# On the exponential time-decay for the one-dimensional wave equation with
variable coefficients

Anton Arnold1∗, Sjoerd Geevers2∗, Ilaria Perugia2∗, Dmitry Ponomarev1,3∗ 1
Institute of Analysis and Scientific Computing, Vienna University of
Technology
Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
2 Faculty of Mathematics, University of Vienna
Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
3 St. Petersburg Department of V. A. Steklov Mathematical Institute, RAS,
Fontanka 27, 191023 St. Petersburg, Russia

###### Abstract.

We consider the initial-value problem for the one-dimensional, time-dependent
wave equation with positive, Lipschitz continuous coefficients, which are
constant outside a bounded region. Under the assumption of compact support of
the initial data, we prove that the local energy decays exponentially fast in
time, and provide the explicit constant to which the solution converges for
large times. We give explicit estimates of the rate of this exponential decay
by two different techniques. The first one is based on the definition of a
modified, weighted local energy, with suitably constructed weights. The second
one is based on the integral formulation of the problem and, under a more
restrictive assumption on the variation of the coefficients, allows us to
obtain improved decay rates.

*A. Arnold, S. Geevers, and I. Perugia have been funded by the Austrian Science Fund (FWF) through the project F 65 “Taming Complexity in Partial Differential Systems”. I. Perugia has also been funded by the FWF through the project P 29197-N32. A. Arnold and D. Ponomarev were supported by the bi-national FWF-project I3538-N32

Keywords wave equation with variable coefficients, local energy decay, long-
time asymptotics
Mathematics Subject Classification  35L05, 35L10, 35B40

##  1\. Introduction

The paper is dedicated to the investigation of the time-decay of the solution
of the following initial-value problem

(1)
$\begin{cases}\beta\left(x\right)\partial_{t}^{2}u\left(x,t\right)-\partial_{x}\left(\alpha\left(x\right)\partial_{x}u\left(x,t\right)\right)=0,&x\in\mathbb{R},\>t>0,\\\
u\left(x,0\right)=u_{0}\left(x\right),\hskip
10.00002pt\partial_{t}u\left(x,0\right)=u_{1}\left(x\right),&x\in\mathbb{R},\end{cases}$

where the coefficients $\alpha$, $\beta$ are positive, Lipschitz, and constant
outside a bounded domain, and the initial data $u_{0}$, $u_{1}$ are compactly
supported. Studying such time-decay is important for practical applications
and it shall be used in our follow-up paper [2].

Most relevant works, such as the recent ones [10] and [3], are focussed on
proving the decay in time of the solution derivatives in $L^{2}$-norm taken
over bounded sets. Such local energy decay is usually obtained in spatial
dimensions $d\geq 2$. Another recent paper [4] contains a time-decay result in
the one-dimensional case under certain restrictions on $\alpha$, $\beta$ and
localisation of the initial data $u_{0}$, $u_{1}$.

It is the purpose of the present work to improve such $L^{2}$-estimates of the
solution derivatives by showing that, in case of compactly supported initial
data, the decay of the solution in time is exponential rather than algebraic.
We also prove that, in this case, an auxiliary condition on $\alpha$, $\beta$
used in [4] can be avoided. Moreover, we generalise the result to prove the
decay not only of derivatives but also of the solution function itself.
Namely, we conclude that the solution converges exponentially fast in time to
a constant, which is explicitly computable in terms of $\alpha$, $\beta$ and
$u_{1}$.

Let us denote $\mathbb{R}_{+}:=\left\\{x\in\mathbb{R}:x\geq 0\right\\}$ and
start with a well-posedness result for problem (1).

######  Proposition 1.1.

Let $\alpha\in W^{1,\infty}\left(\mathbb{R}\right)$ and $\beta\in
L^{\infty}\left(\mathbb{R}\right)$ with
$\alpha\left(x\right)\geq\alpha_{\min}>0$ and
$\beta\left(x\right)\geq\beta_{\min}>0$ for $x\in\mathbb{R}$. Furthermore, let
$u_{0}\in H^{1}\left(\mathbb{R}\right)$ and $u_{1}\in
L^{2}\left(\mathbb{R}\right)$. Then, problem (1) has a unique solution $u\in
C^{1}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)\cap
C\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right)$, with the
continuous dependence on the initial data, according to the following estimate

(2)
$\left\|u\left(\cdot,t\right)\right\|_{H^{1}\left(\mathbb{R}\right)}+\left\|\partial_{t}u\left(\cdot,t\right)\right\|_{L^{2}\left(\mathbb{R}\right)}\leq
C\left(\left\|u_{0}\right\|_{H^{1}\left(\mathbb{R}\right)}+\left\|u_{1}\right\|_{L^{2}\left(\mathbb{R}\right)}\right),\hskip
10.00002pt0\leq t\leq T,$

valid for any $T>0$, with some constant $C=C\left(T,\alpha,\beta\right)>0$.

Moreover, if $u_{0}\in H^{2}\left(\mathbb{R}\right)$ and $u_{1}\in
H^{1}\left(\mathbb{R}\right)$, then we have $u\in
C^{2}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)\cap
C^{1}\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right)\cap
C\left(\mathbb{R}_{+},H^{2}\left(\mathbb{R}\right)\right)$.

For the long-time analysis, we shall now make more restrictive assumptions.

######  Assumption 1.2.

Let $\alpha$, $\beta\in W^{1,\infty}\left(\mathbb{R}\right)$ be real-valued
functions such that $\alpha(x)\geq\alpha_{\min}$, $\beta(x)\geq\beta_{\min}$
for $x\in\mathbb{R}$, and $\alpha(x)\equiv\alpha_{0}$,
$\beta(x)\equiv\beta_{0}$ for $x\in\mathbb{R}\backslash\Omega_{in}$, with some
open bounded interval $\Omega_{in}\subset\mathbb{R}$ and constants
$\alpha_{\min}$, $\beta_{\min}$, $\alpha_{0}$, $\beta_{0}>0$.

######  Assumption 1.3.

Suppose that $u_{0}\in H^{1}\left(\mathbb{R}\right)$, $u_{1}\in
L^{2}\left(\mathbb{R}\right)$ are real-valued functions with the supports
$\mathrm{supp}\,u_{0}$, $\mathrm{supp}\,u_{1}\subset\Omega_{0}$ for some open
bounded interval $\Omega_{0}\subset\mathbb{R}$.

Our main results are given by the following two theorems.

######  Theorem 1.4.

Assume that $\alpha$, $\beta$, $\Omega_{in}$ are as in Assumption 1.2, and
$u_{0}$, $u_{1}$, $\Omega_{0}$ are as in Assumption 1.3. Then, for any open
bounded interval $\Omega\subset\mathbb{R}$, the solution of (1) obeys the
decay estimate

(3)
$\left\|u\left(\cdot,t\right)-u_{\infty}\right\|_{H^{1}\left(\Omega\right)}+\left\|\partial_{t}u\left(\cdot,t\right)\right\|_{L^{2}\left(\Omega\right)}\leq
Ce^{-\Lambda t},\hskip 10.00002ptt\geq 0,$

for some constant
$C=C\left(u_{0},u_{1},\alpha,\beta,\left|\Omega_{0}\right|,\left|\Omega_{in}\right|\right){>0}$,
with $\left|\Omega_{0}\right|$, $\left|\Omega_{in}\right|$ denoting the
Lebesgue measure of the sets $\Omega_{0}$, $\Omega_{in}$, respectively, and

(4)
$u_{\infty}:=\frac{1}{2\left(\alpha_{0}\beta_{0}\right)^{1/2}}\int_{\Omega_{0}}u_{1}\left(x\right)\beta\left(x\right)dx,\hskip
10.00002pt\Lambda:=\frac{\gamma_{0}}{\sqrt{2}\,t_{0}}e^{-2\gamma_{0}}\left(1-4\gamma_{0}e^{-4\gamma_{0}}\right)^{1/2},$
(5)
$\gamma_{0}:=\frac{t_{0}}{4}\left\|\frac{\alpha^{\prime}}{\left(\alpha\beta\right)^{1/2}}+\left(\frac{\alpha}{\beta^{3}}\right)^{1/2}\beta^{\prime}\right\|_{L^{\infty}\left(\Omega_{in}\right)},\hskip
10.00002ptt_{0}:=\int_{\Omega_{in}}\left(\frac{\beta\left(x\right)}{\alpha\left(x\right)}\right)^{1/2}dx.$

We remark that the steady state $u_{\infty}$ given in (4) generalises the one
resulting from the d’Alembert formula for constant $\alpha$ and $\beta$, and
for $u_{0}$, $u_{1}$ with compact support:

$u_{\infty}=\frac{1}{2}\sqrt{\frac{\beta_{0}}{\alpha_{0}}}\int_{\mathbb{R}}u_{1}(x)\,dx.$

Note that even though the decay (3) is exponential, the decay rate $\Lambda$
might be be rather small in case of significantly varying coefficients
$\alpha$, $\beta$ or large $\Omega_{in}$. In particular, the estimate for the
decay rate given by the second equation of (4) is exponentially small for
large values of $\gamma_{0}$. Therefore, an alternative estimate is provided
by the following theorem, which employs a different idea and yields a better
decay rate, but in return requires an assumption on the variation of $\alpha$
and $\beta$.

######  Theorem 1.5.

Assume that $\alpha$, $\beta$, $\Omega_{in}$ are as in Assumption 1.2,
$u_{0}$, $u_{1}$, $\Omega_{0}$ are as in Assumption 1.3, and $t_{0}$ is
defined in (5). Moreover, suppose that

(6)
$b_{0}:=\left\|\alpha\right\|_{L^{\infty}\left(\Omega_{in}\right)}^{3/2}\left\|\beta\right\|_{L^{\infty}\left(\Omega_{in}\right)}^{1/2}\left\|\left(\frac{1}{\left(\alpha\beta\right)^{1/2}}\right)^{\prime}\right\|_{L^{2}\left(\Omega_{in}\right)}^{2}<\frac{1}{t_{0}}.$

Then, for the solution of (1), estimate (3) holds true, and the decay rate is
given by

(7) $\Lambda=\frac{1}{2t_{0}}\left|\log\left(b_{0}t_{0}\right)\right|.$

######  Remark 1.6.

Theorems 1.4 and 1.5 are valid also for complex-valued $u_{0}$, $u_{1}$ in
Assumption 1.3. Indeed, due to the linearity of problem (1) and the real-
valuedness of $\alpha$, $\beta$, both theorems could be applied separately to
the real and imaginary parts of the initial data.

Theorems 1.4 and 1.5 are to be compared with the following recent result from
[4].

######  Proposition 1.7.

[4, Thm. 1.2 (for $d=1$)] Assume that $\alpha\left(x\right)\equiv\alpha_{0}$
and that $\beta$ satisfies Assumption 1.2. Moreover, suppose that $\beta$ is
such that

(8)
$\eta:=\left\|\beta\right\|_{L^{\infty}\left(\Omega_{in}\right)}^{1/2}\left\|\left(\frac{1}{\beta^{1/2}}\right)^{\prime}\right\|_{L^{\infty}\left(\Omega_{in}\right)}{\left|\Omega_{in}\right|}<1.$

Then, for any bounded $\Omega\subset\mathbb{R}$, the solution of (1) satisfies

(9)
$\left\|\partial_{x}u\left(\cdot,t\right)\right\|_{L^{2}\left(\Omega\right)}+\left\|\partial_{t}u\left(\cdot,t\right)\right\|_{L^{2}\left(\Omega\right)}\leq\frac{C}{t^{\left(1-\eta\right)/2}},\hskip
10.00002ptt\geq t_{1},$

for some constant $C>0$ and sufficiently large $t_{1}>0$.

Note that, when the initial data are compactly supported, the algebraic decay
in (9) appears to be sub-optimal compared to both Theorem 1.4 and Theorem 1.5,
which show the exponential decay of the solution derivatives. As observed in
[4, Rem. 1.2], the smallness assumption (8) on $\eta$ measuring the relative
perturbation of the coefficients is essential in the proof there, but it is
not needed in our proof of Theorem 1.4.

For the wave equation (1) with _constant_ coefficients and compactly supported
initial data, the solution converges on all bounded domains $\Omega$ to the
constant $u_{\infty}$ in _finite time_. However, for variable coefficients,
convergence in finite time does not hold and exponential decay is the generic
scenario: as an example, consider (1) with $\alpha(x)\equiv\alpha_{0}$ in
$\mathbb{R}$ and $\alpha$ Lipschitz with $\beta(x)=\beta_{1}$ on a closed
subinterval of some open set $\Omega_{in}$, and $\beta(x)=\beta_{0}$ in
$\mathbb{R}\setminus\Omega_{in}$. A localised, travelling wave packet starting
within $\Omega_{in}$ gets partly reflected and partly transmitted. Hence, its
local norm
$\left\|u\left(\cdot,t\right)-u_{\infty}\right\|_{H^{1}\left(\Omega_{in}\right)}$
decays by a constant factor at each of these reflections, giving rise to an
exponential decay. On the other hand, we note that, for non-compactly
supported initial data, the decay of the derivatives of the solution in time
is generally related to the decay of the initial data at infinity. This can be
easily seen already for the constant-coefficient case: the solution furnished
explicitly by the d’Alembert formula decays to some $u_{\infty}$ only
algebraically fast if the assumed decay at infinity of $u_{0}$ or $u_{1}$ is
algebraic.

The following Sections 2, 3, and 4 are devoted to the proof of Proposition
1.1, Theorem 1.4, and Theorem 1.5, respectively. Some conclusions are drawn in
Section 5, and technical lemmas used in the proof of Theorem 1.4 are deferred
to the Appendix.

##  2\. Proof of Proposition 1.1

One standard approach to prove the well-posedness of problem (1) is to
construct the solution $u$, for coefficients $\alpha$ and $\beta$ sufficiently
regular, by a Galerkin approximation (e.g. see [5, Sect. 7.2], or [9, Par.
11.2]). Here, we shall, however, use semigroup theory tools, since they
require lower regularity assumptions on $\alpha$ and $\beta$. Although this
approach is quite standard (e.g. see [8, Par. 2], [11, Par. 3], or [9, Sect.
12.3.2]), we still outline it, adapting to our particular setting.

With the notation

$\phi\left(x,t\right):=\left(u\left(x,t\right),v\left(x,t\right)\right)^{T},\hskip
10.00002pt\hskip 10.00002ptv\left(x,t\right):=\partial_{t}u\left(x,t\right),$
$A=A\left(x\right):=i\left(\begin{array}[]{cc}0&1\\\
\frac{1}{\beta\left(x\right)}\partial_{x}\left(\alpha\left(x\right){\partial_{x}\,\cdot}\,\right)-\frac{1}{\beta\left(x\right)}&0\end{array}\right),\hskip
10.00002pt\hskip 10.00002ptC=C\left(x\right):=\left(\begin{array}[]{cc}0&0\\\
\frac{1}{\beta\left(x\right)}&0\end{array}\right),$

problem (1) is equivalent to the first order linear system

(10)
$\begin{cases}\partial_{t}\phi\left(x,t\right)={\left(-iA\left(x\right)+C\left(x\right)\right)}\phi\left(x,t\right),&x\in\mathbb{R},\hskip
10.00002ptt>0,\\\
\phi\left(x,0\right)=\phi_{0}\left(x\right):=\left(u_{0}\left(x\right),u_{1}\left(x\right)\right)^{T},&x\in\mathbb{R}.\end{cases}$

We split the proof into 2 steps.

### Step 1. Functional setting.

We set
$L^{2}_{\beta}\left(\mathbb{R}\right):=L^{2}\left(\mathbb{R};\beta\left(x\right)dx\right)$.
Due to the assumption $\beta$, $1/\beta\in L^{\infty}\left(\mathbb{R}\right)$,
the space ${L^{2}_{\beta}\left(\mathbb{R}\right)}$ actually consists of the
same functions as $L^{2}\left(\mathbb{R}\right)$. We define the operator
$B^{2}:=-\frac{1}{\beta\left(x\right)}\partial_{x}\left(\alpha\left(x\right){\partial_{x}\,\cdot\,}\right)+\frac{1}{\beta\left(x\right)}$
in ${L^{2}_{\beta}\left(\mathbb{R}\right)}$. Since $\alpha$, $\beta$ are
assumed to be positive and bounded away from zero, the operator $B^{2}$ is
strictly positive. We claim that this operator is self-adjoint with its
(maximal) domain being $\operatorname{Dom}B^{2}=H^{2}\left(\mathbb{R}\right)$.
Indeed, the symmetry of the operator $B^{2}$ is evident, and $w\in
H^{2}\left(\mathbb{R}\right)$ implies $B^{2}w\in
L^{2}_{\beta}\left(\mathbb{R}\right)$. By [12, Lem 2.3], it remains to verify
its surjectivity on ${L^{2}_{\beta}\left(\mathbb{R}\right)}$ or, equivalently,
on $L^{2}\left(\mathbb{R}\right)$. The latter follows from the fact that, for
$\alpha\in W^{1,\infty}\left(\mathbb{R}\right)$, the operator
$\widetilde{B}^{2}:=-\partial_{x}\left(\alpha\left(x\right){\partial_{x}\,\cdot\,}\right)+1$
from $H^{2}\left(\mathbb{R}\right)$ into $L^{2}\left(\mathbb{R}\right)$ is
strictly positive, and hence invertible.

Next, we define in ${L^{2}_{\beta}\left(\mathbb{R}\right)}$ the positive
square root $B:=\sqrt{B^{2}}$, whose (maximal) domain is
$\operatorname{Dom}B=H^{1}\left(\mathbb{R}\right)$. In fact, for any $w\in
H^{1}\left(\mathbb{R}\right)$, we have

$\displaystyle\left\|Bw\right\|_{{L^{2}_{\beta}\left(\mathbb{R}\right)}}^{2}$
$\displaystyle=\left\langle
Bw,Bw\right\rangle_{{L^{2}_{\beta}\left(\mathbb{R}\right)}}=\int_{\mathbb{R}}\left(B^{2}w\right)\left(x\right)w\left(x\right)\beta\left(x\right)dx$
$\displaystyle=\int_{\mathbb{R}}\left[\alpha\left(x\right)\left(w^{\prime}\left(x\right)\right)^{2}+w^{2}\left(x\right)\right]dx,$

which, due to positivity and boundedness of $\alpha$, shows the equivalence
between $\left\|B\,\cdot\,\right\|_{{L^{2}_{\beta}\left(\mathbb{R}\right)}}$
and $\left\|\,\cdot\,\right\|_{H^{1}\left(\mathbb{R}\right)}$.

With this preparation, we can introduce the Hilbert space
$X:=H^{1}\left(\mathbb{R}\right)\times L^{2}\left(\mathbb{R}\right)$ equipped
with the inner product $\langle\phi,\widetilde{\phi}\rangle_{X}:=\langle
Bu,B\widetilde{u}\rangle_{L^{2}_{\beta}\left(\mathbb{R}\right)}+\langle
v,\widetilde{v}\rangle_{L^{2}_{\beta}\left(\mathbb{R}\right)}$. Note that we
have $\phi_{0}\in X$.

### Step 2. Evolution semigroup.

Observe that the matrix operator $A$ is symmetric in $X$, since for any
$\phi$, $\widetilde{\phi}\in H^{2}\left(\mathbb{R}\right)\times
H^{1}\left(\mathbb{R}\right)$, we have

$\displaystyle\left\langle A\phi,\widetilde{\phi}\right\rangle_{X}$
$\displaystyle=i\left\langle
Bv,B\widetilde{u}\right\rangle_{{L^{2}_{{\beta}}\left(\mathbb{R}\right)}}+i\left\langle-B^{2}u,\widetilde{v}\right\rangle_{{L^{2}_{{\beta}}\left(\mathbb{R}\right)}}$
$\displaystyle=i\left\langle
v,B^{2}\widetilde{u}\right\rangle_{{L^{2}_{{\beta}}\left(\mathbb{R}\right)}}-i\left\langle
Bu,B\tilde{v}\right\rangle_{{L^{2}_{{\beta}}\left(\mathbb{R}\right)}}=\left\langle\phi,A\widetilde{\phi}\right\rangle_{X}.$

In fact, $A$ is self-adjoint in $X$ with (maximal) domain
$\operatorname{Dom}A=H^{2}\left(\mathbb{R}\right)\times
H^{1}\left(\mathbb{R}\right)$. The latter statement follows from the
maximality of $\operatorname{Dom}B^{2}=H^{2}\left(\mathbb{R}\right)$.

By Stone’s theorem [7, Ch. 1 Thm. 10.8], the operator $-iA$ generates a
$C_{0}$-semigroup of unitary operators on $X$, and the operator $C$ is a
bounded perturbation. Therefore, using results on the perturbation of
semigroups [7, Sect. 3.1 Thm. 1.1], the operator $-iA+C$, which defines
problem (10), generates a $C_{0}$-semigroup of bounded operators on $X$.
Consequently, we have the following two solution concepts for problem (10),
and thus for problem (1), depending on the regularity of
$\phi_{0}=\left(u_{0},u_{1}\right)^{T}$ (see [7, p. 105] for this discussion):

  * _i)_

For $\phi_{0}\in X$, problem (10) has a unique mild solution, i.e. $\phi\in
C\left(\mathbb{R}_{+},X\right)$, and hence $u\in
C\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right)\cap
C^{1}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)$,
$v=\partial_{t}u\in C\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)$

  * _ii)_

For $\phi_{0}\in\text{Dom }A,$ problem (10) has a unique classical solution
(in the semigroup sense), i.e. $\phi\in C^{1}\left(\mathbb{R}_{+},X\right)\cap
C\left(\mathbb{R}_{+},\text{Dom }A\right)$, and hence $u\in
C\left(\mathbb{R}_{+},H^{2}\left(\mathbb{R}\right)\right)\cap
C^{1}\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right)$,
$v=\partial_{t}u\in
C^{1}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)$.

Finally, estimate (2) is an automatic consequence of the semigroup approach
due to [7, Sect. 1.2 Thm. 2.2]. ∎

##  3\. Proof of Theorem 1.4

Without loss of generality, let us take
$\Omega=\Omega_{in}=\Omega_{0}=\left(0,x_{0}\right)$ for some $x_{0}>0$. This
is possible since the decay in the region of interest $\Omega$ can be deduced
by enlarging it (if necessary) to an interval containing both $\Omega_{0}$ and
$\Omega_{in}$.

While the result of this theorem is stated for the mild solution

(11) $u\in C^{1}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)\cap
C\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right),$

its proof requires the regularity of classical solutions established in
Proposition 1.1. Hence we approximate the initial data $(u_{0},\,u_{1})^{T}\in
X{:=H^{1}\left(\mathbb{R}\right)\times L^{2}\left(\mathbb{R}\right)}$ by a
sequence of functions $(u_{0}^{n},\,u_{1}^{n})^{T}_{n\in\mathbb{N}}\subset
H^{2}\left(\mathbb{R}\right)\times H^{1}\left(\mathbb{R}\right)$ with support
in $\Omega_{0}$ and converging in $X$ to $(u_{0},\,u_{1})^{T}$. Due to
Proposition 1.1, each approximate initial condition $(u_{0}^{n},\,u_{1}^{n})$
gives rise to a classical solution

(12) $u^{n}\in
C^{2}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)\cap
C^{1}\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right)\cap
C\left(\mathbb{R}_{+},H^{2}\left(\mathbb{R}\right)\right).$

By the linearity of problem (1) and bound (2), these classical solutions
converge uniformly to the mild solution $u$ on any finite time interval
$[0,T]$:

(13) $u^{n}\>\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\>u\quad\mbox{in
}C^{1}([0,T],L^{2}(\mathbb{R}))\cap C([0,T],H^{1}(\mathbb{R})).$

We recall that the notions of classical and mild solutions are discussed in
[7, p. 105].

The proof is performed in 3 steps. In Step 1, for any fixed $n\in\mathbb{N}$,
we consider problem (1) for initial data $(u_{0}^{n},\,u_{1}^{n})\in
H^{2}\left(\mathbb{R}\right)\times H^{1}\left(\mathbb{R}\right)$ and classical
solution $u^{n}$ as in (12). We transform (1) into an auxiliary problem, for
which we construct a weighted, local energy functional that would admit an
exponential decay under certain conditions on weight functions. In Step 2, we
construct these weight functions such that the rate of decay of the energy
functional can be estimated explicitly. In Step 3, we use the local energy
decay and the convergence in (13) to prove that the mild solution $u$ of (1),
for large times, converges to a constant uniformly on $\left[0,x_{0}\right]$.
Moreover, we identify this constant as $u_{\infty}$ defined in (4).

### Step 1. Auxiliary local energy functional.

As anticipated, here and in Step 2, we establish the exponential-in-time decay
of any classical solution $u^{n}$, i.e. for fixed index $n\in\mathbb{N}$.

Let us perform the change of variable

(14) $x\mapsto
y\left(x\right):=\dfrac{1}{t_{0}}\int_{0}^{x}\sqrt{\beta\left(\xi\right)/\alpha\left(\xi\right)}d\xi,\hskip
10.00002ptx\left(s\right)=y^{-1}\left(s\right),$ (15)
$t\mapsto\tau\left(t\right):=t/t_{0},\hskip
10.00002ptt\left(\tau\right)=t_{0}\tau,\hskip
10.00002ptt_{0}:=\int_{0}^{x_{0}}\sqrt{\beta\left(\xi\right)/\alpha\left(\xi\right)}d\xi,$

and denote
$v\left(y,\tau\right):={u^{n}}\left(x\left(y\right),t_{0}\tau\right)$,
$v_{0}\left(y\right):={u^{n}_{0}}\left(x\left(y\right)\right)$,
$v_{1}\left(y\right):=t_{0}\,{u^{n}_{1}}\left(x\left(y\right)\right)$,

(16)
$\rho\left(y\right):=\frac{1}{\sqrt{\alpha\left(x\left(y\right)\right)\beta\left(x\left(y\right)\right)}},\hskip
10.00002pt\gamma\left(y\right):=\frac{\rho^{\prime}\left(y\right)}{\rho\left(y\right)},\hskip
10.00002pt\rho_{0}:=\rho\left(0\right)=\rho\left(1\right)=\frac{1}{\sqrt{\alpha_{0}\beta_{0}}}.$

Due to Assumption 1.2, the map $x\mapsto y\left(x\right)$ is
$W^{2,\infty}\left(\mathbb{R}\right)$, and hence $v$ inherits the regularity
of $u^{n}$ given by (12):

(17) $v\in C^{2}\left(\mathbb{R}_{+},L^{2}\left(\mathbb{R}\right)\right)\cap
C^{1}\left(\mathbb{R}_{+},H^{1}\left(\mathbb{R}\right)\right)\cap
C\left(\mathbb{R}_{+},H^{2}\left(\mathbb{R}\right)\right).$

We thus arrive at a problem equivalent to (1):

(18)
$\begin{cases}\partial_{\tau}^{2}v\left(y,\tau\right)-\partial_{y}^{2}v\left(y,\tau\right)+\gamma\left(y\right)\partial_{y}v\left(y,\tau\right)=0,&y\in\mathbb{R},\hskip
10.00002pt\tau>0,\\\ v\left(y,0\right)=v_{0}\left(y\right),\hskip
10.00002pt\partial_{\tau}v\left(y,0\right)=v_{1}\left(y\right),&y\in\mathbb{R},\end{cases}$

where $\gamma\left(y\right)\equiv 0$ for
$y\in\mathbb{R}\backslash\left(0,1\right)$ and $\text{supp }v_{0}$,
$\text{supp }v_{1}\subseteq(0,1)$.

Motivated by the form of the global energy introduced in [6] for the wave
equation in (18), we consider the local energy functional

$\displaystyle\mathcal{E}_{loc}\left(\tau\right):=$
$\displaystyle\frac{1}{2}\int_{0}^{1}\left[\left(\partial_{\tau}v\left(y,\tau\right)\right)^{2}+\left(\partial_{y}v\left(y,\tau\right)\right)^{2}\right]\frac{dy}{\rho\left(y\right)}$
(19) $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{0}^{1}\left[\left(\partial_{\tau}v\left(y,\tau\right)+\partial_{y}v\left(y,\tau\right)\right)^{2}+\left(\partial_{\tau}v\left(y,\tau\right)-\partial_{y}v\left(y,\tau\right)\right)^{2}\right]\frac{dy}{\rho\left(y\right)}>0.$

This quantity is decaying since, by direct calculations employing integration
by parts, using (16) and the wave equation in (18), we have

$\displaystyle\mathcal{E}^{\prime}_{loc}\left(\tau\right)=$
$\displaystyle\int_{0}^{1}\left[\partial_{\tau}v\left(y,\tau\right)\partial^{2}_{\tau}v\left(y,\tau\right)+\partial_{y}v\left(y,\tau\right){\partial_{y}\partial_{\tau}}v\left(y,\tau\right)\right]\frac{dy}{\rho\left(y\right)}$
(20) $\displaystyle{=}$
$\displaystyle{\int_{0}^{1}\left[\partial_{\tau}v\left(y,\tau\right)\left(\partial^{2}_{\tau}v\left(y,\tau\right)-\partial_{y}^{2}v\left(y,\tau\right)+\frac{\rho^{\prime}\left(y\right)}{\rho\left(y\right)}\partial_{y}v\left(y,\tau\right)\right)\right]\frac{dy}{\rho\left(y\right)}}$
$\displaystyle{+\left[\partial_{y}v\left(1,\tau\right)\partial_{\tau}v\left(1,\tau\right)-\partial_{y}v\left(0,\tau\right)\partial_{\tau}v\left(0,\tau\right)\right]\frac{1}{\rho_{0}}}$
$\displaystyle=$
$\displaystyle\left[\partial_{y}v\left(1,\tau\right)\partial_{\tau}v\left(1,\tau\right)-\partial_{y}v\left(0,\tau\right)\partial_{\tau}v\left(0,\tau\right)\right]\frac{1}{\rho_{0}}\leq
0.$

Note that we have used here the regularity of a classical solution. Moreover,
in order to deduce the sign in the last line, we have used the exact outflow
boundary conditions (well-defined for any $t\geq 0$ due to (17)):

(21)
$\left.\left(\partial_{y}-\partial_{\tau}\right)v\left(y,\tau\right)\right|_{y=0}=0=\left.\left(\partial_{y}+\partial_{\tau}\right)v\left(y,\tau\right)\right|_{y=1},$

which are due to the support properties of the functions $v_{0}$, $v_{1}$ and
$\gamma$.

We note, in passing, that the global energy, defined in the same way as (19)
but with the integration range replaced by $\mathbb{R}$, is conserved.

Even though the conventional local energy functional (19) can (and, in the
proof of Theorem 1.5, will) be used to show the exponential decay of the
solution derivatives under some quantitative restriction on the coefficients
$\alpha$ and $\beta$, here, we proceed with an alternative strategy that does
not require such a restriction. To this effect, we consider now a modified
local energy functional in the spirit of [1, Sec. 3]:

(22)
$E_{loc}\left(\tau\right):=\frac{1}{2}\int_{0}^{1}\left[\phi_{1}\left(y\right)\left(\partial_{\tau}v\left(y,\tau\right)+\partial_{y}v\left(y,\tau\right)\right)^{2}+\phi_{2}\left(y\right)\left(\partial_{\tau}v\left(y,\tau\right)-\partial_{y}v\left(y,\tau\right)\right)^{2}\right]\frac{dy}{\rho\left(y\right)},$

where $\phi_{1}\left(y\right)$, $\phi_{2}\left(y\right)$ are some strictly
positive weight functions on $\left[0,1\right]$, which are yet to be chosen.

Similarly to (3), we differentiate under the integral sign to obtain
(suppressing the arguments of all functions for the sake of brevity)

(23) $\displaystyle E_{loc}^{\prime}\left(\tau\right)=$
$\displaystyle\int_{0}^{1}\left[\phi_{1}\left(\partial_{\tau}v+\partial_{y}v\right)\left(\partial_{\tau}^{2}v+{\partial_{y}\partial_{\tau}}v\right)+\phi_{2}\left(\partial_{\tau}v-\partial_{y}v\right)\left(\partial_{\tau}^{2}v-{\partial_{y}\partial_{\tau}}v\right)\right]\frac{dy}{\rho}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\left[\phi_{1}\left(\partial_{\tau}v+\partial_{y}v\right)\left(\partial_{y}^{2}v-\gamma\partial_{y}v+{\partial_{y}\partial_{\tau}}v\right)+\phi_{2}\left(\partial_{\tau}v-\partial_{y}v\right)\left(\partial_{y}^{2}v-\gamma\partial_{y}v-{\partial_{y}\partial_{\tau}}v\right)\right]\frac{dy}{\rho}$
$\displaystyle=$
$\displaystyle{\frac{1}{2}}\int_{0}^{1}\left[\phi_{1}\partial_{y}\frac{\left(\partial_{\tau}v+\partial_{y}v\right)^{2}}{\rho}-\phi_{2}\partial_{y}\frac{\left(\partial_{\tau}v-\partial_{y}v\right)^{2}}{\rho}+\left(\phi_{1}-\phi_{2}\right)\gamma\frac{\left(\partial_{\tau}v\right)^{2}-\left(\partial_{y}v\right)^{2}}{\rho}\right]dy,$

where we used the wave equation in (18) to eliminate second-order time
derivatives, and the definition of $\gamma$ in (16). Integrating (23) by parts
and employing boundary conditions (21), we obtain

$\displaystyle E_{loc}^{\prime}\left(\tau\right)=$
$\displaystyle\frac{1}{2}\int_{0}^{1}\left[-\phi_{1}^{\prime}\left(y\right)\left(\partial_{\tau}v\left(y,\tau\right)+\partial_{y}v\left(y,\tau\right)\right)^{2}+\phi_{2}^{\prime}\left(y\right)\left(\partial_{\tau}v\left(y,\tau\right)-\partial_{y}v\left(y,\tau\right)\right)^{2}\right.$
$\displaystyle\left.+\left(\phi_{1}\left(y\right)-\phi_{2}\left(y\right)\right)\gamma\left(y\right)\left(\left(\partial_{\tau}v\left(y,\tau\right)\right)^{2}-\left(\partial_{y}v\left(y,\tau\right)\right)^{2}\right)\right]\frac{dy}{\rho\left(y\right)}$
$\displaystyle-\left[\phi_{2}\left(1\right)\left(\partial_{y}v\left(1,\tau\right)\right)^{2}+\phi_{1}\left(0\right)\left(\partial_{y}v\left(0,\tau\right)\right)^{2}\right]\frac{2}{\rho_{0}}.$

We estimate

$\displaystyle\left(\phi_{1}-\phi_{2}\right)\gamma\left(\left(\partial_{\tau}v\right)^{2}-\left(\partial_{y}v\right)^{2}\right)$
$\displaystyle=\left(\phi_{1}-\phi_{2}\right)\gamma\left(\partial_{\tau}v+\partial_{y}v\right)\left(\partial_{\tau}v-\partial_{y}v\right)$
$\displaystyle\leq\frac{1}{2}\left|\phi_{1}-\phi_{2}\right|\left|\gamma\right|\left(\left(\partial_{\tau}v+\partial_{y}v\right)^{2}+\left(\partial_{\tau}v-\partial_{y}v\right)^{2}\right),$

and therefore, since $\rho>0$, we arrive at

$\displaystyle E_{loc}^{\prime}\left(\tau\right)\leq$
$\displaystyle-\frac{1}{2}\int_{0}^{1}\left[\left(\phi_{1}^{\prime}\left(y\right)-\frac{\left|\gamma\left(y\right)\right|}{2}\left|\phi_{1}\left(y\right)-\phi_{2}\left(y\right)\right|\right)\left(\partial_{\tau}v\left(y,\tau\right)+\partial_{y}v\left(y,\tau\right)\right)^{2}\right.$
(24)
$\displaystyle\left.+\left(-\phi_{2}^{\prime}\left(y\right)-\frac{\left|\gamma\left(y\right)\right|}{2}\left|\phi_{1}\left(y\right)-\phi_{2}\left(y\right)\right|\right)\left(\partial_{\tau}v\left(y,\tau\right)-\partial_{y}v\left(y,\tau\right)\right)^{2}\right]\frac{dy}{\rho\left(y\right)}.$

We aim to obtain an estimate of the form
$E_{loc}^{\prime}\left(\tau\right)\leq-\lambda_{0}E_{loc}\left(\tau\right)$
with some $\lambda_{0}>0$, and explore the possibility of choosing the weight
functions $\phi_{1}\left(y\right)$, $\phi_{2}\left(y\right)$ that would lead
to such an estimate. In [1] it is shown in a more general setting that such
weights do exist theoretically. Here, we present an explicit construction of
$\phi_{1}$, $\phi_{2}$, which will permit us to obtain an explicit estimate of
the decay rate of the solution.

### Step 2. Construction of the weights $\phi_{1}$, $\phi_{2}$.

Motivated by (24), we shall now construct functions $\phi_{1}\left(y\right)$,
$\phi_{2}\left(y\right)>0$ in (22) that satisfy, on $\left[0,1\right]$, the
following differential inequalities

(25)
$\begin{cases}\phi_{1}^{\prime}\left(y\right)-\frac{\left|\gamma\left(y\right)\right|}{2}\left|\phi_{1}\left(y\right)-\phi_{2}\left(y\right)\right|\geq\lambda\phi_{1}\left(y\right),\hskip
10.00002pty\in\left(0,1\right),\\\
-\phi_{2}^{\prime}\left(y\right)-\frac{\left|\gamma\left(y\right)\right|}{2}\left|\phi_{1}\left(y\right)-\phi_{2}\left(y\right)\right|\geq\lambda\phi_{2}\left(y\right),\hskip
10.00002pty\in\left(0,1\right),\\\
\phi_{1}\left(0\right)=\phi_{2}\left(0\right)=1,\end{cases}$

for some constant $\lambda>0$ to be chosen. To this end, we consider functions
$\varphi_{1}\left(y\right)$, $\varphi_{2}\left(y\right)>0$ solving the linear
ODE system

(26)
$\begin{cases}\varphi_{1}^{\prime}\left(y\right)-\gamma_{0}\left(\varphi_{1}\left(y\right)-\varphi_{2}\left(y\right)\right)=\lambda\varphi_{1}\left(y\right),\hskip
10.00002pty\in\left(0,1\right),\\\
-\varphi_{2}^{\prime}\left(y\right)-\gamma_{0}\left(\varphi_{1}\left(y\right)-\varphi_{2}\left(y\right)\right)=\lambda\varphi_{2}\left(y\right),\hskip
10.00002pty\in\left(0,1\right),\\\
\varphi_{1}\left(0\right)=\varphi_{2}\left(0\right)=1,\end{cases}$

where
$\gamma_{0}:=\frac{1}{2}\left\|\gamma\right\|_{L^{\infty}\left(0,1\right)}$.

It is easy to see that any solution of (26) automatically satisfies (25),
provided that $\varphi_{1}\left(y\right)$, $\varphi_{2}\left(y\right)>0$ for
all $y\in\left[0,1\right]$. Indeed, it suffices to verify that

(27)
$0\leq\varphi_{1}\left(y\right)-\varphi_{2}\left(y\right)=\left|\varphi_{1}\left(y\right)-\varphi_{2}\left(y\right)\right|,\hskip
10.00002pty\in\left[0,1\right].$

Summing both ODEs of (26), multiplying by $e^{-2\gamma_{0}y}$, and integrating
on $(0,y)$, we obtain

$\varphi_{1}\left(y\right)-\varphi_{2}\left(y\right)=\lambda\int_{0}^{y}e^{2\gamma_{0}\left(y-\xi\right)}\left[\varphi_{1}\left(\xi\right)+\varphi_{2}\left(\xi\right)\right]d\xi,\hskip
10.00002pty\in\left[0,1\right],$

and hence positivity of the integrand directly implies (27).

Note that it is possible to consider more general versions of (25) (and hence
(26)), with equations involving different parameters $\lambda_{1}$,
$\lambda_{2}$ on the right-hand sides, and initial conditions given by a
positive constant different from $1$. However, it can be shown that no
advantage could be gained from such generalisations.

We shall now estimate the largest possible value $\lambda>0$ (or, more
precisely, its supremum) such that $\varphi_{1}$, $\varphi_{2}>0$ in
$\left[0,1\right]$. According to the first equation in (26) and (27),
$\varphi_{1}\left(y\right)$ is a monotonically increasing function of $y$ for
$y\in\left(0,1\right)$ and thus $\varphi_{1}\left(0\right)=1$ implies
$\varphi_{1}\left(y\right)>1$ for $y\in\left(0,1\right]$. Similarly, it
follows from the second equation of (26) that $\varphi_{2}\left(y\right)$
decreases monotonically in $\left[0,1\right]$ starting from the value
$\varphi_{2}\left(0\right)=1$. Therefore, to guarantee the positivity of
$\varphi_{2}\left(y\right)$ in $\left[0,1\right]$, it suffices to require that
$\varphi_{2}\left(1\right)>0$. We can write the solution of (26) explicitly:

(28)
$\varphi_{1}\left(y\right)=\frac{e^{\gamma_{0}y}}{\sqrt{\lambda^{2}+\gamma_{0}^{2}}}\left[\sqrt{\lambda^{2}+\gamma_{0}^{2}}\cosh\left(\sqrt{\lambda^{2}+\gamma_{0}^{2}}y\right)+\left(\lambda-\gamma_{0}\right)\sinh\left(\sqrt{\lambda^{2}+\gamma_{0}^{2}}y\right)\right],$
(29)
$\varphi_{2}\left(y\right)=\frac{e^{\gamma_{0}y}}{\sqrt{\lambda^{2}+\gamma_{0}^{2}}}\left[\sqrt{\lambda^{2}+\gamma_{0}^{2}}\cosh\left(\sqrt{\lambda^{2}+\gamma_{0}^{2}}y\right)-\left(\lambda+\gamma_{0}\right)\sinh\left(\sqrt{\lambda^{2}+\gamma_{0}^{2}}y\right)\right],$

Then, the above requirement $\varphi_{2}\left(1\right)>0$ is equivalent to

(30)
$\tanh\left({\sqrt{\gamma_{0}^{2}+\lambda^{2}}}\right)<\frac{\sqrt{\gamma_{0}^{2}+\lambda^{2}}}{\gamma_{0}+\lambda}.$

To determine the interval of admissible values of $\lambda$ satisfying (30),
we consider the corresponding equality

(31)
$\tanh\left({\sqrt{\gamma_{0}^{2}+\lambda^{2}_{*}}}\right)=\frac{\sqrt{\gamma_{0}^{2}+\lambda^{2}_{*}}}{\gamma_{0}+\lambda_{*}},$

which implicitly defines a function $\lambda_{*}(\gamma_{0})$ in
$\mathbb{R}_{+}$. The proof of the unique solvability of (31) for
$\lambda_{*}$, as well as the fact that (30) then holds for all
$\lambda\in[0,\lambda_{*})$, is actually deferred to Lemma A.1 in the
Appendix. The function $\lambda_{*}(\gamma_{0})$ is illustrated in Figure 1.
We see that $\lambda_{*}(\gamma_{0})$ is monotonically decreasing with
$\lambda_{*}\to 0$ as $\gamma_{0}\to\infty$ (this can also be verified
analytically by implicit differentiation of (31)).

While the parameter choice $\lambda=\lambda_{*}$ is not admissible for the
construction of the weight functions (since $\varphi_{2}\left(1\right)=0$ if
$\lambda=\lambda_{*}$), any value $\lambda\in(0,\lambda_{*})$ is admissible,
and therefore guarantees positivity of $\varphi_{1}$, $\varphi_{2}$.

Figure 1. Blue: function $\lambda_{*}(\gamma_{0})$, implicitly defined by
(31). Red: function $\lambda_{0}(\gamma_{0})$, defined in (32).

Since the function $\lambda_{*}(\gamma_{0})$ is not defined explicitly, we
give an example of an explicit lower bound that eventually yields the explicit
(but suboptimal) decay rate stated in (4). The function

(32)
$\lambda_{0}(\gamma_{0}):=\sqrt{2}\gamma_{0}e^{-2\gamma_{0}}\sqrt{1-4\gamma_{0}e^{-4\gamma_{0}}}$

satisfies (30) and $\lambda_{0}(\gamma_{0})<\lambda_{*}(\gamma_{0})$ for any
$\gamma_{0}>0$, and it is depicted in Figure 1. The proof of the inequality
$\lambda_{0}(\gamma_{0})<\lambda_{*}(\gamma_{0})$ is given in Lemma A.2 in the
Appendix.

Therefore, we can take $\lambda=\lambda_{0}:=\lambda_{0}(\gamma_{0})$, which
satisfies inequality (30). Hence the weights
$\phi_{1}\left(y\right):=\varphi_{1}\left(y\right)>0$,
$\phi_{2}\left(y\right):=\varphi_{2}\left(y\right)>0$ satisfy (25).
Consequently, we obtain from (24) and (22)

(33)
$E_{loc}^{\prime}\left(\tau\right)\leq-\lambda_{0}E_{loc}\left(\tau\right),\hskip
10.00002pt\tau>0\hskip 10.00002pt\Longrightarrow\hskip
10.00002ptE_{loc}\left(\tau\right)\leq
E_{loc}\left(0\right)e^{-\lambda_{0}\tau},\hskip 10.00002pt\tau\geq 0.$

### Step 3. Convergence of the solution and identification of the constant
$u_{\infty}$.

The local energy functional (22) can be rewritten in the original variables
$\left(x,t\right)$, and we emphasise here that it is a functional of the
approximate classical solution $u^{n}$:

$\displaystyle{\widetilde{E}_{loc}[u^{n}]}\left(t\right):=E_{loc}\left(\frac{t}{t_{0}}\right)=$
$\displaystyle\frac{t_{0}}{2}\int_{0}^{x_{0}}\left[{\widetilde{\varphi}_{1}}\left(x\right)\left(\partial_{t}u^{n}\left(x,t\right)+\sqrt{\frac{\alpha\left(x\right)}{\beta\left(x\right)}}\partial_{x}u^{n}\left(x,t\right)\right)^{2}\right.$
$\displaystyle\left.+{\widetilde{\varphi}_{2}}\left(x\right)\left(\partial_{t}u^{n}\left(x,t\right)-\sqrt{\frac{\alpha\left(x\right)}{\beta\left(x\right)}}\partial_{x}u^{n}\left(x,t\right)\right)^{2}\right]\beta\left(x\right)dx,$

where
${\widetilde{\varphi}_{1}}\left(x\right):=\varphi_{1}\left(y\left(x\right)\right)$,
${\widetilde{\varphi}_{1}}\left(x\right):=\varphi_{2}\left(y\left(x\right)\right)$
are given by (28)–(29), with $\lambda=\lambda_{0}$ and the mapping $x\mapsto
y\left(x\right)$ given by (14). Then, from (33), we obtain

${\widetilde{E}_{loc}[u^{n}]}\left(t\right)\leq{\widetilde{E}_{loc}[u^{n}]}\left(0\right)e^{-\frac{\lambda_{0}}{t_{0}}t},\hskip
10.00002ptt\geq 0,$

where the decay rate $\lambda_{0}/t_{0}$ is independent of the index $n$. Due
to the convergence (13), we can pass to the limit $n\to\infty$ and obtain the
same inequality for the mild solution $u$:

(34)
$\widetilde{E}_{loc}[u]\left(t\right)\leq\widetilde{E}_{loc}[u]\left(0\right)e^{-\frac{\lambda_{0}}{t_{0}}t},\hskip
10.00002ptt\geq 0{,}$

and, due to the regularity of $u$ in (11), we have $\widetilde{E}_{loc}[u]\in
C\left(\mathbb{R}_{+}\right)$.

Setting

$C_{0}:={\min}\left\\{\underset{y\in\left(0,1\right)}{\inf}\varphi_{1}\left(y\right),\underset{y\in\left(0,1\right)}{\inf}\varphi_{2}\left(y\right)\right\\}={\min}\left\\{\varphi_{1}\left(0\right),\varphi_{2}\left(1\right)\right\\}=\varphi_{2}\left(1\right)>0,$

we have

${\widetilde{E}_{loc}[u]}\left(t\right)\geq
C_{0}t_{0}\int_{0}^{x_{0}}\left[\beta\left(x\right)\left(\partial_{t}u\left(x,t\right)\right)^{2}+\alpha\left(x\right)\left(\partial_{x}u\left(x,t\right)\right)^{2}\right]dx,$

and thus deduce that

(35)
$\left\|\partial_{t}u\left(\cdot,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}\leq\frac{\left\|1/\beta\right\|_{L^{\infty}\left(0,x_{0}\right)}}{C_{0}t_{0}}{\widetilde{E}_{loc}[u]}\left(t\right),\hskip
10.00002pt\left\|\partial_{x}u\left(\cdot,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}\leq\frac{\left\|1/\alpha\right\|_{L^{\infty}\left(0,x_{0}\right)}}{C_{0}t_{0}}{\widetilde{E}_{loc}[u]}\left(t\right).$

This, together with (34), yields the exponential decay of
$\left\|\partial_{t}u\right\|_{L^{2}\left(0,x_{0}\right)}$ and
$\left\|\partial_{x}u\right\|_{L^{2}\left(0,x_{0}\right)}$ claimed in (3). In
order to complete the proof, it remains to identify $u_{\infty}$, and prove
that, for all $t\geq 0$, $u(\cdot,t)$ converges in $H^{1}(0,x_{0})$ to
$u_{\infty}$.

Using the regularity of $u$ given by (11), we have, for
$x\in\left(0,x_{0}\right)$, $t>0$,

$u\left(x,t\right)=u\left(0,t\right)+\int_{0}^{x}\partial_{x}u\left(s,t\right)ds=u\left(x_{0},t\right)-\int_{x}^{x_{0}}\partial_{x}u\left(s,t\right)ds,$

which implies

(36)
$\left\|u\left(\cdot,t\right)-u\left(0,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2},\,\left\|u\left(\cdot,t\right)-u\left(x_{0},t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}\leq
x_{0}^{2}\left\|\partial_{x}u\left(\cdot,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}.$

Let us now integrate the wave equation in (1) over the rectangle
$\left(0,x_{0}\right)\times\left(0,t\right)$, $t>0$. Using the outflow
boundary conditions (11)

$\left.\left(\partial_{x}-\sqrt{\frac{\beta_{0}}{\alpha_{0}}}\partial_{t}\right)u\left(x,t\right)\right|_{x=0}=0,\hskip
10.00002pt\left.\left(\partial_{x}+\sqrt{\frac{\beta_{0}}{\alpha_{0}}}\partial_{t}\right)u\left(x,t\right)\right|_{x=x_{0}}=0,$

and the compact support of $u_{0}$, $u_{1}$, we obtain

(37)
$\int_{0}^{x_{0}}\beta\left(s\right)\partial_{t}u\left(s,t\right)ds-\int_{0}^{x_{0}}\beta\left(s\right)u_{1}\left(s\right)ds=-\sqrt{\alpha_{0}\beta_{0}}\left(u\left(0,t\right)+u\left(x_{0},t\right)\right).$

Due to the weak regularity of the mild solution $u$, the same identity
actually first needs to be derived for the classical solutions $u^{n}$ from
(12), and then (37) follows by passing to the limit $n\to\infty$. From (37),
we deduce

$\displaystyle
u\left(x,t\right)-\frac{1}{2\sqrt{\alpha_{0}\beta_{0}}}\int_{0}^{x_{0}}\beta\left(s\right)u_{1}\left(s\right)ds=$
$\displaystyle\frac{1}{2}\left[u\left(x,t\right)-u\left(0,t\right)\right]+\frac{1}{2}\left[u\left(x,t\right)-u\left(x_{0},t\right)\right]$
(38)
$\displaystyle-\frac{1}{2\sqrt{\alpha_{0}\beta_{0}}}\int_{0}^{x_{0}}\beta\left(s\right)\partial_{t}u\left(s,t\right)ds$

for all $x\in\left(0,x_{0}\right)$ and all $t\geq 0$. For the first two terms
on the right-hand side of (38), we have

$\left\|u\left(x,t\right)-u\left(0,t\right)\right\|_{H^{1}\left(0,x_{0}\right)}^{2}=\left\|u\left(x,t\right)-u\left(0,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}+\left\|\partial_{x}u\left(x,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2},$
$\left\|u\left(x,t\right)-u\left(x_{0},t\right)\right\|_{H^{1}\left(0,x_{0}\right)}^{2}=\left\|u\left(x,t\right)-u\left(x_{0},t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}+\left\|\partial_{x}u\left(x,t\right)\right\|_{L^{2}\left(0,x_{0}\right)}^{2}.$

Moreover, since the integral term on the right-hand side of (38) is
independent of $x$, we also have

$\begin{split}\left\|\frac{1}{2\sqrt{\alpha_{0}\beta_{0}}}\int_{0}^{x_{0}}\beta\left(s\right)\partial_{t}u\left(s,t\right)ds\right\|_{H^{1}\left(0,x_{0}\right)}&=\left\|\frac{1}{2\sqrt{\alpha_{0}\beta_{0}}}\int_{0}^{x_{0}}\beta\left(s\right)\partial_{t}u\left(s,t\right)ds\right\|_{L^{2}\left(0,x_{0}\right)}\\\
&\leq\frac{1}{2}\sqrt{\frac{x_{0}}{\alpha_{0}\beta_{0}}}\left\|\beta\right\|_{L^{2}\left(0,x_{0}\right)}\left\|\partial_{t}u\right\|_{L^{2}\left(0,x_{0}\right)}.\end{split}$

Therefore, taking the $H^{1}\left(0,x_{0}\right)$ norm of the identity in
(38), with $u_{\infty}$ defined as in (4), we deduce from (36) that

$\displaystyle\left\|u\left(\cdot,t\right)-u_{\infty}\right\|_{H^{1}\left(0,x_{0}\right)}\leq$
$\displaystyle\frac{1}{2}\left\|u\left(x,t\right)-u\left(0,t\right)\right\|_{H^{1}\left(0,x_{0}\right)}+\frac{1}{2}\left\|u\left(x,t\right)-u\left(x_{0},t\right)\right\|_{H^{1}\left(0,x_{0}\right)}$
$\displaystyle+{\frac{1}{2}\sqrt{\frac{x_{0}}{\alpha_{0}\beta_{0}}}\left\|\beta\right\|_{L^{2}\left(0,x_{0}\right)}\left\|\partial_{t}u\right\|_{L^{2}\left(0,x_{0}\right)}}$
$\displaystyle\leq$
$\displaystyle\sqrt{1+x_{0}^{2}}\left\|\partial_{x}u\right\|_{L^{2}\left(0,x_{0}\right)}+\frac{1}{2}\sqrt{\frac{x_{0}}{\alpha_{0}\beta_{0}}}\left\|\beta\right\|_{L^{2}\left(0,x_{0}\right)}\left\|\partial_{t}u\right\|_{L^{2}\left(0,x_{0}\right)}.$

Finally, employing (35), we obtain

$\displaystyle\left\|u\left(\cdot,t\right)-u_{\infty}\right\|_{H^{1}\left(0,x_{0}\right)}\leq$
$\displaystyle\frac{1}{2\left(C_{0}t_{0}\right)^{1/2}}\left[2\left(\left(1+x_{0}^{2}\right)\left\|1/\alpha\right\|_{L^{\infty}\left(0,x_{0}\right)}\right)^{1/2}\right.$
$\displaystyle+\left.\left(\frac{x_{0}}{\alpha_{0}\beta_{0}}\left\|1/\beta\right\|_{L^{\infty}\left(0,x_{0}\right)}\right)^{1/2}\left\|\beta\right\|_{L^{2}\left(0,x_{0}\right)}\right]{\widetilde{E}_{loc}[u]^{1/2}}\left(t\right)$

which, together with (34), leads to the bound of
$\left\|u\left(\cdot,t\right)-u_{\infty}\right\|_{H^{1}\left(0,x_{0}\right)}$
in (3). From (32) and (34), we find the claimed decay rate
$\Lambda=\frac{\gamma_{0}}{\sqrt{2}\,t_{0}}e^{-2\gamma_{0}}\left(1-4\gamma_{0}e^{-4\gamma_{0}}\right)^{1/2}$.

Since
$\left\|u\left(\cdot,t\right)-u_{\infty}\right\|_{H^{1}\left(0,x_{0}\right)}$
decays to zero, $u_{\infty}$ is uniquely defined as in (4). This concludes the
proof. ∎

In the proof of Theorem 1.4, modified weight functions could also have been
obtained by imposing the initial condition for (25) at an internal point
$y_{0}$ in $(0,1)$ instead of at $y_{0}=0$, and then solving the two
corresponding linear ODEs forward and backward. Potentially, this may lead to
an increased maximal decay rate, as compared to that we have deduced from
(31).

We also remark that the decay rate given by $\lambda_{0}(\gamma_{0})$ is
exponentially small for large $\gamma_{0}$. This is not an artefact of the
approximation of $\lambda_{*}(\gamma_{0})$ by $\lambda_{0}(\gamma_{0})$ (see
Figure 1), but is rather characteristic to the weight function approach in the
modified local energy.

##  4\. Proof of Theorem 1.5

As in the proof of Theorem 1.4 in Section 3, we assume, without loss of
generality, that $\Omega=\Omega_{in}=\Omega_{0}=\left(0,x_{0}\right)$ for some
$x_{0}>0$.

To prove the claim, we shall modify Step 1 of the proof of Theorem 1.4.
Namely, instead of working with the modified local energy functional (22), we
shall focus on its conventional counterpart (19) and estimate the quantities
that enter it. Since we shall not differentiate the energy functional in time,
we work here directly with the mild solution $u$. We consider again equation
(18) in the variables $(y,\tau)$ given by (14)–(15). By treating the third
term on the left-hand side as a (known) source term, it results in a wave
equation with constant coefficients, to which we apply the d’Alembert formula:

(39)
$v\left(y,\tau\right)=\frac{1}{2}\left[v_{0}\left(y+\tau\right)+v_{0}\left(y-\tau\right)\right]+\frac{1}{2}\int_{y-\tau}^{y+\tau}v_{1}\left(z\right)dz-\frac{1}{2}\int_{0}^{\tau}\int_{y-s}^{y+s}\gamma\left(z\right)\partial_{y}v\left(z,\tau-s\right)dzds.$

Therefore, we obtain

$\displaystyle\partial_{y}v\left(y,\tau\right)=$
$\displaystyle\frac{1}{2}\left[v_{0}^{\prime}\left(y+\tau\right)+v_{0}^{\prime}\left(y-\tau\right)\right]+\frac{1}{2}\left[v_{1}\left(y+\tau\right)-v_{1}\left(y-\tau\right)\right]$
(40)
$\displaystyle+\frac{1}{2}\int_{0}^{\tau}\left[\gamma\left(y-s\right)\partial_{y}v\left(y-s,\tau-s\right)-\gamma\left(y+s\right)\partial_{y}v\left(y+s,\tau-s\right)\right]ds,$
$\displaystyle\partial_{\tau}v\left(y,\tau\right)=$
$\displaystyle\frac{1}{2}\left[v_{0}^{\prime}\left(y+\tau\right)-v_{0}^{\prime}\left(y-\tau\right)\right]+\frac{1}{2}\left[v_{1}\left(y+\tau\right)+v_{1}\left(y-\tau\right)\right]$
(41)
$\displaystyle-\frac{1}{2}\int_{0}^{\tau}\left[\gamma\left(y-s\right)\partial_{y}v\left(y-s,\tau-s\right)+\gamma\left(y+s\right)\partial_{y}v\left(y+s,\tau-s\right)\right]ds.$

Identity (41) becomes evident by making a change of variable $s\mapsto\tau-s$
in the double-integral term in (39) before and after the differentiation with
respect to $\tau$:

$\displaystyle\partial_{\tau}\int_{0}^{\tau}\int_{y-s}^{y+s}\gamma\left(z\right)\partial_{y}v\left(z,\tau-s\right)dzds=\partial_{\tau}\int_{0}^{\tau}\int_{y-\tau+s}^{y+\tau-s}\gamma\left(z\right)\partial_{y}v\left(z,s\right)dzds$
$\displaystyle=\int_{0}^{\tau}\left[\gamma\left(y+\tau-s\right)\partial_{y}v\left(y+\tau-s,s\right)+\gamma\left(y-\tau+s\right)\partial_{y}v\left(y-\tau+s,s\right)\right]ds$
$\displaystyle=\int_{0}^{\tau}\left[\gamma\left(y+s\right)\partial_{y}v\left(y+s,\tau-s\right)+\gamma\left(y-s\right)\partial_{y}v\left(y-s,\tau-s\right)\right]ds.$

Note that, for $\tau\geq 1$, the terms in (40) and (41) that do not contain
the integral vanish for all $y$ in $\left(0,1\right)$. To estimate
$\mathcal{E}_{loc}$ for $\tau\geq 1$, we insert (40) and (41) in the
definition of $\mathcal{E}_{loc}\left(\tau\right)$ in (19) and use that
$\textrm{supp }\gamma\subset\left(0,1\right)$:

(42) $\displaystyle\mathcal{E}_{loc}\left(\tau\right)=$
$\displaystyle\frac{1}{4}\int_{0}^{1}\left[\left(\int_{y}^{1}\gamma\left(r\right)\partial_{y}v\left(r,\tau+y-r\right)dr\right)^{2}\right.$
$\displaystyle\left.+\left(\int_{0}^{y}\gamma\left(r\right)\partial_{y}v\left(r,\tau-y+r\right)dr\right)^{2}\right]\frac{dy}{\rho\left(y\right)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{4}\int_{0}^{1}\left[\int_{y}^{1}\rho\left(\tilde{r}\right)\gamma^{2}\left(\tilde{r}\right)d\tilde{r}\int_{y}^{1}\left(\partial_{y}v\left(r,\tau+y-r\right)\right)^{2}\frac{dr}{\rho\left(r\right)}\right.$
$\displaystyle\left.+\int_{0}^{y}\rho\left(\tilde{r}\right)\gamma^{2}\left(\tilde{r}\right)d\tilde{r}\int_{0}^{y}\left(\partial_{y}v\left(r,\tau-y+r\right)\right)^{2}\frac{dr}{\rho\left(r\right)}\right]\frac{dy}{\rho\left(y\right)}.$

Denoting

$b_{0}:=\left\|\alpha\right\|_{L^{\infty}\left(0,x_{0}\right)}^{3/2}\left\|\beta\right\|_{L^{\infty}\left(0,x_{0}\right)}^{1/2}\left\|\left(1/\left(\alpha\beta\right)^{1/2}\right)^{\prime}\right\|_{L^{2}\left(0,x_{0}\right)}^{2},$

as in (6), we have

$\max\left\\{\sup_{y\in\left(0,1\right)}\frac{1}{\rho\left(y\right)}\int_{0}^{y}\rho\left(s\right)\gamma^{2}\left(s\right)ds,\sup_{y\in\left(0,1\right)}\frac{1}{\rho\left(y\right)}\int_{y}^{1}\rho\left(s\right)\gamma^{2}\left(s\right)ds\right\\}\leq
b_{0}t_{0}.$

Therefore, we continue with (42), and estimate

(43) $\displaystyle\mathcal{E}_{loc}\left(\tau\right)$
$\displaystyle\leq\frac{b_{0}t_{0}}{4}\left[\int_{0}^{1}\int_{0}^{y}\left(\partial_{y}v\left(r,\tau-y+r\right)\right)^{2}\frac{1}{\rho\left(r\right)}drdy\right.$
$\displaystyle\phantom{=}\qquad\quad\left.+\int_{0}^{1}\int_{y}^{1}\left(\partial_{y}v\left(r,\tau+y-r\right)\right)^{2}\frac{1}{\rho\left(r\right)}drdy\right].$

In the two terms on the right-hand side of (43), the integration involves the
values of $\partial_{y}v$ on two right-angled triangles with vertices
$\left\\{\left(0,\tau-1\right),\left(0,\tau\right),\left(1,\tau\right)\right\\}$
and
$\left\\{\left(1,\tau-1\right),\left(0,\tau\right),\left(1,\tau\right)\right\\}$.
These triangles are parametrised by the sets of lines parallel to their
hypotenuses, which are the two characteristic families of the wave equation in
(18). We now change the integration variables to have parametrisation of these
triangles by the sets of horizontal lines. Consequently, for the first
integral, denoting $\sigma:=\tau-y+r$, we have that $\sigma\in(\tau-1,\tau)$
and $r\in(0,\sigma-\tau+1)$. Similarly, for the second integral, denoting
$\sigma:=\tau+y-r$, we have that $\sigma\in(\tau-1,\tau)$ and
$r\in(\tau-\sigma,1)$. Therefore, rewriting the integrals in (43) as integrals
in the variables $\sigma$ and $r$, we obtain

$\begin{split}&\int_{0}^{1}\int_{0}^{y}\left(\partial_{y}v\left(r,\tau-y+r\right)\right)^{2}\frac{1}{\rho\left(r\right)}drdy+\int_{0}^{1}\int_{y}^{1}\left(\partial_{y}v\left(r,\tau+y-r\right)\right)^{2}\frac{1}{\rho\left(r\right)}drdy\\\
&\qquad=\int_{\tau-1}^{\tau}\int_{0}^{\sigma-\tau+1}\left(\partial_{y}v\left(r,\sigma\right)\right)^{2}\frac{1}{\rho\left(r\right)}drd\sigma+\int_{\tau-1}^{\tau}\int_{\tau-\sigma}^{1}\left(\partial_{y}v\left(r,\sigma\right)\right)^{2}\frac{1}{\rho\left(r\right)}drd\sigma.\end{split}$

With this identity, taking into account the positivity of the integrands, we
obtain from (43) that

(44)
$\displaystyle\mathcal{E}_{loc}\left(\tau\right)\leq\frac{b_{0}t_{0}}{2}\int_{\tau-1}^{\tau}\int_{0}^{1}\left(\partial_{y}v\left(r,\sigma\right)\right)^{2}\frac{1}{\rho\left(r\right)}drd\sigma\leq
b_{0}t_{0}\int_{\tau-1}^{\tau}\mathcal{E}_{loc}\left(\sigma\right)d\sigma.$

Since $\mathcal{E}_{loc}\left(\tau\right)$ is a positive non-increasing
function (recall (3)), it follows from (44) that

$\mathcal{E}_{loc}\left(\tau\right)\leq
b_{0}t_{0}\,\mathcal{E}_{loc}\left(\tau-1\right),\hskip 10.00002pt\tau\geq 1.$

Iterating this inequality $\left\lfloor\tau\right\rfloor$ times (where
$\left\lfloor X\right\rfloor$ denotes the integer part of $X$), and using the
assumption $b_{0}t_{0}<1$, we obtain, for $\tau\geq 1$,

$\displaystyle\mathcal{E}_{loc}\left(\tau\right)$ $\displaystyle\leq$
$\displaystyle\mathcal{E}_{loc}\left(\lfloor\tau\rfloor\right)\leq(b_{0}t_{0})^{2}\mathcal{E}_{loc}\left(\lfloor\tau\rfloor-2\right)\leq\ldots\leq{(b_{0}t_{0})^{\lfloor\tau\rfloor}\mathcal{E}_{loc}\left(0\right)}$
$\displaystyle=$
$\displaystyle\mathcal{E}_{loc}\left(0\right)e^{\lfloor\tau\rfloor\log(b_{0}t_{0})}\leq\mathcal{E}_{loc}\left(0\right)e^{(\tau-1)\log(b_{0}t_{0})}=\mathcal{E}_{loc}\left(0\right)\frac{e^{\tau\log(b_{0}t_{0})}}{b_{0}t_{0}},$

which, due to $\log(b_{0}t_{0})<0$, shows an exponential decay in $\tau$.

When scaling back to the original variables $(x,t)$ we obtain

$\tilde{\mathcal{E}}_{loc}\left(t\right):=\mathcal{E}_{loc}\left(\frac{t}{t_{0}}\right)\leq\tilde{\mathcal{E}}_{loc}\left(0\right)\frac{e^{\frac{t}{t_{0}}\log(b_{0}t_{0})}}{b_{0}t_{0}},\hskip
10.00002ptt\geq t_{0}.$

Step 3 of the proof of Theorem 1.4 can then be repeated (with a slight
modification of the multiplicative constants), yielding the exponential
convergence (with rate (7)) of the solution to the constant $u_{\infty}$. ∎

##  5\. Conclusion

For the homogeneous wave equation in $\mathbb{R}$ with compactly supported
initial data, in the case of constant coefficients, the solution converges to
the steady state in finite time. However, exponential convergence is the
generic scenario in the case of variable coefficients.

We proved two different results for this convergence in local energy. The
first one employs a multiplier technique, constructing appropriate weight
functions to define a modified local energy that satisfies a first order
differential inequality. This allowed us to prove the exponential decay. Using
the solution balance over space-time rectangles, we were able to identify the
constant steady state. The second result improves the first one, under an
additional assumption on the variation of the coefficients. After rescaling
the equation, its integral form allowed us to estimate the local energy by
iterating in time, leading to an exponential decay with improved rates.

Our results complement those of algebraic time-decay presented in [4], where
the initial data were assumed to be sufficiently localised but not compactly
supported. In that case, exponential decay is generally not expected.

## Appendix

######  Lemma A.1.

For any fixed $\gamma_{0}>0$, the equation

(45)
$\tanh\left({\sqrt{\gamma_{0}^{2}+\lambda^{2}_{*}}}\right)=\frac{\sqrt{\gamma_{0}^{2}+\lambda^{2}_{*}}}{\gamma_{0}+\lambda_{*}}$

for $\lambda_{*}>0$ has a unique solution. Moreover, the corresponding strict
inequality (30) holds true for $\lambda\in[0,\lambda_{*})$.

###### Proof.

Since the right-hand side of (45), for $\lambda_{*}>0$, is smaller than 1,
equation (45) is equivalent to

(46)
$f_{\gamma_{0}}(\tilde{\lambda}):=\gamma_{0}\sqrt{1+\tilde{\lambda}^{2}}-\operatorname{artanh}\left({\frac{\sqrt{1+\tilde{\lambda}^{2}}}{1+\tilde{\lambda}}}\right)=0,$

where $\tilde{\lambda}:=\lambda_{*}/\gamma_{0}$. We have

$f_{\gamma_{0}}^{\prime}(\tilde{\lambda})=\frac{2\gamma_{0}\tilde{\lambda}^{2}-\tilde{\lambda}+1}{2\tilde{\lambda}\sqrt{1+\tilde{\lambda}^{2}}},\qquad\lim_{\tilde{\lambda}\to+\infty}f^{\prime}_{\gamma_{0}}(\tilde{\lambda})=\gamma_{0}>0.$

Hence
$\lim_{\tilde{\lambda}\to+\infty}f_{\gamma_{0}}(\tilde{\lambda})=+\infty$ and,
moreover, $\lim_{\tilde{\lambda}\to
0^{+}}f_{\gamma_{0}}(\tilde{\lambda})=-\infty$.

Case 1: For $\gamma_{0}\geq 1/8$, we have
$f^{\prime}_{\gamma_{0}}(\tilde{\lambda})\geq 0$ for any $\tilde{\lambda}>0$.
Hence, $f_{\gamma_{0}}$ is strictly increasing and $f_{\gamma_{0}}$ has a
unique zero at some $\tilde{\lambda}_{*}>0$ and
$f_{\gamma_{0}}(\tilde{\lambda})<0$ for
$\tilde{\lambda}\in(0,\tilde{\lambda}_{*})$.

Case 2: For $\gamma_{0}\in(0,1/8)$, we consider
$f_{\gamma_{0}}(\tilde{\lambda}_{1})$, where
$\tilde{\lambda}_{1}:=\frac{1-\sqrt{1-8\gamma_{0}}}{4\gamma_{0}}$ is the first
zero of $f_{\gamma_{0}}^{\prime}$. Define
$g(\gamma_{0}):=\frac{1-\sqrt{1-8\gamma_{0}}}{4\gamma_{0}}$. From
$g^{\prime}(\gamma_{0})>0$ for all $\gamma_{0}\in(0,1/8)$,
$\lim_{\gamma_{0}\to 0^{+}}g(\gamma_{0})=1$, and $g(1/8)=2$, we deduce that
$g$ is strictly increasing and $\tilde{\lambda}_{1}=g(\gamma_{0})\in(1,2)$ for
all $\gamma_{0}\in(0,1/8)$. Since
$\operatorname{artanh}\left({\frac{\sqrt{1+\tilde{\lambda}^{2}}}{1+\tilde{\lambda}}}\right)$
is an increasing function of $\tilde{\lambda}$ in $(1,2)$, we obtain from (46)
that
$f_{\gamma_{0}}(\tilde{\lambda}_{1})\leq\frac{\sqrt{1+4}}{8}-\operatorname{artanh}\left(\frac{\sqrt{2}}{2}\right)\simeq-0.6019$.
This shows that $f_{\gamma_{0}}(\tilde{\lambda}_{1})<0$ for
$\gamma_{0}\in(0,1/8)$. Hence, $f_{\gamma_{0}}$ also has a unique zero
$\tilde{\lambda}_{*}{>\tilde{\lambda}_{2}}>0$ in this case, where
$\tilde{\lambda}_{2}:=\frac{1+\sqrt{1-8\gamma_{0}}}{4\gamma_{0}}$ is the
second zero of $f_{\gamma_{0}}^{\prime}$.

Using the strict monotonicity of the $\tanh$ function, this implies that (30)
holds true for $\lambda\in[0,\lambda_{*})$ (in both cases). ∎

######  Lemma A.2.

The function

(47)
$\lambda_{0}(\gamma_{0}):=\sqrt{2}\gamma_{0}e^{-2\gamma_{0}}\sqrt{1-4\gamma_{0}e^{-4\gamma_{0}}}$

satisfies $\lambda_{0}(\gamma_{0})<\lambda_{*}(\gamma_{0})$, where
$\lambda_{*}(\gamma_{0})$ is defined via (31). Moreover, each
$\lambda=\lambda_{0}(\gamma_{0})$ satisfies inequality (30).

###### Proof.

To show that $\lambda{:=}\lambda_{0}(\gamma_{0})$ satisfies (30) we first
derive the following auxiliary inequality:

$\displaystyle\sqrt{1+\left(\lambda/\gamma_{0}\right)^{2}}$
$\displaystyle<1+\frac{1}{2}\left(\lambda/\gamma_{0}\right)^{2}=1+e^{-4\gamma_{0}}\left(1-4\gamma_{0}e^{-4\gamma_{0}}\right)$
$\displaystyle<1+\dfrac{e^{4\gamma_{0}}+4\gamma_{0}-e^{2\gamma_{0}}\sqrt{e^{4\gamma_{0}}+8\gamma_{0}}}{8\gamma_{0}^{2}},$

where, in the first two steps, we used $\sqrt{1+x}<1+\frac{x}{2},\,x>0$ and
definition (47), respectively. In the last step, we used the estimate
$\sqrt{1+x}<1+x/2-x^{2}/8+x^{3}/16$, with $x=8\gamma_{0}e^{-4\gamma_{0}}>0$,
which follows from the Taylor expansion of the square root.

The first and last term in this chain of inequalities form exactly the
assumption of Lemma A.3 below with
$q:=\sqrt{1+\left(\lambda/\gamma_{0}\right)^{2}}>1$ and $a:=\gamma_{0}$. The
assertion of Lemma A.3 is equivalent to the validity of (30).

Finally, Lemma A.1 also yields
$\lambda_{0}(\gamma_{0})<\lambda_{*}(\gamma_{0})$ ∎

######  Lemma A.3.

For any $a>0$ and $1<q<1+\dfrac{e^{4a}+4a-e^{2a}\sqrt{e^{4a}+8a}}{8a^{2}}$, we
have

(48) $\frac{q}{\tanh\left(aq\right)}>1+\sqrt{q^{2}-1}.$

###### Proof.

The proof is split into two steps.
Step 1. We shall first analyse for which values $q_{0}\geq 1$ the following
auxiliary inequality holds:

(49) $\left(1+2a\right)e^{-2a}-2ae^{-2a}q_{0}^{2}\geq\sqrt{q_{0}^{2}-1}.$

Using $Q:=q_{0}^{2}\geq 1$ we rewrite (49) as

(50) $1+2a-2aQ\geq e^{2a}\sqrt{Q-1}.$

This is equivalent to

(51)
$4a^{2}Q^{2}-\left(4a\left(1+2a\right)+e^{4a}\right)Q+\left(1+2a\right)^{2}+e^{4a}\geq
0,$

as long as both sides of (50) are nonnegative, i.e. for
$Q\in\left[1,\frac{1+2a}{2a}\right]$.

The left-hand side of (51) is a quadratic expression (in $Q$) which is
positive for $Q\in\mathbb{R}\backslash\left[Q_{1},Q_{2}\right]$, with

$\displaystyle Q_{1}:=Q_{0}-\frac{e^{2a}\sqrt{e^{4a}+8a}}{8a^{2}},\hskip
10.00002ptQ_{2}:=Q_{0}+\frac{e^{2a}\sqrt{e^{4a}+8a}}{8a^{2}},$ $\displaystyle
Q_{0}:=1+\frac{e^{4a}+4a}{8a^{2}}.$

It can be checked easily that $Q_{1}\in\left(1,\frac{1+2a}{2a}\right)$, and
therefore any $Q\in\left[1,Q_{1}\right]$ satisfies (50). Consequently, any
$q_{0}\in\left[1,\sqrt{Q_{1}}\right]$ satisfies (49).

Step 2. We observe that since $e^{-bx}\geq e^{-b}\left(1+b-bx\right)$ for any
$b,\,x\in\mathbb{R}$ (by convexity of $e^{-bx}$), we have (with $b:=2a$,
$x:=q_{0}$)

$\displaystyle\frac{q_{0}}{\tanh\left(aq_{0}\right)}=$
$\displaystyle\frac{q_{0}\left(e^{aq_{0}}+e^{-aq_{0}}\right)}{e^{aq_{0}}-e^{-aq_{0}}}>q_{0}\left(1+e^{-2aq_{0}}\right)$
$\displaystyle\geq$ $\displaystyle
q_{0}\left[1+e^{-2a}\left(1+2a-2aq_{0}\right)\right]$ $\displaystyle\geq$
$\displaystyle 1+\left(1+2a\right)e^{-2a}-2ae^{-2a}q_{0}^{2}.$

Therefore, every $q_{0}>1$ satisfying (49) also satisfies (48). ∎

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