\right)^{2}-\frac{A\Gamma}{2\lambda k}$

which must be at least zero. In the sequel, we choose it zero. On the other
hand $\varepsilon_{2}$ is positive, then

(3.18) $\Lambda-\frac{A}{4\lambda
k}\frac{\tau}{\varepsilon_{3}}=\sqrt{\frac{A\Gamma}{2\lambda k}}$

or equivalently

(3.19) $\frac{A}{4\lambda
k}\frac{\tau}{\varepsilon_{3}}=\Lambda-\sqrt{\frac{A\Gamma}{2\lambda k}}.$

It is obvious that the left hand side of the last equation is positive, that
is

(3.20) $\frac{A}{k}<2\lambda\frac{\Lambda^{2}}{\Gamma}.$

Moreover, since $A>1$ and $0<k<1$ we have

(3.21) $1<\frac{A}{k}<2\lambda\frac{\Lambda^{2}}{\Gamma}.$

Finally, note that

(3.22) $\varepsilon_{2}=\sqrt{2\lambda\Gamma\frac{k}{A}}.$

Second part: it concerns the equivalence between $V(t)$ and $\tilde{V}(t).$
Let $\varepsilon_{5}>0$ and $\varepsilon_{6}>0,$ we have

$\left|N_{5}V_{5}\right|\leq\frac{N_{5}}{2\varepsilon_{5}}V_{4}+\frac{N_{5}\Phi\varepsilon_{5}}{2}\left\|u_{x}\right\|^{2}$

and

$\left|N_{6}V_{6}\right|\leq\frac{N_{6}\varepsilon_{6}}{2}c_{p}\left\|u_{x}\right\|^{2}+\frac{N_{6}}{2\varepsilon_{6}}\left\|u_{t}\right\|^{2}.$

For $V(t)$ to be equivalent to $\tilde{V}(t)$ it is sufficient that

(3.23)
$\displaystyle\frac{N_{6}}{2\varepsilon_{6}}<\frac{N_{1}}{2},\;\;\;\;\;\;\frac{N_{5}}{2\varepsilon_{5}}<N_{4},$
(3.24)
$\displaystyle\frac{N_{6}\varepsilon_{6}}{2}c_{p}+\frac{N_{5}\Phi\varepsilon_{5}}{2}<\frac{N_{2}\alpha}{2}.$

Using $N_{6}=\frac{ab\Psi}{\alpha\tau}N_{5}$ and $N_{5}=abN_{1}$ in(3.23) we
get

$\varepsilon_{6}>\frac{ab\Psi}{\tau\alpha},\;\;\;\;\;\;\varepsilon_{5}>\frac{b}{2}.$

We choose

$\varepsilon_{6}=2\frac{ab\Psi}{\tau\alpha},\;\;\;\;\;\;\varepsilon_{5}=b.$

Using again $N_{6}=\frac{\Psi}{\alpha\tau}N_{5}$ and
$N_{2}=\frac{\beta\Psi}{\alpha^{2}\tau}N_{5},$ inequality (3.24) becomes

(3.25) $\frac{b\Psi^{2}}{\beta\tau}c_{p}+\Phi b<\frac{\beta\Psi}{\alpha\tau}.$

Third part: It is enough to examine the equations (3.21), (3.8), (3.14) and
(3.25).

We take $h:=\frac{\Psi}{\Gamma},$ then
$\Lambda=\Psi+\Gamma=(1+h)\Gamma=(1+h)\frac{1-e^{-2\lambda}}{2\lambda}$.

First step. We begin by assumption (3.21) which can be translated into

(3.26) $1<\frac{A}{k}<(1-e^{-2\lambda})(1+h)^{2}$

We choose $h:=e^{-2\lambda}.$ Then

$(1-e^{-2\lambda})(1+h)^{2}=(1-h)(1+h)^{2}=1+h-h^{2}-h^{3}>1$

for $\lambda$ large enough. We choose $A=1+h-h^{2}-2h^{3}-4h^{4}$ and
$k=1-h^{4}.$ We have, for $\lambda$ large enough, $A>1$, $0<k<1$ and (3.26) is
satisfied since

(3.27) $1<\frac{A}{k}=1+h-h^{2}-2h^{3}+o(h^{3})<(1-h)(1+h)^{2}$

Second step. Estimate of $\varepsilon_{2},\;b$ and $\varepsilon_{3}$ according
to $h$ and $\lambda$ for $\lambda$ large enough. We have

$\displaystyle\frac{k}{A}=\frac{1+o(h^{2})}{1+h-h^{2}+o(h^{2})}$ (3.28)
$\displaystyle=1-h+2h^{2}+o(h^{2})$

and

$\Gamma=\frac{1}{2\lambda}(1-h)$

then

$\displaystyle 2\lambda\Gamma\frac{k}{A}$ $\displaystyle=$
$\displaystyle(1-h)(1-h+2h^{2}+o(h^{2}))$ $\displaystyle=$ $\displaystyle
1-2h+3h^{2}+o(h^{2}).$

Hence we obtain, using (3.22),

$\displaystyle\varepsilon_{2}$ $\displaystyle=$ $\displaystyle
1-h+\frac{3}{2}h^{2}-\frac{1}{2}h^{2}+o(h^{2})$ $\displaystyle=$
$\displaystyle 1-h+h^{2}+o(h^{2}).$

We evaluate $b$. First,

$\displaystyle\frac{1}{2\varepsilon_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2(1-h+h^{2}+o(h^{2}))}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+h-h^{2}+(h-h^{2})^{2}+o(h^{2})\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(1+h+o(h^{2})\right),$

then

$\displaystyle 1+h-\frac{1}{2\varepsilon_{2}}=\frac{1}{2}(1+h+o(h^{2})),$

hence

$\displaystyle\frac{1}{1+h-\frac{1}{2\varepsilon_{2}}}$ $\displaystyle=$
$\displaystyle 2(1-h+h^{2}+o(h^{2})).$

Finally, from(3.11) and using that $\Lambda=(1+h)\Gamma$ we have

(3.29) $\displaystyle b=\frac{A}{\Gamma(1+h-\frac{1}{2\varepsilon_{2}})}$
$\displaystyle=$ $\displaystyle
4\lambda\frac{(1+h-h^{2}+o(h^{2}))(1-h+h^{2}+o(h^{2}))}{1-h}$ $\displaystyle=$
$\displaystyle 4\lambda(1+h+o(h^{2})).$

Now we evaluate $\varepsilon_{3}:$

First, recall that

(3.30) $\frac{A}{2\lambda
k}=\frac{1}{2\lambda}(1+h-h^{2}-2h^{3}-3h^{4}+o(h^{4})),$

then

$\frac{A\Gamma}{2\lambda
k}=\frac{1}{4\lambda^{2}}(1-2h^{2}-h^{3}-h^{4}+o(h^{4}))$

and

(3.31) $\sqrt{\frac{A\Gamma}{2\lambda
k}}=\frac{1}{2\lambda}(1-h^{2}-\frac{1}{2}h^{3}-h^{4}+o(h^{4}))$

Using (3.19), (3.30), (3.31) and that $\Lambda=\frac{1}{2\lambda}(1-h^{2})$,
we have

(3.32) $\varepsilon_{3}=\frac{\tau}{h^{3}}(1-h+o(h)).$

Third step. Interpretation of Inequality (3.8). First, we need to express
$\Phi$ according to $h$ and $\lambda.$

Since

$\displaystyle f(\rho)$ $\displaystyle=$ $\displaystyle
e^{\lambda\rho}\left(h\Gamma+\int_{\rho}^{1}e^{-2\lambda s}ds\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\lambda}e^{\lambda\rho}\left(e^{-2\lambda}(1-e^{-2\lambda})+(e^{-2\lambda\rho}-e^{-2\lambda})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\lambda}e^{\lambda\rho}\left(e^{-2\lambda\rho}-e^{-4\lambda}\right)$

then,

$\displaystyle f^{2}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{4\lambda^{2}}e^{2\lambda\rho}\left(e^{-4\lambda\rho}-2e^{-4\lambda}e^{-2\lambda\rho}+e^{-8\lambda}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4\lambda^{2}}\left(e^{-2\lambda\rho}-2e^{-4\lambda}+e^{-8\lambda}e^{2\lambda\rho}\right).$

Hence,

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\int_{0}^{1}f^{2}(\rho)d\rho$ $\displaystyle=$
$\displaystyle\frac{1}{4\lambda^{2}}\left(\frac{1-e^{-2\lambda}}{2\lambda}-2e^{-4\lambda}+\frac{1}{2\lambda}\left(e^{-6\lambda}-e^{-8\lambda}\right)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{8\lambda^{3}}(1-h-4\lambda
h^{2}+o(h^{2})).$

Now, inequality (3.8) can be rewritten as:

$2\alpha
c_{p}(1-h^{2}+o(h^{2}))+\frac{\tau^{2}\alpha^{2}}{4\lambda^{2}h^{4}}(1-h+o(h))<\beta^{2}.$

Then, we take

(3.33)
$2\left(\frac{c_{p}}{\alpha\tau^{2}}+\frac{e^{8\lambda}}{8\lambda^{2}}\right)<\left(\frac{\beta}{\alpha\tau}\right)^{2}$

with $\lambda$ large enough.

Fourth step. Condition (3.14) and existence of $\varepsilon_{4}.$ Inequality
(3.14) can be rewritten as:

$\frac{\alpha\gamma}{2\beta\kappa}(1-h^{2}+o(h^{2}))<\varepsilon_{4}<\frac{\alpha}{\gamma
c_{p}}(1-h+o(h)).$

Then, we take

(3.34) $\frac{\gamma^{2}}{2\kappa}c_{p}<\beta(1-h+o(h))$

and $\varepsilon_{4}$ can be taken equal to
$\frac{\alpha}{2}\left(\frac{\gamma}{2\beta\kappa}+\frac{1}{\gamma
c_{p}}(1-h+o(h))\right),$ with $\lambda$ large enough.

Fifth step. Interpretation of assumption (3.25). It can be rewritten as:

$2h\frac{c_{p}}{\tau\beta}(1-h^{2}+o(h^{2}))+\frac{1}{\lambda
h}(1+h+o(h))<\frac{\beta}{\tau\alpha}.$

It suffices to take

(3.35)
$\left(2e^{-2\lambda}\frac{c_{p}}{\tau\beta}+\frac{1}{\lambda}(e^{2\lambda}+1+o(1))\right)<\frac{\beta}{\alpha\tau}.$

with $\lambda$ large enough.

Note that for $\lambda$ large enough, $\beta=\alpha\tau e^{4\lambda}$
satisfies the three conditions (3.33), (3.34) and (3.35). Moreover, there
exists $\beta_{0}>0$ such that every $\beta>\beta_{0}$ satisfies the three
conditions (3.33), (3.34) and (3.35).

For every $\beta>\beta_{0}$ we have

(3.36) $\dot{V}(t)\leq-n_{0}\tilde{V}(t),$

where $n_{0}=min\\{n_{1},n_{2},\frac{1}{c_{p}}n_{3},\frac{1}{c_{p}}n_{4}\\}.$
Recall that $V(t)$, $\tilde{V}(t)$ and $E(t)$ are equivalent then, there
exists $a_{0}>0,C>0$ such that

$E(t)<Ce^{-a_{0}t}.$

∎

## Comments

We can replace the Neumann conditions for $\theta$

$\theta_{x}(0,t)=\theta_{x}(\ell,t)=0$

by the Dirichlet conditions

$\theta(0,t)=\theta(\ell,t)=0,$

we then obtain the same results.

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spectroscopic classifications from the HECATE catalog (see text). In this
subplot, we also show LINERs together with AGN and “Composite” galaxies (red),
and the passive together with star-forming ones (green). _Bottom panel_ : same
as the other panels, but galaxies are color-coded with a probabilistic
estimate on the presence of an AGN, from photometric and spectroscopic
classifications from the HECATE catalog (see text).

The bottom panel of Fig. 4 and in Fig. 8 do not include the classification of
the galaxies based on optical spectra. Here, we investigate the dependency of
the observed X-ray weakness on the classification of the galaxy based on
optical photometry and spectroscopy, as an independent proxy compared to the
UVOIR selection. However, we note that the UVOIR variability method is
knowingly selecting AGN candidates in galaxies classified as inactive
(Baldassare et al. 2018, 2020). First, we retrieved the galaxy classification
of our sample from the Reference Catalog of galaxy SEDs
(RCSEDv2111111https://dev-rcsed2.voxastro.org; Chilingarian & Zolotukhin 2012;
Chilingarian et al. 2017), between $z=0.01-0.1$. The lower end is chosen to
avoid aperture biases, the higher end is chosen to limit the analysis to the
range in which X-ray non-detections were stacked. A handful of sources which
were either missing in the database or had spectra with poor quality were
excluded. This analysis was limited to 99 galaxies. We show in the top panel
of Fig. 12 the equivalent of the bottom panel of Fig. 4, to which we added
subpanels with histograms and a different color coding. We highlight in green
(squares for detections, arrows for non-detections) the galaxies classified as
star-forming from the BPT narrow lines diagnostics (Baldwin et al. 1981),
whilst in red (diamonds and arrows) those classified as Composite or as AGN.
In addition, we highlight with orange contours the galaxies classified as
star-forming, but for which RCSED reports a significant detection of a broad
$H\alpha$ line.

Furthermore, we also estimate the activity classifications with the updated
version of HECATE catalog (Kyritsis et al., in prep.). The classifications are
based on two different methods. The first one is an advanced data-driven
version of the traditional BPT diagrams, which utilizes a soft clustering
scheme for classifying emission-line galaxies in different activity classes
using simultaneously four emission-line ratios (Stampoulis et al. 2019). The
second one is based on the application of the Random Forest machine learning
algorithm on mid/IR (W1-W2, W2-W3; WISE) and optical (g-r; SDSS) colors and
can discriminate galaxies into 5 activity classes (i.e star-forming, AGN,
“Composite”, “LINER”, and “Passive”; Daoutis et al. 2023). Both activity
classification methods are probabilistic, meaning that they provide the
probability of a galaxy to belong in each class, and an example of their
application is presented in the work of Kyritsis et al. (in prep.) for the
selection of all the bona-fide star-forming galaxies which were observed by
the eRASS1 all-sky survey. First, we confirmed that the two methods yielded
similar results from one-another, compatibly with the top panel of Fig. 12.
Then, in the middle panel of Fig. 12 we show the combination of the two above-
mentioned methods from the HECATE catalog: we represent a galaxy in red
(green) if either the emission-lines diagnostics or the Random Forest consider
it as AGN, “LINER” or “Composite” (star-forming or passive). Similarly to the
top panel of the same figure, there is no significant different among the two
sets of classifications. Finally, in the bottom panel of Fig. 12 we color-code
the plot with the probability of a galaxy to host an AGN based on the Random
Forest algorithm. Again, we identify no major bias: x-ray detections are found
at all $P_{AGN}$ and non-detections do not seem to strongly depend on
$P_{AGN}$ either. This tests highlights that there is no significant
difference between the X-ray weakness of galaxies classified as star-forming,
compared to those classified as active.

Furthermore, we have checked the impact of X-ray variability, although it is
expected to yield a scatter in both brighter and fainter directions and not
the latter only. As a matter of fact, we have crossmatched the eRASS:4 low-z
galaxies with the fourth _XMM-Newton_ serendipitous source catalog (Webb et
al. 2020) and the _Chandra_ Source Catalog (Evans et al. 2020). We added a
handful of sources from Messick et al. (2023), which were not included in the
catalogs (namely NSA IDs 156688, 104881, 51928, 67333, 124477). We show in
Fig. 13 the resulting comparison, where the 1:1 (with related 0.5 dex scatter)
is show with a solid (dashed) line. Different energy bands might have been
used across different sources, although consistent bands are used between
eROSITA and other missions for the individual source. Different symbols are
used for _XMM-Newton_ (squares) and _Chandra_ (circles), while different
colors highlight eROSITA detected (green) and non-detected (gray) sources.
Detections with _XMM-Newton_ and _Chandra_ are highlighted with green contours
for visualization purpose. All the sources detected by eROSITA and either
_XMM-Newton_ or _Chandra_ (with observations taken between 2015 and 2022) show
compatible fluxes across the different epochs. All eROSITA upper limits (apart
from one) are brighter than the detection with _XMM-Newton_ or _Chandra_ ,
therefore they are compatible with the 1:1 and were not supposed to be
detected by eROSITA. Upper limits in both missions (gray data points with
black contours) are, by definition, compatible with the 1:1. Therefore, we
confirm that the impact of variability or a low duty cycle in these galaxies
has to be minimal.

Figure 13: Comparison between eROSITA eRASS:4 X-ray fluxes and archival _XMM-
Newton_ (Webb et al. 2020) and _Chandra_ data (Evans et al. 2020; Messick et
al. 2023) of the same galaxy.

In order to quantify how the X-ray weakness compares with the radio properties
of the MBHs, we cross-matched our low-z sample (Fig. 8) with radio
archives121212Link to radio catalog, the Rapid ASKAP Continuum Survey
(McConnell et al. 2020; Hale et al. 2021) and the second data release of the
LOFAR Two-metre Sky Survey (Shimwell et al. 2022). We then convert the
observed radio fluxes to $5\,$GHz luminosities assuming both a spectrum with
radio spectral index -1 (top panel of Fig. 14) and a flat spectrum (bottom
panel of Fig. 14). We estimated the black hole masses from the stellar masses
of the galaxies (Reines & Volonteri 2015) and plotted our sources in the
fundamental plane of black hole accretion (Merloni et al. 2003). We show this
in Fig. 14. To be conservative, we draw the main conclusions from the top
panel as it shows the faintest 5 GHz luminosity from the extrapolations.
Realistically, radio spectra of these sources would be a mixed bag between
slopes of minus one and zero, therefore between the two panels. We note that
both X-ray and radio fluxes are likely contaminated by the galaxy. Therefore
we computed the radio luminosity at $5\,$GHz as predicted by star-formation in
the galaxy (Ranalli et al. 2003). In Fig. 14, we highlight in orange MBHs with
a SFR estimate available from the MPA-JHU catalog (Brinchmann et al. 2004) and
with a radio luminosity greater than that predicted for the galaxy alone
(Ranalli et al. 2003). The sample statistic is now very low, although X-ray
weak $3\sigma$ upper limits remain. This is more evident if a flat radio
spectrum is assumed. Hence, MBHs appear to be X-ray weaker even compared to
the fundamental plane, including the large intrinsic scatter of $\sim
0.88\,$dex of the relation. This is at odds with the interpretation that X-ray
weakness is simply due to the low-luminosity nature of these MBHs.

Figure 14: The fundamental plane of black hole accretion (Merloni et al. 2003)
is shown with the solid line, with its $\sim 0.88\,$dex scatter. We show all
the sources in our low-z sample which can be matched to an archival radio
observation. We highlight in orange galaxies with a SFR estimate and a radio
luminosity brighter than that predicted by SFR. The _top panel_ shows radio
fluxes extrapolated assuming a radio spectral index of -1, the _bottom_ using
a flat slope.

##  Appendix B Tables

We present eRASS1 and eRASS:4 results on all MBH candidates in Table
LABEL:tab:results and more details on the detected sources in Table 2.

Table 1: Input MBH candidates from optical/IR/UV variability and related eROSITA information from aperture photometry and spectroscopy. Input | eRASS1 | eRASS:4
---|---|---
RA | Dec | $z$ | $M*$ |  Ref.a |  $P_{B}$b |  $L_{0.2-2.0\,keV}$c |  $P_{B}$b |  $L_{0.2-2.0\,keV}$c
48.25895 | -0.686379 | 0.131 | 9.8 | Ba18 | 2.62e-32 | $\mathbf{43.41_{43.29}^{43.53}\,(_{43.11}^{43.67})}$ | 6.19e-66 | $\mathbf{43.19_{43.11}^{43.28}\,(_{43.0}^{43.39})}$
49.42965 | 0.326904 | 0.069 | 9.7 | Ba18 | 1.51e-13 | $\mathbf{42.32_{42.15}^{42.49}\,(_{41.91}^{42.7})}$ | 3.23e-28 | $\mathbf{42.19_{42.08}^{42.3}\,(_{41.94}^{42.46})}$
47.61596 | -0.830791 | 0.08 | 9.74 | Ba18 | 0 | $\mathbf{44.2_{44.18}^{44.22}\,(_{44.15}^{44.25})}$ | 0 | $\mathbf{44.12_{44.11}^{44.14}\,(_{44.09}^{44.15})}$
55.87608 | -7.58542 | 0.036 | 9.84 | Ba20 | 1 | $<39.78\,(40.85)$ | 0.94 | $<39.36\,(40.35)$
154.78365 | 19.98218 | 0.039 | 9.88 | Ba20 | 1 | $<39.94\,(41.06)$ | 0.44 | $<39.85\,(40.8)$
57.34663 | -11.99095 | 0.032 | 9.36 | Ba20 | 0 | $\mathbf{44.02_{44.01}^{44.02}\,(_{44.0}^{44.04})}$ | 0 | $\mathbf{43.86_{43.85}^{43.86}\,(_{43.85}^{43.87})}$
123.4507 | 37.83548 | 0.029 | 9.07 | Ba20 | 1 | $<39.88\,(40.98)$ | 0.23 | $<40.1\,(40.87)$
184.55599 | 20.07689 | 0.046 | 9.33 | Ba20 | 0.34 | $<40.76\,(41.44)$ | 0.61 | $<39.99\,(40.9)$
140.52167 | 35.425 | 0.025 | 9.82 | Ba20 | 0.05 | $<40.82\,(41.28)$ | 0.2 | $<39.53\,(40.39)$
115.81143 | 19.49506 | 0.046 | 9.81 | Ba20 | 0.08 | $<41.35\,(41.88)$ | 0.12 | $<40.76\,(41.36)$
186.69977 | 7.67155 | 0.002 | 8.24 | Ba20 | 1 | $<37.21\,(38.33)$ | 0.83 | $<36.96\,(37.93)$
120.11895 | 15.4531 | 0.015 | 9.04 | Ba20 | 0.04 | $<40.48\,(40.88)$ | 0.05 | $<40.16\,(40.43)$
132.63486 | 29.09562 | 0.049 | 9.66 | Ba20 | 0.24 | $<40.96\,(41.73)$ | 0.11 | $<40.95\,(41.4)$
191.64688 | 2.36911 | 0.048 | 9.87 | Ba20 | 0 | $\mathbf{43.95_{43.93}^{43.97}\,(_{43.91}^{43.99})}$ | 0 | $\mathbf{43.91_{43.9}^{43.92}\,(_{43.89}^{43.93})}$
132.75134 | 28.76201 | 0.034 | 9.91 | Ba20 | 1 | $<39.93\,(41.04)$ | 1 | $<39.51\,(40.52)$
198.98767 | 7.28419 | 0.044 | 9.5 | Ba20 | 1 | $<39.93\,(40.96)$ | 0.81 | $<39.76\,(40.7)$
126.87772 | 19.36663 | 0.036 | 9.76 | Ba20 | 0.3 | $<40.66\,(41.46)$ | 0.13 | $<40.75\,(41.17)$
124.54954 | 27.7192 | 0.038 | 9.59 | Ba20 | 1 | $<39.96\,(41.09)$ | 0.15 | $<40.28\,(40.99)$
145.72186 | 9.49338 | 0.011 | 8.85 | Ba20 | 0.32 | $<39.55\,(40.31)$ | 0.05 | $<39.54\,(40.09)$
148.93346 | 35.61131 | 0.051 | 9.82 | Ba20 | 1 | $<40.12\,(41.26)$ | 0.75 | $<40.0\,(40.92)$
204.26228 | 18.17054 | 0.026 | 8.45 | Ba20 | 1 | $--$ | 1 | $--$
119.84415 | 9.61993 | 0.009 | 9.75 | Ba20 | 1 | $<38.66\,(39.74)$ | 0.4 | $<39.01\,(39.69)$
202.4247 | 10.61023 | 0.027 | 8.77 | Ba20 | 1 | $<39.47\,(40.51)$ | 1 | $<39.02\,(40.03)$
183.06861 | 17.24661 | 0.028 | 8.59 | Ba20 | 0.43 | $<40.21\,(41.08)$ | 0.61 | $<39.42\,(40.4)$
210.45343 | 9.22321 | 0.021 | 9.7 | Ba20 | 1 | $<39.15\,(40.25)$ | 0.76 | $<39.06\,(39.97)$
139.06332 | 17.44624 | 0.031 | 9.48 | Ba20 | 1 | $<39.8\,(40.98)$ | 0.13 | $<40.3\,(40.96)$
144.11431 | 23.31915 | 0.028 | 9.41 | Ba20 | 1 | $<39.78\,(40.86)$ | 0.74 | $<39.56\,(40.44)$
220.05293 | 2.79542 | 0.03 | 9.49 | Ba20 | 1.93e-05 | $\mathbf{41.22_{41.01}^{41.39}\,(_{40.61}^{41.59})}$ | 5.23e-09 | $\mathbf{41.05_{40.91}^{41.19}\,(_{40.67}^{41.38})}$
217.73633 | 6.08396 | 0.029 | 9.11 | Ba20 | 0.63 | $<40.0\,(40.81)$ | 0.88 | $<39.37\,(40.31)$
221.62021 | 3.23674 | 0.026 | 9.49 | Ba20 | 1 | $<39.51\,(40.42)$ | 0.87 | $<39.29\,(40.16)$
189.25945 | 6.92527 | 0.005 | 8.7 | Ba20 | 0.6 | $<38.47\,(39.46)$ | 0.77 | $<38.0\,(38.93)$
125.11033 | 22.3844 | 0.026 | 9.57 | Ba20 | 1 | $<39.83\,(40.88)$ | 0.27 | $<40.23\,(40.83)$
184.1644 | 14.26028 | 0.023 | 8.4 | Ba20 | 1 | $<39.41\,(40.61)$ | 3.82e-03 | $<40.59\,(40.87)$
189.60913 | 6.04994 | 0.049 | 9.8 | Ba20 | 1 | $<40.47\,(41.35)$ | 0.58 | $<40.11\,(40.95)$
180.92748 | 14.37081 | 0.043 | 9.7 | Ba20 | 0.39 | $<40.54\,(41.46)$ | 0.84 | $<40.0\,(40.92)$
146.08088 | 9.9848 | 0.01 | 9.41 | Ba20 | 0.28 | $<39.4\,(40.34)$ | 0.76 | $<38.6\,(39.5)$
192.19534 | 26.74864 | 0.045 | 8.27 | Ba20 | 1 | $<39.95\,(41.05)$ | 0.88 | $<39.62\,(40.64)$
159.31828 | 38.10653 | 0.051 | 9.65 | Ba20 | 1 | $<40.16\,(41.29)$ | 0.45 | $<40.27\,(41.11)$
187.72858 | 10.8227 | 0.049 | 9.84 | Ba20 | 1 | $<40.0\,(41.06)$ | 0.87 | $<39.92\,(40.8)$
179.26696 | 22.31264 | 0.023 | 9.03 | Ba20 | 1 | $<39.58\,(40.61)$ | 1 | $<38.92\,(40.02)$
68.17618 | -4.38251 | 0.015 | 9.43 | Ba20 | 1 | $<39.27\,(40.24)$ | 0.36 | $<39.36\,(40.05)$
184.28861 | 12.45432 | 0.007 | 9.43 | Ba20 | 0.4 | $<39.48\,(40.01)$ | 4.27e-10 | $\mathbf{39.86_{39.74}^{39.96}\,(_{39.54}^{40.1})}$
155.09803 | 21.34426 | 0.04 | 9.62 | Ba20 | 0.26 | $<40.78\,(41.39)$ | 0.15 | $<40.68\,(41.17)$
126.84973 | 23.18012 | 0.018 | 7.66 | Ba20 | 1 | $<39.45\,(40.5)$ | 0.71 | $<39.11\,(40.07)$
151.03704 | 4.27028 | 0.054 | 9.98 | Ba20 | 0.04 | $<41.68\,(42.01)$ | 0.11 | $<40.99\,(41.37)$
144.8969 | 6.41444 | 0.025 | 9.58 | Ba20 | 0.03 | $<40.95\,(41.36)$ | 7.50e-03 | $<40.65\,(41.02)$
160.27304 | 25.13649 | 0.054 | 9.82 | Ba20 | 0.35 | $<40.96\,(41.74)$ | 0.06 | $<41.07\,(41.47)$
189.54408 | 4.7583 | 0.049 | 9.37 | Ba20 | 0.56 | $<40.47\,(41.39)$ | 0.02 | $<40.92\,(41.37)$
138.56138 | 17.11853 | 0.028 | 9.59 | Ba20 | 0.23 | $<40.51\,(41.12)$ | 0.64 | $<39.67\,(40.55)$
146.04343 | 34.04196 | 0.043 | 9.4 | Ba20 | 1 | $<40.0\,(41.1)$ | 0.81 | $<39.73\,(40.73)$
182.67692 | 13.31345 | 0.023 | 9.27 | Ba20 | 0.38 | $<39.89\,(40.75)$ | 0.72 | $<39.75\,(40.71)$
185.39202 | 4.77953 | 0.007 | 9.67 | Ba20 | 0.46 | $<38.41\,(39.44)$ | 0.72 | $<37.87\,(38.84)$
120.75392 | 8.64568 | 0.034 | 9.83 | Ba20 | 1 | $<39.89\,(40.96)$ | 1 | $<39.36\,(40.46)$
117.19945 | 28.23974 | 0.028 | 9.95 | Ba20 | 1 | $<39.81\,(40.89)$ | 1 | $<39.3\,(40.26)$
148.56008 | 10.47307 | 0.04 | 9.82 | Ba20 | 1 | $<39.97\,(41.0)$ | 1 | $<39.49\,(40.58)$
52.479801 | -28.73320 | 0.04 | 8.6 | B22 | 1 | $<39.54\,(40.63)$ | 0.37 | $<39.59\,(40.46)$
53.036098 | -27.52000 | 1.3 | 9.8 | B22 | 1 | $--$ | 1 | $--$
9.2741003 | -44.66830 | 0.77 | 8.0 | B22 | 0.01 | $<44.24\,(44.67)$ | 1.93e-05 | $\mathbf{43.88_{43.63}^{44.13}\,(_{43.02}^{44.54})}$
52.871101 | -27.39349 | 0.346 | 9.4 | B22 | 0.17 | $<42.36\,(43.07)$ | 1 | $--$
8.8598003 | -44.63510 | 1.15 | 9.4 | B22 | 1 | $<43.19\,(44.51)$ | 0.54 | $<43.16\,(44.09)$
52.512298 | -27.54680 | 0.527 | 10.0 | B22 | 1.66e-05 | $\mathbf{43.9_{43.63}^{44.11}\,(_{43.2}^{44.35})}$ | 8.07e-26 | $\mathbf{44.05_{43.93}^{44.17}\,(_{43.77}^{44.32})}$
53.168899 | -28.60619 | 0.322 | 9.9 | B22 | 0.18 | $<42.31\,(43.16)$ | 0.17 | $<42.36\,(42.91)$
9.079099 | -44.21459 | 0.25 | 8.9 | B22 | 0.42 | $<42.16\,(43.06)$ | 0.84 | $<41.41\,(42.45)$
8.4623 | -43.9351 | 0.85 | 9.9 | B22 | 0.05 | $<44.18\,(44.76)$ | 7.04e-03 | $<44.05\,(44.48)$
53.1301 | -27.6189 | 0.123 | 9.4 | B22 | 1 | $<40.68\,(41.65)$ | 1 | $<40.25\,(41.21)$
52.180198 | -28.8197 | 3.354 | 7.5 | B22 | 1 | $<43.64\,(44.9)$ | 0.43 | $<44.15\,(45.21)$
53.03739 | -28.749099 | 0.28 | 8.4 | B22 | 1 | $<41.51\,(42.52)$ | 0.22 | $<41.73\,(42.48)$
8.4967 | -43.444499 | 0.80 | 9.9 | B22 | 1 | $<42.75\,(43.95)$ | 0.21 | $<43.13\,(44.05)$
9.263999 | -44.610099 | 0.25 | 8.9 | B22 | 0.43 | $<42.06\,(43.03)$ | 0.86 | $<41.37\,(42.37)$
8.6218 | -43.3488 | 0.57 | 9.8 | B22 | 1.99e-03 | $<44.28\,(44.64)$ | 1.47e-08 | $\mathbf{43.94_{43.73}^{44.14}\,(_{43.42}^{44.41})}$
52.18659 | -27.2649 | 0.21 | 9.5 | B22 | 1 | $<41.73\,(42.75)$ | 0.43 | $<41.64\,(42.48)$
52.10599 | -28.895299 | 0.65 | 8.2 | B22 | 9.6e-02 | $<43.42\,(43.96)$ | 3.81e-03 | $<43.58\,(43.91)$
10.3934 | -44.239398 | 0.15 | 7.2 | B22 | 0.35 | $<41.64\,(42.68)$ | 0.53 | $<41.13\,(42.0)$
51.79159 | -28.720199 | 0.39 | 9.8 | B22 | 1 | $<41.76\,(42.84)$ | 0.75 | $<42.0\,(42.95)$
52.25859 | -28.0694 | 0.1528 | 8.9 | B22 | 1 | $<40.85\,(41.87)$ | 0.79 | $<40.63\,(41.65)$
8.7887 | -44.539398 | 0.16 | 9.3 | B22 | 0.4 | $<41.88\,(42.65)$ | 0.50 | $<41.49\,(42.25)$
9.054499 | -43.0345 | 0.19 | 8.8 | B22 | 0.02 | $<42.82\,(43.25)$ | 0.01 | $<42.41\,(42.74)$
52.472599 | -28.883699 | 2.0135 | 7.5 | B22 | 0.58 | $<43.61\,(44.93)$ | 0.22 | $<43.7\,(44.68)$
53.326499 | -28.563499 | 0.16 | 9.0 | B22 | 0.49 | $<41.35\,(42.28)$ | 0.93 | $<40.7\,(41.63)$
52.5881 | -27.4328 | 0.206 | 9.7 | B22 | 0.04 | $<42.36\,(42.76)$ | 0.12 | $<41.41\,(42.15)$
52.1963 | -27.264999 | 2.98 | 7.3 | B22 | 8.45e-03 | $<45.6\,(46.0)$ | 0.01 | $<45.22\,(45.61)$
10.315899 | -43.223701 | 0.65 | 9.7 | B22 | 1 | $<42.61\,(43.79)$ | 0.57 | $<42.4\,(43.48)$
10.2382 | -44.521598 | 0.32 | 8.9 | B22 | 0.28 | $<43.02\,(43.9)$ | 1 | $--$
9.846699 | -44.125801 | 0.85 | 7.4 | B22 | 0.08 | $<44.22\,(44.86)$ | 0.11 | $<43.72\,(44.19)$
52.1265 | -28.419099 | 0.22 | 9.9 | B22 | 0.26 | $<41.5\,(42.49)$ | 0.56 | $<41.1\,(42.04)$
52.18939 | -27.1881 | 1.3 | 9.7 | B22 | 1.08e-03 | $<44.72\,(45.13)$ | 1.58e-03 | $<44.16\,(44.6)$
53.3412 | -27.6804 | 0.148 | 9.6 | B22 | 0.58 | $<41.0\,(42.01)$ | 0.17 | $<41.22\,(41.87)$
52.1851 | -27.2276 | 0.13 | 9.5 | B22 | 1 | $<40.82\,(41.91)$ | 1 | $--$
52.933998 | -28.083499 | 0.45 | 9.7 | B22 | 1 | $<42.01\,(43.03)$ | 0.59 | $<42.08\,(42.97)$
52.4207 | -28.089399 | 0.081 | 7.9 | B22 | 1 | $<40.19\,(41.19)$ | 0.94 | $<39.93\,(40.82)$
10.387399 | -43.4743 | 0.24 | 9.7 | B22 | 1 | $<41.62\,(42.62)$ | 0.06 | $<42.39\,(42.86)$
10.4139 | -43.7233 | 0.6 | 9.7 | B22 | 0.14 | $<43.43\,(44.09)$ | 1.27e-04 | $\mathbf{43.90_{43.61}^{44.45}\,(_{43.07}^{44.45})}$
52.2588 | -28.068799 | 0.153 | 8.9 | B22 | 1 | $<40.93\,(41.77)$ | 0.56 | $<40.71\,(41.69)$
9.0171 | -43.297901 | 0.33 | 9.3 | B22 | 0.03 | $<43.33\,(43.85)$ | 1.37e-03 | $<43.2\,(43.53)$
10.013799 | -44.398601 | 0.24 | 9.4 | B22 | 0.44 | $<41.96\,(42.98)$ | 0.58 | $<41.53\,(42.43)$
52.16749 | -27.786399 | 0.24 | 8.6 | B22 | 1 | $<41.35\,(42.34)$ | 0.50 | $<41.78\,(42.65)$
8.464699 | -44.4332 | 0.16 | 8.7 | B22 | 0.37 | $<41.84\,(42.62)$ | 0.49 | $<41.35\,(42.13)$
52.84619 | -28.5128 | 0.28 | 9.3 | B22 | 1 | $<41.46\,(42.44)$ | 0.78 | $<41.43\,(42.38)$
10.145099 | -44.459499 | 0.8 | 10.0 | B22 | 0.161 | $<44.0\,(44.7)$ | 7.63e-03 | $<44.21\,(44.58)$
8.9757 | -43.5718 | 0.21 | 8.7 | B22 | 0.07 | $<42.55\,(43.08)$ | 7.59e-03 | $<42.63\,(42.97)$
52.4536 | -29.074499 | 0.0713 | 9.7 | B22 | 0.22 | $<40.85\,(41.55)$ | 0.82 | $<40.0\,(40.89)$
190.636092 | -1.350718 | 0.004 | 8.75 | W22Z | 0.6 | $<38.14\,(39.13)$ | 0.43 | $<38.06\,(38.86)$
116.775433 | 37.61536 | 0.036 | 9.54 | W22Z | 1 | $<40.04\,(41.12)$ | 0.43 | $<39.71\,(40.74)$
187.780908 | 0.46318 | 0.021 | 9.51 | W22Z | 0.66 | $<39.67\,(40.62)$ | 0.71 | $<39.15\,(40.16)$
189.182179 | -3.020882 | 0.008 | 8.12 | W22Z | 0.53 | $<38.89\,(39.75)$ | 0.74 | $<38.44\,(39.31)$
154.950492 | 3.521796 | 0.032 | 9.05 | W22Z | 1 | $<39.92\,(40.99)$ | 0.37 | $<40.03\,(40.74)$
206.293083 | -3.022434 | 0.047 | 9.42 | W22Z | 0.01 | $<41.48\,(41.8)$ | 0.08 | $<40.68\,(41.21)$
149.581283 | 1.9923499 | 1.332 | 7.58 | Ki20 | 0.37 | $<43.76\,(44.85)$ | 0.03 | $<44.09\,(44.73)$
150.018203 | 2.2594499 | 0.045 | 7.95 | Ki20 | 1 | $<40.15\,(41.31)$ | 0.66 | $<40.02\,(40.82)$
150.300659 | 2.5982298 | 0.041 | 8.04 | Ki20 | 0.09 | $<41.09\,(41.61)$ | 0.02 | $<41.16\,(41.43)$
150.708801 | 2.6656498 | 0.165 | 8.34 | Ki20 | 1 | $<41.38\,(42.39)$ | 1 | $<40.83\,(41.81)$
150.696624 | 1.9857900 | 0.291 | 8.6 | Ki20 | 1 | $<41.93\,(42.96)$ | 1 | $<41.57\,(42.63)$
150.455932 | 1.7164499 | 1.733 | 8.78 | Ki20 | 1 | $<43.49\,(44.9)$ | 1 | $<43.17\,(44.41)$
150.407669 | 1.8676999 | 0.068 | 8.84 | Ki20 | 1 | $<40.43\,(41.55)$ | 1 | $<39.91\,(41.03)$
150.391540 | 2.5911500 | 0.72 | 8.88 | Ki20 | 1 | $<42.81\,(43.96)$ | 1 | $<42.43\,(43.47)$
149.856933 | 2.6380300 | 0.516 | 8.92 | Ki20 | 0.37 | $<42.96\,(43.94)$ | 0.18 | $<42.97\,(43.67)$
150.496139 | 2.3275001 | 1.661 | 8.95 | Ki20 | 1 | $<43.55\,(44.98)$ | 0.78 | $<43.45\,(44.63)$
150.467742 | 1.7152700 | 1.764 | 8.95 | Ki20 | 1 | $<43.49\,(44.99)$ | 0.64 | $<43.44\,(44.61)$
149.694030 | 2.3343899 | 0.497 | 9.07 | Ki20 | 1 | $<43.22\,(44.09)$ | 1 | $<42.39\,(43.54)$
150.383422 | 1.8542599 | 0.716 | 9.14 | Ki20 | 0.3 | $<43.31\,(44.21)$ | 0.66 | $<42.53\,(43.57)$
149.970794 | 2.7495799 | 1.29 | 9.38 | Ki20 | 1 | $<43.29\,(44.54)$ | 0.75 | $<43.12\,(44.15)$
149.532913 | 1.9588700 | 0.565 | 9.42 | Ki20 | 0.38 | $<42.89\,(43.99)$ | 0.77 | $<42.25\,(43.28)$
150.233535 | 2.6826100 | 1.355 | 9.52 | Ki20 | 1 | $<43.39\,(44.71)$ | 0.7 | $<43.15\,(44.3)$
150.714324 | 2.0632100 | 2.658 | 9.54 | Ki20 | 0.26 | $<44.18\,(45.7)$ | 0.66 | $<43.55\,(45.06)$
149.682632 | 2.4382600 | 0.328 | 9.68 | Ki20 | 1 | $<41.95\,(43.13)$ | 1 | $--$
150.143249 | 2.6062099 | 0.731 | 9.72 | Ki20 | 1 | $<42.79\,(43.91)$ | 0.19 | $<42.9\,(43.72)$
150.578582 | 2.1759700 | 2.045 | 9.73 | Ki20 | 0.35 | $<44.34\,(45.44)$ | 5.11e-03 | $<45.12\,(45.57)$
150.067337 | 2.4851500 | 1.322 | 9.76 | Ki20 | 0.35 | $<43.83\,(44.8)$ | 0.76 | $<43.08\,(44.32)$
150.717987 | 2.4684998 | 3.204 | 9.76 | Ki20 | 1 | $<44.1\,(45.51)$ | 0.7 | $<43.71\,(45.08)$
150.112960 | 2.1648600 | 1.927 | 9.78 | Ki20 | 0.28 | $<44.41\,(45.42)$ | 0.04 | $<44.91\,(45.47)$
149.957504 | 2.6875700 | 1.377 | 9.79 | Ki20 | 1 | $<43.46\,(44.61)$ | 1 | $<42.82\,(44.14)$
150.232666 | 2.5442700 | 1.404 | 9.8 | Ki20 | 1 | $<43.26\,(44.65)$ | 1 | $<42.93\,(44.16)$
149.963470 | 2.2414400 | 2.147 | 9.83 | Ki20 | 1 | $<43.87\,(45.03)$ | 0.43 | $<43.9\,(45.01)$
150.034317 | 2.6411199 | 0.517 | 9.84 | Ki20 | 0.06 | $<43.13\,(44.01)$ | 0.18 | $<43.14\,(43.69)$
150.584503 | 2.0216801 | 1.91 | 9.85 | Ki20 | 1 | $<43.73\,(45.04)$ | 1 | $<43.34\,(44.55)$
150.249649 | 1.9529399 | 1.107 | 9.88 | Ki20 | 1 | $<43.14\,(44.45)$ | 1 | $--$
150.022613 | 2.1401100 | 0.837 | 9.91 | Ki20 | 1 | $<42.95\,(44.19)$ | 0.44 | $<42.94\,(43.93)$
150.327224 | 1.9285999 | 0.525 | 9.91 | Ki20 | 9.60e-03 | $<43.63\,(44.16)$ | 2.50e-03 | $<43.55\,(43.93)$
150.175231 | 2.7417199 | 2.313 | 9.95 | Ki20 | 0.32 | $<44.67\,(45.57)$ | 0.14 | $<44.96\,(45.51)$
150.322402 | 2.0412800 | 2.771 | 9.95 | Ki20 | 1 | $<44.54\,(46.11)$ | 1 | $--$
150.629684 | 2.4763898 | 2.278 | 9.99 | Ki20 | 0.07 | $<45.12\,(45.77)$ | 0.01 | $<45.0\,(45.54)$
150.468978 | 2.3317399 | 0.337 | 10.0 | Ki20 | 1 | $<42.0\,(43.15)$ | 1 | $<41.42\,(42.59)$
153.667574 | 19.413615 | 0.029 | 8.48 | Sh22 | 1 | $<39.63\,(40.86)$ | 0.72 | $<39.63\,(40.53)$
170.81565 | 24.034789 | 0.025 | 9.09 | Sh22 | 1 | $<39.52\,(40.65)$ | 0.04 | $<40.3\,(40.68)$
132.857567 | 39.594941 | 0.041 | 8.98 | Sh22 | 3.09e-03 | $<41.63\,(41.91)$ | 2.11e-06 | $\mathbf{41.38_{41.19}^{41.54}\,(_{40.81}^{41.73})}$
123.110779 | 39.538888 | 0.033 | 9.74 | W22W | 0.26 | $<40.78\,(41.49)$ | 0.1 | $<40.62\,(41.14)$
175.712442 | 20.442126 | 0.019 | 9.61 | W22W | 0.33 | $<39.93\,(40.76)$ | 9.75e-03 | $<40.33\,(40.65)$
188.282729 | 10.248929 | 0.053 | 9.61 | W22W | 0.7 | $<40.32\,(41.3)$ | 0.09 | $<41.17\,(41.55)$
212.720667 | 1.369554 | 0.026 | 9.73 | W22W | 0.26 | $<40.07\,(40.92)$ | 0.89 | $<39.18\,(40.12)$
172.611858 | 30.464468 | 0.059 | 9.66 | W22W | 0.37 | $<40.89\,(41.69)$ | 0.02 | $<41.26\,(41.62)$
196.890929 | 14.008647 | 0.055 | 9.72 | W22W | 0.65 | $<40.44\,(41.49)$ | 0.09 | $<41.26\,(41.62)$
137.389817 | 38.710721 | 0.056 | 9.69 | W22W | 1 | $<40.68\,(41.67)$ | 0.16 | $<40.93\,(41.58)$
200.81925 | 21.301658 | 0.022 | 9.61 | W22W | 1 | $<39.33\,(40.43)$ | 0.21 | $<40.04\,(40.44)$
198.352746 | 8.040035 | 0.024 | 9.71 | W22W | 0.01 | $<40.76\,(41.19)$ | 0.05 | $<40.5\,(40.76)$
125.056017 | 16.211627 | 0.044 | 9.56 | W22W | 1 | $<40.12\,(41.06)$ | 1 | $<39.57\,(40.67)$
141.164 | 17.663327 | 0.014 | 9.6 | W22W | 1 | $<39.16\,(40.11)$ | 1 | $<38.81\,(39.74)$
197.119175 | 10.669054 | 0.025 | 9.68 | W22W | 0.12 | $<40.68\,(41.11)$ | 0.64 | $<39.49\,(40.32)$
177.135242 | 12.705296 | 0.015 | 9.52 | W22W | 0.32 | $<39.69\,(40.56)$ | 0.28 | $<39.2\,(39.99)$
194.5911 | 8.541609 | 0.029 | 9.56 | W22W | 1 | $<39.48\,(40.63)$ | 1 | $<39.18\,(40.19)$
193.533008 | 10.006581 | 0.037 | 9.51 | W22W | 0.62 | $<40.21\,(41.08)$ | 0.74 | $<39.55\,(40.54)$
139.321046 | 26.449779 | 0.025 | 9.56 | W22W | 1 | $<39.57\,(40.63)$ | 1 | $<39.21\,(40.16)$
124.961629 | 24.787822 | 0.026 | 9.65 | W22W | 5.51e-03 | $<41.05\,(41.44)$ | 0.17 | $<39.7\,(40.54)$
187.264621 | 29.779443 | 0.081 | 9.71 | W22W | 3.56e-06 | $\mathbf{42.2_{41.95}^{42.41}\,(_{41.54}^{42.67})}$ | 2.85e-26 | $\mathbf{42.4_{42.3}^{42.49}\,(_{42.16}^{42.62})}$
194.833679 | 9.187163 | 0.028 | 9.66 | W22W | 1 | $<39.85\,(40.84)$ | 0.78 | $<39.41\,(40.34)$
189.051083 | 26.756133 | 0.025 | 9.71 | W22W | 1 | $<39.42\,(40.56)$ | 0.54 | $<39.34\,(40.26)$
196.822679 | 13.646658 | 0.027 | 9.66 | W22W | 4.63e-04 | $<41.29\,(41.48)$ | 3.84e-17 | $\mathbf{41.27_{41.16}^{41.37}\,(_{41.01}^{41.49})}$
148.661629 | 40.534554 | 0.067 | 9.52 | W22W | 4.26e-16 | $\mathbf{42.4_{42.25}^{42.53}\,(_{42.04}^{42.7})}$ | 4.69e-197 | $\mathbf{42.97_{42.93}^{43.01}\,(_{42.87}^{43.06})}$
130.368625 | 16.278657 | 0.073 | 9.72 | W22W | 1 | $<40.51\,(41.68)$ | 1 | $<40.06\,(41.19)$
214.864575 | 4.753834 | 0.143 | 9.66 | W22W | 4.65e-27 | $\mathbf{43.26_{43.15}^{43.37}\,(_{42.99}^{43.52})}$ | 2.23e-135 | $\mathbf{43.36_{43.31}^{43.41}\,(_{43.24}^{43.48})}$
199.247375 | 3.888836 | 0.045 | 9.71 | W22W | 0.51 | $<40.34\,(41.22)$ | 0.21 | $<40.38\,(41.06)$
198.475725 | 16.727696 | 0.022 | 9.5 | W22W | 0.52 | $<39.81\,(40.68)$ | 0.61 | $<39.16\,(40.15)$
224.726117 | 1.990925 | 0.03 | 9.49 | W22W | 0.72 | $<39.74\,(40.79)$ | 0.79 | $<39.4\,(40.45)$
172.116967 | 27.902114 | 0.068 | 9.62 | W22W | 1 | $<40.49\,(41.44)$ | 1 | $<40.0\,(40.98)$
126.338879 | 15.289808 | 0.033 | 9.49 | W22W | 1 | $<39.85\,(41.0)$ | 0.8 | $<39.55\,(40.62)$
135.710296 | 14.235639 | 0.051 | 9.48 | W22W | 1 | $<40.36\,(41.44)$ | 0.08 | $<40.94\,(41.45)$
121.752554 | 5.626982 | 0.052 | 9.47 | W22W | 1 | $<40.23\,(41.36)$ | 1 | $<39.74\,(40.75)$
165.316779 | 10.271646 | 0.036 | 9.47 | W22W | 0.27 | $<40.79\,(41.39)$ | 0.05 | $<40.74\,(41.13)$
180.957258 | 29.715598 | 0.01 | 9.46 | W22W | 1 | $<38.85\,(39.94)$ | 0.76 | $<38.48\,(39.35)$
49.858571 | -6.121106 | 0.008 | 9.45 | W22W | 0.59 | $<38.82\,(39.72)$ | 8.69e-04 | $<39.7\,(39.87)$
124.31605 | 31.651983 | 0.045 | 9.72 | W22W | 1 | $<40.18\,(41.31)$ | 0.36 | $<40.22\,(40.98)$
160.860308 | 11.090063 | 0.047 | 9.64 | W22W | 2.18e-197 | $\mathbf{43.15_{43.11}^{43.19}\,(_{43.06}^{43.25})}$ | 0 | $\mathbf{43.16_{43.14}^{43.18}\,(_{43.11}^{43.21})}$
148.741533 | 36.097484 | 0.049 | 9.39 | W22W | 0.39 | $<41.3\,(41.72)$ | 0.49 | $<40.58\,(41.28)$
121.075004 | 15.344019 | 0.039 | 9.73 | W22W | 0.24 | $<41.23\,(41.68)$ | 0.25 | $<40.57\,(41.14)$
166.066846 | 5.27521 | 0.117 | 9.33 | W22W | 1 | $<41.02\,(42.17)$ | 0.14 | $<41.69\,(42.24)$
120.55615 | 22.434016 | 0.03 | 9.28 | W22W | 0.22 | $<40.67\,(41.31)$ | 0.75 | $<39.66\,(40.65)$
179.147133 | 28.490349 | 0.012 | 9.28 | W22W | 1 | $<38.84\,(39.96)$ | 1 | $<38.63\,(39.78)$
192.946992 | 9.859572 | 0.031 | 9.27 | W22W | 0.62 | $<39.85\,(40.88)$ | 0.4 | $<39.74\,(40.63)$
197.315425 | 13.385648 | 0.027 | 9.23 | W22W | 0.64 | $<40.04\,(40.9)$ | 0.98 | $<39.07\,(40.12)$
216.470175 | 6.5375 | 0.024 | 9.21 | W22W | 0.3 | $<39.96\,(40.75)$ | 0.37 | $<39.45\,(40.29)$
176.516967 | 11.581268 | 0.01 | 9.18 | W22W | 1 | $<38.78\,(39.95)$ | 0.84 | $<38.52\,(39.38)$
177.927437 | 6.85848 | 0.102 | 9.18 | W22W | 0.26 | $<41.51\,(42.29)$ | 0.17 | $<41.41\,(41.88)$
157.367592 | 9.833651 | 0.022 | 9.13 | W22W | 0.04 | $<40.8\,(41.19)$ | 0.32 | $<40.07\,(40.65)$
186.326062 | 5.742002 | 0.004 | 9.11 | W22W | 0.02 | $<39.17\,(39.5)$ | 8.18e-03 | $<38.98\,(39.2)$
116.980812 | 34.036707 | 0.016 | 9.1 | W22W | 1 | $<39.34\,(40.46)$ | 1.18e-03 | $<40.4\,(40.68)$
199.418587 | 13.940189 | 0.028 | 9.01 | W22W | 0.34 | $<40.12\,(40.92)$ | 0.58 | $<39.5\,(40.42)$
146.598954 | 34.281131 | 0.075 | 9.01 | W22W | 0.31 | $<41.13\,(41.92)$ | 0.42 | $<40.51\,(41.36)$
137.393212 | 25.222989 | 0.008 | 8.98 | W22W | 1 | $<39.06\,(40.06)$ | 0.75 | $<38.57\,(39.42)$
198.4167 | 16.657694 | 0.022 | 8.98 | W22W | 1 | $<39.42\,(40.3)$ | 0.43 | $<39.02\,(40.03)$
158.025108 | 22.989391 | 0.058 | 8.97 | W22W | 1 | $<40.63\,(41.57)$ | 0.39 | $<40.46\,(41.3)$
198.461625 | 23.254982 | 0.012 | 8.97 | W22W | 0.55 | $<39.31\,(40.1)$ | 0.2 | $<39.23\,(39.76)$
211.625433 | 0.327646 | 0.106 | 8.93 | W22W | 0.53 | $<41.08\,(41.96)$ | 0.94 | $<40.35\,(41.35)$
130.393062 | 2.188855 | 0.029 | 8.93 | W22W | 1 | $<39.9\,(40.97)$ | 0.41 | $<39.54\,(40.46)$
159.934579 | 0.857951 | 0.025 | 8.88 | W22W | 1 | $<39.62\,(40.79)$ | 0.2 | $<40.32\,(40.89)$
142.826833 | 2.781206 | 0.115 | 8.83 | W22W | 0.26 | $<41.68\,(42.49)$ | 0.67 | $<41.12\,(41.94)$
194.222692 | 8.161578 | 0.009 | 8.75 | W22W | 0.64 | $<38.82\,(39.72)$ | 0.89 | $<38.21\,(39.17)$
151.34625 | 19.271905 | 0.013 | 8.73 | W22W | 1 | $<39.31\,(40.53)$ | 0.15 | $<39.57\,(40.2)$
188.150687 | 18.023079 | 0.003 | 8.67 | W22W | 0.45 | $<38.21\,(39.03)$ | 0.39 | $<37.61\,(38.46)$
151.795479 | 12.651689 | 0.009 | 8.47 | W22W | 1 | $<38.76\,(39.83)$ | 0.12 | $<39.37\,(39.85)$
192.982012 | 8.878221 | 0.004 | 8.45 | W22W | 0.34 | $<38.3\,(39.15)$ | 0.38 | $<37.98\,(38.79)$
200.175829 | 20.910312 | 0.009 | 8.44 | W22W | 0.47 | $<38.82\,(39.77)$ | 0.72 | $<38.4\,(39.28)$
199.919987 | 3.409397 | 0.022 | 8.27 | W22W | 1 | $<39.27\,(40.36)$ | 0.45 | $<39.37\,(40.23)$
196.733792 | 14.807446 | 0.003 | 8.1 | W22W | 1 | $<37.65\,(38.71)$ | 1 | $<37.17\,(38.2)$
209.6864 | 12.596371 | 0.024 | 9.2 | S20 | 0.69 | $<39.69\,(40.64)$ | 0.72 | $<39.21\,(40.14)$
40.217499 | -8.474285 | 0.082 | 8.46 | Ha23 | 1 | $<40.41\,(41.49)$ | 0.46 | $<40.67\,(41.42)$
149.321659 | 2.147833 | 0.086 | 9.8 | Was22 | 0.35 | $<41.1\,(42.02)$ | 0.15 | $<41.17\,(41.82)$
148.575538 | 2.741297 | 0.084 | 9.39 | Was22 | 0.07 | $<41.74\,(42.32)$ | 0.15 | $<41.03\,(41.7)$

Sources which were masked out (see Sect. 3) are shown with dashes.

  * a

Reference for input coordinates, redshift and stellar mass $M_{*}$: Ba18
stands for Baldassare et al. (2018); Ba20 for Baldassare et al. (2020); B22
for Burke et al. (2022); W22Z for ZTF-selected sources from Ward et al.
(2022); Ki20 for Kimura et al. (2020); Sh22 for Shin et al. (2022); W22W for
Wise-selected sources from Ward et al. (2022); S20 for Secrest & Satyapal
(2020); Ha23 for Harish et al. (2023); Was22 for Wasleske et al. (2022).

  * b

No-source probability $P_{B}$ (Eq. 1). Sources are considered detected at
$P_{B}<=0.0003$ (and are highlighted in bold).

  * c

Logarithmic X-ray luminosity in the rest-frame 0.2-2.0 keV range, in units of
$\log(\,$erg s${}^{-1})$. For detected sources (in bold), median and 16th,
84th percentile values are shown first, with 1st and 99th in parenthesis. For
non detected sources, 84th and 99th percentile values are shown as $1\sigma$
and $3\sigma$ upper limits, respectively.

Table 2: eRASS:4 detections matched with _XMM-Newton_ , _Chandra_ , _ROSAT_
and _Swift-XRT_. Sources with no previous X-ray detections are highlighted in
bold.

RA | Dec | $L_{0.2-2.0\,keV}$ | XMM | Chandra | ROSAT | Swift | Comments
---|---|---|---|---|---|---|---
48.25895 | -0.686379 | $\mathbf{43.19_{43.11}^{43.28}\,(_{43.0}^{43.39})}$ | – | – | – | – |
49.42965 | 0.326904 | $\mathbf{42.19_{42.08}^{42.3}\,(_{41.94}^{42.46})}$ | – | – | – | – |
47.61596 | -0.830791 | $44.12_{44.11}^{44.14}\,(_{44.09}^{44.15})$ | Y | – | Y | – | AGN
57.34663 | -11.99095 | $43.86_{43.85}^{43.86}\,(_{43.85}^{43.87})$ | Y | – | Y | Y | BL Lac
191.64688 | 2.36911 | $43.91_{43.9}^{43.92}\,(_{43.89}^{43.93})$ | Y | – | Y | Y | AGN
220.05293 | 2.79542 | $41.05_{40.91}^{41.19}\,(_{40.67}^{41.38})$ | Y | Y | Y | – | AGN
184.28861 | 12.45432 | $\mathbf{39.86_{39.74}^{39.96}\,(_{39.54}^{40.1})}$ | – | – | – | – |
9.2741003 | -44.66830 | $43.88_{43.63}^{44.13}\,(_{43.02}^{44.54})$ | Y | – | – | – | XMM serendipitous
52.512298 | -27.54680 | $44.05_{43.93}^{44.17}\,(_{43.77}^{44.32})$ | Y | – | Y | – | AGN
132.857567 | 39.594941 | $41.38_{41.19}^{41.54}\,(_{40.81}^{41.73})$ | – | Y | – | – | AGN
187.264621 | 29.779443 | $42.4_{42.3}^{42.49}\,(_{42.16}^{42.62})$ | – | Y | – | – | AGN
196.822679 | 13.646658 | $\mathbf{41.27_{41.16}^{41.37}\,(_{41.01}^{41.49})}$ | – | – | – | – |
148.661629 | 40.534554 | $42.97_{42.93}^{43.01}\,(_{42.87}^{43.06})$ | Y | Y | Y | – | AGN
214.864575 | 4.753834 | $43.36_{43.31}^{43.41}\,(_{43.24}^{43.48})$ | – | – | – | Y | BL Lac
160.860308 | 11.090063 | $43.16_{43.14}^{43.18}\,(_{43.11}^{43.21})$ | – | – | – | Y | AGN
8.6218 | -43.3488 | $43.94_{43.73}^{44.14}\,(_{43.42}^{44.41})$ | Y | Y | Y | Y |
10.4139 | -43.7233 | $43.90_{43.61}^{44.45}\,(_{43.07}^{44.45})$ | Y | Y | Y | Y |  QSO $z\sim 2$