# Regularity results for degenerate wave equations in a neighborhood of the
boundary

B. S. V. Araújo<EMAIL_ADDRESS>R. Demarque<EMAIL_ADDRESS>L.
Viana<EMAIL_ADDRESS>Unidade Acadêmica de Matemática, Universidade
Federal de Campina Grande, Campina Grande, PB, Brazil  Departamento de
Ciências da Natureza, Universidade Federal Fluminense, Rio das Ostras, RJ,
Brazil  Departamento de Análise, Universidade Federal Fluminense, Niterói, RJ,
Brazil

###### Abstract

In this paper we establish some regularity results concerning the behavior of
weak solutions and very weak solutions of the degenerate wave equation near
the boundary. For the nondegenerate case, the correponding results were
originally obtained by Fabre and Puel (J. of Diff. Eq. 106, 1993). This kind
of results is closely related to the exact boundary controllability for the
wave equation as the limit of internal controllability.

###### keywords:

Degenerate wave equation, regularity of solutions, behavior near the boundary.

###### MSC:

[2020]35L05, 35L80, 35B65, 35B40

††journal: Arxiv

blc

##  1 Introduction and Statements of the main results

In this work we are interested in studying the behavior, near the boundary
point $x=1$, of the weak and very weak solutions of the following degenerate
wave equation:

$\begin{cases}u_{tt}-\displaystyle\left(x^{\alpha}u_{x}\right)_{x}=f,&(t,x)\in
Q,\\\ u(t,1)=0,&\text{ in }(0,T),\\\ \begin{cases}u(t,0)=0,&\text{if
}\alpha\in(0,1),\\\ \text{or}&\\\ (x^{\alpha}u_{x})(t,0)=0,&\text{if
}\alpha\in[1,2),\end{cases}&t\in(0,T),\\\ u(0,x)=u_{0}(x)\text{ and
}u_{t}(0,x)=u_{1}(x)&x\in(0,1),\end{cases}$ (1.1)

where $T>0$, $Q=(0,T)\times(0,1)$, $\alpha\in(0,2)$ and the data
$(f,u_{0},u_{1})$ belongs to spaces that will determine the regularity of the
solution. To develop this study, the $L^{2}$ norm of the solution will be
analyzed in an $\varepsilon$-neighborhood of the boundary point $x=1$.

For the nondegenerate wave equation, an analogous investigation has been
considered by Fabre and Puel in [6]. Their results have played a key role in
[5], where an exact boundary controllability is achieved as the limit of a
sequence of internal controllability problems, set in $\varepsilon$\-
neighborhoods of the boundary, as $\varepsilon\to 0$. Here, we are supposed to
remark that the previous work arises from [9], where Zuazua has used Lion’s
Hilbert uniqueness method (HUM) to assure the exact controllability for the
nondegenerate wave equation when the distributed control acts on an
$\varepsilon$-neighborhood $\omega_{\varepsilon}$ of
$\Gamma_{0}\subset\Gamma$, where $\Gamma$ denotes the boundary of the domain.
So that, in [5], a passage to the limit procedure can be established, by
studying the convergence of solutions of

$\psi_{\varepsilon
tt}-\Delta\psi_{\varepsilon}=\frac{1}{\varepsilon^{3}}{\varphi}_{\varepsilon}\chi_{\omega_{\varepsilon}\times(0,T)},\text{
in }Q,$ (1.2)

where there exists a singular right-hand side with respect to $\varepsilon$
and $\frac{1}{\varepsilon^{3}}{\varphi}_{\varepsilon}$ is the control given by
HUM as a solution of an adjoint problem.

Recently, in [4], Chaves-Silva, Puel and Santos have extended the research
contained in [5] for the nondegenerate heat equation context. Shortly
afterwards, in [3], we have proposed a natural continuation for the subject of
[4] and [5], taking into consideration the one-dimensional degenerate heat
equation. At this point, asking ourselves if those results presented in [3, 4,
5] can be proved for the one-dimensional degenerate wave equation seems a
realistic sequel.

Concerning the nondegenerate heat and wave equations, we would also contrast
the following facts: even though [4] and [5] communicate similar theorems,
their obtainment does not come from the same techniques. This is really
expected, since, in the whole Control Theory, parabolic and hyperbolic
equations are usually dealt with different approaches. Both problems are
solved using a bound for the $L^{2}$-norms of weak solutions in
$\varepsilon$-neighborhoods of the boundary. However, in [4], the desired
bound is achieved from specific weighted Carleman estimates, while, in [5], it
can be found relying on the regularity results presented in [6]. In fact,
while passing to the limit, such a regularity results are crucial in order to
check the continuity of a certain functional. As pointed out by Fabre in [6],
equations like (1.2) can be stated regardless the exact controllability
context, which means that the mentioned regularity results can be useful in
other more general situations. Having in mind this initial discussion, before
trying to extend [3] for the degenerate wave equation context, our analysis
starts from this current paper, where we will obtain regularity theorems which
are analogous to that ones proved in [6].

Next, before stating our main results, let us present some important
functional spaces, introduced in [1].

######  Definition 1.1 (Weighted Sobolev spaces).

Consider $\alpha\in(0,1)$, for the weakly degenerate case (WDC), or
$\alpha\in[1,2)$, for the strongly degenerate case (SDC).

  * (I)

For the (WDC), we set

$H_{\alpha}^{1}:=\\{u\in L^{2}(0,1);\ u\mbox{ is absolutely continuous in
}[0,1],x^{\alpha/2}u_{x}\in L^{2}(0,1)\mbox{ and }u(1)=u(0)=0\\},$

equipped with the natural norm

$\|u\|_{H_{\alpha}^{1}}:=\left(\|u\|_{L^{2}(0,1)}^{2}+\|x^{\alpha/2}u_{x}\|_{L^{2}(0,1)}^{2}\right)^{1/2};$

  * (II)

For the (SDC),

$H_{\alpha}^{1}:=\\{u\in L^{2}(0,1);\ u\text{ is {locally} absolutely
continuous in }(0,1],x^{\alpha/2}u_{x}\in L^{2}(0,1)\mbox{ and }u(1)=0\\},$

and the norm keeps the same;

  * (III)

In both situations, the (WDC) and the (SDC),

$H_{\alpha}^{2}:=\\{u\in H_{\alpha}^{1};\ x^{\alpha/2}u_{x}\in H^{1}(0,1)\\}$

with the norm
$\|u\|_{H_{\alpha}^{2}}:=\left(\|u\|_{H_{\alpha}^{1}}^{2}+\|(x^{\alpha/2}u_{x})_{x}\|_{L^{2}(0,1)}^{2}\right)^{1/2}$.

Another important space in this context is
$H_{\alpha}^{-1}=(H_{\alpha}^{1})^{\prime}$ (the dual space of
$H_{\alpha}^{1}$). For $u\in H_{\alpha}^{-1}$, from Lax-Milgram Theorem, there
exists a unique $\tilde{u}\in H_{\alpha}^{1}$ such that

$\langle
u,v\rangle_{H_{\alpha}^{-1}}=\int_{0}^{1}x^{\alpha}\tilde{u}_{x}v_{x}\,dx\ \
\forall v\in H_{\alpha}^{1}.$

Hence, $H_{\alpha}^{-1}$ is a Hilbert space equipped with the inner product

$(u,v)_{H_{\alpha}^{-1}}=\int_{0}^{1}x^{\alpha}\tilde{u}_{x}\tilde{v}_{x}\,dx.$

Next, let us specify which kind of solution for (1.1) we will deal with.

######  Definition 1.2.

Given $f\in L^{1}(0,T;L^{2}(0,1))$ and $(u_{0},u_{1})\in H^{1}_{\alpha}\times
L^{2}(0,1)$, we say that

$\displaystyle u\in C([0,T];H^{1}_{\alpha})\cap C^{1}([0,T];L^{2}(0,1))$

is a weak solution of the system (1.1) if the following properties hold:

  * (a)

$u(0,x)=u_{0}(x)$ for all $x\in(0,1)$;

  * (b)

$\int_{0}^{T}\int_{0}^{1}(-u_{t}\varphi_{t}+x^{\alpha}u_{x}\varphi_{x})\,dxdt-\int_{0}^{1}u_{1}\varphi(0,x)\,dx=\int_{0}^{T}\int_{0}^{1}f\varphi\,dxdt,$

for all $\varphi\in L^{2}(0,T;H^{1}_{\alpha})$ satisfying $\varphi_{t}\in
L^{2}(Q)$ and $\varphi(T,\cdot)=0$.

At this moment, we are to state our two main results:

######  Theorem 1.3.

Given $0<\varepsilon_{0}<1$, there exists $C>0$ such that, for all
$(u_{0},u_{1})\in H_{\alpha}^{1}\times L^{2}(0,1)$ and $f\in
L^{1}(0,T;L^{2}(0,1))$, if $u$ is a weak solution to (1.1), then

$\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|u(t,x)|^{2}\,dxdt\leq
C\left(\|f\|_{{L^{1}(0,T;L^{2}(0,1))}}^{2}+\|u_{0}\|_{{H^{1}_{\alpha}}}^{2}+\|u_{1}\|_{{L^{2}(0,1)}}^{2}\right),\
\forall\varepsilon\in(0,\varepsilon_{0}],$

where $C$ only depends on $\varepsilon_{0}$, $\alpha$ and $T$.

######  Theorem 1.4.

Given $0<\varepsilon_{0}<1$, there exists $C>0$ such that, for all
$(u_{0},u_{1})\in H_{\alpha}^{1}\times L^{2}(0,1)$ and $f\in
L^{1}(0,T;L^{2}(0,1))$, if $u$ is a weak solution to (1.1), then

$\frac{1}{\varepsilon}\int_{0}^{T}\int_{1-\varepsilon}^{1}|u_{x}(t,x)|^{2}x^{\alpha}\,dxdt\leq
C\left(\|f\|_{{L^{1}(0,T;L^{2}(0,1))}}^{2}+\|u_{0}\|_{{H^{1}_{\alpha}}}^{2}+\|u_{1}\|_{{L^{2}(0,1)}}^{2}\right),\
\forall\varepsilon\in(0,\varepsilon_{0}],$

where $C$ only depends on $\varepsilon_{0}$, $\alpha$ and $T$.

We know that a weak solution $u$ of problem (1.1) satisfies $u_{x}(\cdot,1)\in
L^{2}(0,T)$, see Proposition 2.9. However, if we take $(u_{0},u_{1})\in
L^{2}(0,1)\times H^{-1}_{\alpha}$ (instead of taking it in
$H^{1}_{\alpha}\times L^{2}(0,1)$), we need to consider solutions
$\tilde{u}=\tilde{u}(t,x)$ in a very weak sense. In this case, we have no
information about the regularity of $\tilde{u}_{x}(\cdot,1)$.

In our next result, as in the nondegenerate case, we will provide this kind of
regularity for a function $\varphi$ that is the limit of a sequence
$(\varphi_{\varepsilon})$ of very weak solutions, under a condition of the
$L^{2}$ norms of $(\varphi_{\varepsilon})$ near the boundary. In the
following, we will precise this result.

Given $(g,z^{0},z^{1})\in L^{1}(0,T;L^{2}(0,1))\times L^{2}(0,1)\times
H^{-1}_{\alpha}$, let us consider the problem

$\begin{cases}z_{tt}-\displaystyle\left(x^{\alpha}z_{x}\right)_{x}=g,&(t,x)\in
Q,\\\ z(t,1)=0,&\text{ in }(0,T),\\\ \begin{cases}z(t,0)=0,&\text{if
}\alpha\in(0,1),\\\ \text{or}&\\\ (x^{\alpha}z_{x})(t,0)=0,&\text{if
}\alpha\in[1,2),\end{cases}&t\in(0,T),\\\ z(0)=z^{0}\text{ and
}z_{t}(0)=z^{1}.\end{cases}$ (1.3)

In the following, we will present the definition of solutions for this
problems that can be found in [7, page 47].

######  Definition 1.5.

Given $g\in L^{1}(0,T;L^{2}(0,1))$ and $(z^{0},z^{1})\in L^{2}(0,1)\times
H^{-1}_{\alpha}$, we say $z\in L^{\infty}(0,T;L^{2}(0,1))$ is a very weak
solution (or a solution by transposition) of (1.3) if, for each
$F\in\mathcal{D}(Q)$,

$\int_{0}^{T}\int_{0}^{1}uF\,dxdt=-(z_{0},\theta^{\prime}(0))+\langle
z_{1},\theta(0)\rangle+\int_{0}^{T}\int_{0}^{1}g\theta\,dxdt,$

where $\theta=\theta(t,x)$ solves

$\begin{cases}\theta_{tt}-\displaystyle\left(x^{\alpha}\theta_{x}\right)_{x}=F,&(t,x)\in
Q,\\\ \theta(t,1)=0,&\text{ in }(0,T),\\\ \begin{cases}\theta(t,0)=0,&\text{if
}\alpha\in(0,1),\\\ \text{or}&\\\ (x^{\alpha}\theta_{x})(t,0)=0,&\text{if
}\alpha\in[1,2),\end{cases}&t\in(0,T),\\\
{\theta(T,x)=\theta_{t}(T,x)=0}&x\in(0,1).\end{cases}$

Above, $\mathcal{D}(Q)$ denotes the real vector space of all smooth and
compactly supported functions defined on $Q$.

For the well-posedness of (1.3), in the sense of transposition, see
Proposition 2.10. At this point, it is important to mention that one of the
main differences between weak and very weak solutions are the following: the
weak solution has the so called “hidden regularity” (Proposition 2.9), while
this property is not true for the very weak solution.

Our next result plays like a reciprocal of Theorem 1.4, but, in fact, it is
not really. Indeed, Theorem 1.6 concerns about a sequence of very weak
solutions $\varphi_{n}$ that converges (in some weak sense) to a very weak
solution $\varphi$. If this sequence satisfies a bound near the boundary,
similar to that one got in the previous Theorem 1.4, then the very weak
solution $\varphi$ has a hidden regularity.

Let us consider a family of functions
$(h_{\varepsilon},\varphi^{0}_{\varepsilon},\varphi^{1}_{\varepsilon})\in
L^{1}(0,T:L^{2}(\Omega))\times L^{2}(\Omega)\times H^{-1}_{\alpha}$ such that

$\displaystyle h_{\varepsilon}\rightharpoonup h$ $\displaystyle\text{ in
}L^{1}(0,T:L^{2}(\Omega)),$
$\displaystyle\varphi_{\varepsilon}^{0}\rightharpoonup\varphi^{0}$
$\displaystyle\text{ in }L^{2}(\Omega),$
$\displaystyle\varphi_{\varepsilon}^{1}\rightharpoonup\varphi^{1}$
$\displaystyle\text{ in }H^{-1}_{\alpha},$

and let $\varphi_{\varepsilon}$ be the solution by transposition of problem
(1.3) with
$(g,z^{0},z^{1})=(h_{\varepsilon},\varphi_{\varepsilon}^{0},\varphi_{\varepsilon}^{1})$.
Then $\varphi_{\varepsilon}\in C^{0}([0,T];L^{2}(0,1))\cap
C^{1}([0,T];H^{-1}_{\alpha})$ and
$\varphi_{\varepsilon}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}\varphi$
in $L^{\infty}(0,T;L^{2}(0,1))$, where $\varphi$ is the solution by
transposition of (1.3) with $(g,z^{0},z^{1})=(h,\varphi^{0},\varphi^{1})$.

######  Theorem 1.6.

Let $(h_{\varepsilon},\varphi^{0}_{\varepsilon},\varphi^{1}_{\varepsilon})\in
L^{1}(0,T:L^{2}(\Omega))\times L^{2}(0,1)\times H^{-1}_{\alpha}$ be a family
of functions as described above. If

$\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|\varphi_{\varepsilon}(t,x)|^{2}\,dxdt\leq
C,$ (1.4)

where $C$ does not depend on $\varepsilon$, then $\varphi_{x}(\cdot,1)\in
L^{2}(0,T)$ and

$\frac{1}{3}\|\varphi_{x}(\cdot,1)\|_{{L^{2}(0,T)}}^{2}\leq\liminf_{\varepsilon\to
0^{+}}\left(\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|\varphi_{\varepsilon}(t,x)|^{2}\,dxdt\right)$
(1.5)

######  Remark 1.7.

In this paper, we consider weakly and strongly degenerate wave equations, with
Dirichlet and Neumann boundary conditions, respectively. However, we should
emphasize that these two different situations can be treated simultaneously.
Indeed, each integration by parts provides boundary terms which vanish in both
cases at $x=0$, while the boundary condition is the same at $x=1$.

The remainder of this paper is organized as follows: in Section 2 we present
some well posedness results concerning the weak solution and the very weak
solution of (1.1). The proof of the Theorems 1.3 and 1.4 are discussed in
Section 3. The paper ends with the Section 4, where we present the proof of
Theorem 1.6.

##  2 Preliminaries

As expected from a paper like this, we start presenting some well-posedness
results for the hiperbolic degenerate equation (1.1). Most of these results
are well known and can be found in [2] or in [8]. The first result is
concerning the weak solution for (1.1) and it was established in [2] using a
semigroup approach.

######  Proposition 2.8.

Given $f\in L^{1}(0,T;L^{2}(0,1))$ and $(u_{0},u_{1})\in H^{1}_{\alpha}\times
L^{2}(0,1)$, there exists a unique weak solution $u\in
C^{0}([0,T];H^{1}_{\alpha})\cap C^{1}([0,T];L^{2}(0,1))$ of (1.1). In
addition, there exists a positive constant $C_{T,\alpha}$ such that

$\sup_{t\in[0,T]}\left(\|u_{t}(t)\|_{{L^{2}(0,1)}}^{2}+\|u(t)\|_{{H_{\alpha}^{1}}}^{2}\right)\leq
C_{T,\alpha}\left(\|f\|_{{L^{1}(0,T;L^{2}(0,1))}}^{2}+\|u_{0}\|_{{H_{\alpha}^{1}}}^{2}+\|u_{1}\|_{{L^{2}(0,1)}}^{2}\right).$
(2.1)

Associated to (1.1), we have the following energy functional

$E(t):=\frac{1}{2}\int_{0}^{1}(|u_{t}(t,x)|^{2}+x^{\alpha}|u_{x}(t,x)|^{2})\,dx,$

where $t\in(0,T)$. The previous result establishes that

$E(t)\leq C[\|f\|_{{L^{1}(0,T;L^{2}(0,1))}}^{2}+E(0)],$

a expected fact for hiperbolic equations. The next result is known by “hidden
regularity”, another inherited property from the hiperbolic equations. Like
the previous one, this results was also discussed in [2] and [8].

######  Proposition 2.9.

For any weak solution $u$ of (1.1), we have $u_{x}(\cdot,1)\in L^{2}(0,T)$ and

$\int_{0}^{T}|u_{x}(t,1)|^{2}\,dt\leq
C\left(\|f\|_{L^{1}(0,T;L^{2}(0,1))}^{2}+E(0)\right)$ (2.2)

###### Proof.

See [8, Proposition 2.5]. ∎

Now we will discuss the very weak solution for (1.1). The way to obtain a well
posedness result with less regular data $(z^{0},z^{1})\in L^{2}(0,1)\times
H^{-1}_{\alpha}$ is very similar to that used in [2]. Despite that, this
result was not analyzed in the papers that we read. So, we will give just a
sketch of the proof. First of all, for $u\in H_{\alpha}^{1}$ we define
$-(x^{\alpha}u_{x})_{x}\in H_{\alpha}^{-1}$ by

$\langle-(x^{\alpha}u_{x})_{x},v\rangle_{H_{\alpha}^{-1}}=\int_{0}^{1}x^{\alpha}u_{x}v_{x}\,dx\
\ \forall v\in H_{\alpha}^{1}.$

Then we define the Hilbert space $Y=L^{2}(0,1)\times H_{\alpha}^{-1}$ and the
operator $B:D(B)\longrightarrow Y$ given by

$B(u,v)=(-v,-(x^{\alpha}u_{x})_{x}),$

where $D(B)=H_{\alpha}^{1}\times L^{2}(0,1)\subset Y$. It is not difficult to
see that $(B(U),U)_{Y}=0\ \forall U\in D(B)$. In particular, $B$ is a
accretive operator. Furthermore, from Lax Milgram Theorem we can deduce that
$B$ is M-accretive. It follows that $B$ is skew-adjoint. From semigroup theory
we deduce that $B$ is a generator of a semigroup of contractions and this lead
us to the following well posedness result:

######  Proposition 2.10.

Given $g\in L^{1}(0,T;L^{2}(0,1))$ and $(z^{0},z^{1})\in L^{2}(0,1)\times
H^{-1}_{\alpha}$, there exists a unique solution by transposition $z\in
C^{0}([0,T];L^{2}(0,1))\cap C^{1}([0,T];H^{-1}_{\alpha})$ of (1.3). In
addition, there exists a positive constant $C_{T,\alpha}$ such that

$\sup_{t\in[0,T]}\left(\|z(t)\|_{{L^{2}(0,1)}}^{2}+\|z_{t}(t)\|_{{H_{\alpha}^{-1}}}\right)\leq
C_{T,\alpha}\left(\|g\|_{{L^{1}(0,T;L^{2}(0,1))}}^{2}+\|z^{1}\|_{{H_{\alpha}^{-1}}}^{2}+\|z^{0}\|_{{L^{2}(0,1)}}^{2}\right).$
(2.3)

##  3 Regularity for weak solutions

The aim of this section is to prove Theorems 1.3 and 1.4. They bring $L^{2}$
estimates for weak solutions of system (1.1) and its derivative, in a
$\varepsilon-$neighborhood of the boundary $x=1$.

We observe that Theorem 1.3 will be obtained as a consequence of Theorem 1.4.
First, let us introduce a suitable function that will be used to prove Theorem
1.4.

Given $\delta>0$ and $\gamma\in(0,\delta)$, for $\kappa=\delta+\gamma$, define
$\rho_{\kappa}:[0,1]\to\mathbb{R}$ by

$\rho_{\kappa}(x)=\begin{cases}0,&0\leq x\leq 1-\kappa;\\\
\frac{1}{2\delta\gamma}(x-(1-\kappa))^{2},&1-\kappa<x<1-\delta;\\\
\frac{1}{\delta}(x-1)+1+\frac{\gamma}{2\delta},&1-\delta\leq x\leq
1.\end{cases}$ (3.1)

$x$$1-\kappa$$1-\delta$$\frac{\gamma}{2\delta}$$1$$1+\frac{\gamma}{2\delta}$
Figure 1: Graph of $\rho_{\kappa}$.

Note that $\rho_{\kappa}\in W^{2,\infty}(0,1)$ and, furthermore

  1. $(i)$

$\rho_{\kappa}$ is a non-decreasing function;

  2. $(ii)$

$\rho_{\kappa}=0$ in $(0,1-\kappa]$, $\rho_{\kappa}\geq 0$ and
$\|\rho_{\kappa}^{\prime}\|_{{L^{\infty}(0,1)}}\leq\displaystyle\frac{2}{\kappa}$;

  3. $(iii)$

$\rho_{\kappa}^{\prime}=\displaystyle\frac{1}{\delta}$ in $(1-\delta,1)$,
$\rho_{\kappa}^{\prime\prime}=\displaystyle\frac{1}{\delta\gamma}$ in
$(1-\varepsilon,1-\delta)$ and $\rho_{\kappa}^{\prime\prime}=0$ in
$(1-\delta,1)$.

###### Proof of Theorem 1.4.

Define

$G(\varepsilon):=\frac{1}{\varepsilon}\int_{0}^{T}\int_{1-\varepsilon}^{1}|u_{x}(t,x)|^{2}x^{\alpha}\,dxdt,\forall\varepsilon\in(0,\varepsilon_{0}]$

and

$G(0)=\int_{0}^{T}|u_{x}(t,1)|^{2}\,dt.$

It is sufficient to prove that $G(\varepsilon)\leq CN_{0}$,
$\forall\varepsilon\in[0,\varepsilon_{0}]$, where

$N_{0}=\|f\|_{{L^{1}(0,T;L^{2}(0,1))}}^{2}+\|u_{0}\|_{{H^{1}_{\alpha}}}^{2}+\|u_{1}\|_{{L^{2}(0,1)}}^{2}.$

Since $G:[0,\varepsilon_{0}]\longrightarrow\mathbb{R}$ is a continuous
function, there exists $\delta_{0}\in[0,\varepsilon_{0}]$ such that

$G(\delta_{0})=\max\\{G(\varepsilon);\varepsilon\in[0,\varepsilon_{0}]\\}.$

Next, we divide the proof into three situations.

Situation 1: $\delta_{0}=0$.

In this case, we directly apply Proposition 2.9 to obtain

$G(\varepsilon)\leq G(0)=\int_{0}^{T}|u_{x}(t,1)|^{2}\,dt\leq CN_{0},\
\forall\varepsilon\in[0,\varepsilon_{0}].$

Situation 2: $\delta_{0}\geq\varepsilon_{0}/2$.

Here, applying Proposition 2.8, we get

$\displaystyle G(\varepsilon)\leq G(\delta_{0})$
$\displaystyle=\frac{1}{\delta_{0}}\int_{0}^{T}\int_{1-\delta_{0}}^{1}|u_{x}(t,x)|^{2}x^{\alpha}\,dxdt\leq\frac{2}{\varepsilon_{0}}\int_{0}^{T}\int_{0}^{1}|u_{x}(t,x)|^{2}x^{\alpha}\,dxdt$
$\displaystyle\leq\frac{2T}{\varepsilon_{0}}\sup_{[0,T]}\|u(t,\cdot)\|_{{H_{\alpha}^{1}}}^{2}\leq
CN_{0}.$

Situation 3: $0<\delta_{0}<\varepsilon_{0}/2$.

The proof of this last case is more delicate. For $\delta=\delta_{0}$, take
$\kappa=\delta+\gamma$, where $\gamma\in(0,\delta)$ will be precise later, and
$\rho_{\kappa}$ is given in (3.1). We observe that

$2G(\delta_{0})=2\int_{0}^{T}\int_{1-\delta_{0}}^{1}\frac{1}{\delta_{0}}|u_{x}|^{2}x^{\alpha}\,dxdt=2\int_{0}^{T}\int_{1-\delta_{0}}^{1}\rho_{\kappa}^{\prime}|u_{x}|^{2}x^{\alpha}\,dxdt\leq
2\int_{0}^{T}\int_{1-\kappa}^{1}\rho_{\kappa}^{\prime}|u_{x}|^{2}x^{\alpha}\,dxdt.$
(3.2)

We will see that the last integral is bounded by
$\frac{3}{2}G(\delta_{0})+CN_{0}$, in order to obtain the desired estimate.
This goal will be achieved by the multiplier method.

Firstly, multiply the equation in (1.1) by $2\rho_{\kappa}u_{x}$ and
integrating over $Q$. Using integration by parts and recalling that
$\rho_{\kappa}=0$ in $[0,1-\kappa]$, we have

$-\int_{0}^{T}\int_{1-\kappa}^{1}2u_{t}u_{tx}\rho_{\kappa}\,dxdt+\int_{1-\kappa}^{1}2u_{t}(T)u_{x}(T)\rho_{\kappa}\,dx-\int_{1-\kappa}^{1}2u_{1}u_{0x}\rho_{\kappa}\,dx+\int_{0}^{T}\int_{1-\kappa}^{1}2x^{\alpha}u_{x}^{2}\rho_{\kappa}^{\prime}\,dxdt\\\
+\int_{0}^{T}\int_{1-\kappa}^{1}2x^{\alpha}u_{x}u_{xx}\rho_{\kappa}\,dxdt-\int_{0}^{T}2u_{x}^{2}(t,1)\rho_{\kappa}(1)\,dt=\int_{0}^{T}\int_{0}^{1}2fu_{x}\rho_{\kappa}\,dxdt.$

Since $\rho_{\kappa}(1-\kappa)=0$ and $u_{t}(t,1)=0$, we have

$\displaystyle-\int_{0}^{T}\int_{1-\kappa}^{1}2u_{t}u_{tx}\rho_{\kappa}\,dxdt=$
$\displaystyle-\int_{0}^{T}\int_{1-\kappa}^{1}\frac{d}{dx}(u_{t}^{2})\rho_{\kappa}\,dxdt=\int_{0}^{T}\int_{1-\kappa}^{1}u_{t}^{2}\rho_{\kappa}^{\prime}\,dxdt-\int_{0}^{T}u_{t}^{2}\rho_{\kappa}\big{|}_{x=1-\kappa}^{1}\,dt$
$\displaystyle=\int_{0}^{T}\int_{1-\kappa}^{1}u_{t}^{2}\rho_{\kappa}^{\prime}\,dxdt.$

Similarly,

$\displaystyle\int_{0}^{T}\int_{1-\kappa}^{1}2x^{\alpha}u_{x}u_{xx}\rho_{\kappa}\,dxdt=-\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}u_{x}^{2}\rho_{\kappa}^{\prime}\,dxdt+\int_{0}^{T}u_{x}^{2}(t,1)\rho_{\kappa}(1)\,dt-\int_{0}^{T}\int_{1-\kappa}^{1}\alpha
x^{\alpha-1}u_{x}^{2}\rho_{\kappa}\,dxdt.$

Combining these last three identities, we get

$\int_{0}^{T}\int_{1-\kappa}^{1}(u_{t}^{2}-x^{\alpha}u_{x}^{2})\rho_{\kappa}^{\prime}\,dxdt+\int_{0}^{T}\int_{1-\kappa}^{1}2x^{\alpha}u_{x}^{2}\rho_{\kappa}^{\prime}\,dxdt\\\
=\int_{0}^{T}u_{x}^{2}(t,1)\rho_{\kappa}(1)\,dt+\int_{0}^{T}\int_{1-\kappa}^{1}\alpha
x^{\alpha-1}u_{x}^{2}\rho_{\kappa}\,dxdt+\int_{0}^{T}\int_{0}^{1}2fu_{x}\rho_{\kappa}\,dxdt\\\
-\int_{1-\kappa}^{1}2u_{t}(T)u_{x}(T)\rho_{\kappa}\,dx+\int_{1-\kappa}^{1}2u_{1}u_{0x}\rho_{\kappa}\,dx.$
(3.3)

Analogously, multiplying the equation in (1.1) by $\rho_{\kappa}^{\prime}u$,
we obtain

$-\int_{0}^{T}\int_{1-\kappa}^{1}(u_{t}^{2}-x^{\alpha}u_{x}^{2})\rho_{\kappa}^{\prime}\,dxdt+\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}u_{x}u\rho_{\kappa}^{\prime\prime}\,dxdt\\\
=\int_{0}^{T}\int_{0}^{1}fu\rho_{\kappa}^{\prime}\,dxdt-\int_{1-\kappa}^{1}u_{t}(T)u(T)\rho_{\kappa}^{\prime}\,dx+\int_{1-\kappa}^{1}u_{1}u_{0}\rho_{\kappa}^{\prime}\,dx.$
(3.4)

Adding (3.3) and (3.4), we get

$\int_{0}^{T}\int_{1-\kappa}^{1}2x^{\alpha}u_{x}^{2}\rho_{\kappa}^{\prime}\,dxdt+\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}u_{x}u\rho_{\kappa}^{\prime\prime}\,dxdt\\\
=\int_{0}^{T}u_{x}^{2}(t,1)\rho_{\kappa}(1)\,dt+\int_{0}^{T}\int_{1-\kappa}^{1}\alpha
x^{\alpha-1}u_{x}^{2}\rho_{\kappa}\,dxdt+\int_{0}^{T}\int_{0}^{1}2fu_{x}\rho_{\kappa}\,dxdt\\\
-\int_{1-\kappa}^{1}2u_{t}(T)u_{x}(T)\rho_{\kappa}\,dx+\int_{1-\kappa}^{1}2u_{1}u_{0x}\rho_{\kappa}\,dx+\int_{0}^{T}\int_{0}^{1}fu\rho_{\kappa}^{\prime}\,dxdt\\\
-\int_{1-\kappa}^{1}u_{t}(T)u(T)\rho_{\kappa}^{\prime}\,dx+\int_{1-\kappa}^{1}2u_{1}u_{0}\rho_{\kappa}^{\prime}\,dx.$
(3.5)

Note that the first integral is the one we want to estimate. Thus, let us
estimate each integral in the right hand side of this identity by $CN_{0}$.

Indeed, Proposition 2.9 gives us

$\int_{0}^{T}u_{x}^{2}(t,1)\rho_{\kappa}(1)\,dt\leq CN_{0}.$

From inequality (2.1),

$\int_{0}^{T}\int_{1-\kappa}^{1}\alpha
x^{\alpha-1}u_{x}^{2}\rho_{\kappa}\,dxdt\leq\frac{\alpha\|\rho_{\kappa}\|_{{L^{\infty}(0,1)}}}{1-\varepsilon_{0}}\int_{0}^{T}\int_{1-\kappa}^{1}u_{x}^{2}x^{\alpha}\,dxdt\leq
C\sup_{[0,T]}\|u\|_{{H_{\alpha}^{1}}}^{2}\leq CN_{0}.$
$\displaystyle\int_{0}^{T}\int_{1-\kappa}^{1}|2fu_{x}\rho_{\kappa}|\,dxdt$
$\displaystyle\leq\frac{2\|\rho_{\kappa}\|_{{L^{\infty}(0,1)}}}{(1-\varepsilon_{0})^{\alpha/2}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|f||u_{x}|x^{\alpha/2}\,dxdt$
$\displaystyle\leq
C\int_{0}^{T}\|f(t)\|_{{L^{2}(0,1)}}\|x^{\alpha/2}u_{x}(t)\|_{{L^{2}(0,1)}}\,dt$
$\displaystyle\leq
C\sup_{[0,T]}\|u(t)\|_{{H_{\alpha}^{1}}}\|f(t)\|_{{L^{1}(0,T;L^{2}(0,1))}}$
$\displaystyle\leq CN_{0}.$

Similarly, we can estimate

$\int_{1-\kappa}^{1}2u_{t}(T)u_{x}(T)\rho_{\kappa}\,dx\text{ and
}\int_{1-\kappa}^{1}2u_{1}u_{0x}\rho_{\kappa}\,dx.$

Before estimate the last three integrals, lets us prove that

$\frac{1}{\varepsilon^{2}}\int_{1-\varepsilon}^{1}u^{2}(t,x)\,dx\leq\frac{E(t)}{2(1-\varepsilon_{0})^{\alpha}},\
\forall t\in[0,T],\ \forall\varepsilon\in(0,\varepsilon_{0}).$ (3.6)

In fact, since $u(t,1)=0$, for any $x\in[1-\varepsilon_{0},1)$, we have

$-u(t,x)=u(t,1)-u(t,x)=\int_{x}^{1}\frac{d}{ds}u(t,s)\,ds$

From Hölder’s inequality, we have

$|u(t,x)|^{2}\leq(1-x)\int_{x}^{1}u_{x}^{2}(t,s)\,ds\leq\frac{1-x}{x^{\alpha}}\int_{x}^{1}u_{x}^{2}(t,s)s^{\alpha}\,ds.$
(3.7)

Thus,

$\displaystyle\frac{1}{\varepsilon^{2}}\int_{1-\varepsilon}^{1}|u(t,x)|^{2}dx$
$\displaystyle\leq\frac{1}{\varepsilon^{2}}\int_{1-\varepsilon}^{1}\frac{1-x}{x^{\alpha}}\left(\int_{x}^{1}u_{x}^{2}(t,s)s^{\alpha}\,ds\right)dx$
$\displaystyle\leq\frac{1}{(1-\varepsilon_{0})^{\alpha}\varepsilon^{2}}\int_{1-\varepsilon}^{1}(1-x)\left(\int_{0}^{1}u_{x}^{2}(t,s)s^{\alpha}\,ds\right)dx$
$\displaystyle\leq\frac{E(t)}{(1-\varepsilon_{0})^{\alpha}\varepsilon^{2}}\int_{1-\varepsilon}^{1}(1-x)\,dx$
$\displaystyle\leq\frac{E(t)}{2(1-\varepsilon_{0})^{\alpha}}.$

Now, let us estimate the remain integrals. From (3.6) and Hölder’s inequality,

$\displaystyle\int_{0}^{T}\int_{0}^{1}|fu||\rho_{\kappa}^{\prime}|\,dxdt$
$\displaystyle\leq\frac{C}{\kappa}\int_{0}^{T}\int_{1-\kappa}^{1}|f||u|\,dxdt\leq
C\int_{0}^{T}\|f(t)\|_{{L^{2}(0,1)}}\left(\frac{1}{\kappa^{2}}\int_{1-\kappa}^{1}u_{x}^{2}(t,x)\,dx\right)^{1/2}dt$
$\displaystyle\leq C\int_{0}^{T}\|f(t)\|_{{L^{2}(0,1)}}E(t)^{1/2}\,dt$
$\displaystyle\leq C\sup_{[0,T]}E(t)^{1/2}\|f\|_{{L^{1}(0,T;L^{2}(0,1))}}\leq
CN_{0}.$

Similarly, we can finally estimate

$\int_{1-\kappa}^{1}u_{t}(T)u(T)\rho_{\kappa}^{\prime}\,dx\text{ and
}\int_{1-\kappa}^{1}2u_{1}u_{0}\rho_{\kappa}^{\prime}\,dx.$

Hence, from (3.5), we have

$2\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}u_{x}^{2}\rho_{\kappa}^{\prime}\,dxdt\leq\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}|u_{x}||u||\rho_{\kappa}^{\prime\prime}|\,dxdt+CN_{0},$
(3.8)

Recalling (3.2), it suffices to prove that

$\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}|u_{x}||u||\rho_{\kappa}^{\prime\prime}|\,dxdt\leq\frac{3}{2}G(\delta_{0})+CN_{0}.$
(3.9)

Indeed, since $\rho_{\kappa}^{\prime\prime}=0$ in $(1-\delta_{0},1)$ and
$\rho_{\kappa}^{\prime\prime}\leq\frac{1}{\gamma\delta_{0}}$ in
$(1-\kappa,1-\delta_{0})$, we have

$\begin{split}\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}|u_{x}||u||\rho_{\kappa}^{\prime\prime}|\,dxdt&\leq\frac{1}{\gamma\delta_{0}}\int_{0}^{T}\int_{1-\kappa}^{1-\delta_{0}}x^{\alpha}|u_{x}||u|\,dxdt\\\
&\leq\left(\int_{0}^{T}\int_{1-\kappa}^{1-\delta_{0}}\frac{1}{\gamma}|u_{x}|^{2}x^{\alpha}\,dxdt\right)^{1/2}\left(\int_{0}^{T}\int_{1-\kappa}^{1-\delta_{0}}\frac{1}{\gamma{\delta_{0}^{2}}}|u|^{2}x^{\alpha}\,dxdt\right)^{1/2}\\\
&=I_{1}^{\frac{1}{2}}\,I_{2}^{\frac{1}{2}}.\end{split}$ (3.10)

Since $\kappa=\gamma+\delta_{0}$, we have

$I_{1}=\frac{1}{\gamma}\left(\kappa
G(\kappa)-\delta_{0}G(\delta_{0})\right)\leq G(\delta_{0}).$

In order to estimate $I_{2}$, proceeding as in the proof of (3.6), we can see
that

$|u|^{2}x^{\alpha}\leq
C\left((1-x)\int_{x}^{1}u_{x}^{2}s^{\alpha}\,ds+(1-x)x^{{\alpha-2}}\int_{x}^{1}u^{2}\,ds\right).$
(3.11)

Besides that, we will need an estimate that the proof is given in the
Appendix. Namely, taking $a=1-\varepsilon_{0}$ in inequality (A.3), we have
that

$\sup_{x\in[1-\varepsilon_{0},1]}|u(t,x)|\leq\max\left\\{\frac{1}{\sqrt{\varepsilon_{0}}},\sqrt{\frac{\varepsilon_{0}}{(1-\varepsilon_{0})^{\alpha}}}\right\\}\|u(t,\cdot)\|_{{H^{1}_{\alpha}}}.$

Hence,

$\displaystyle I_{2}$
$\displaystyle\leq\frac{C}{\gamma{\delta_{0}^{2}}}\int_{0}^{T}\int_{1-\kappa}^{1-\delta_{0}}(1-x)\int_{x}^{1}u_{x}^{2}s^{\alpha}\,ds\,dxdt+\frac{C}{\gamma{\delta_{0}^{2}}}\int_{0}^{T}\int_{1-\kappa}^{1-\delta_{0}}(1-x)x^{\alpha-2}\int_{x}^{1}u^{2}\,ds\,dxdt$
$\displaystyle\leq\frac{C}{\gamma{\delta_{0}^{2}}}\int_{1-\kappa}^{1-\delta_{0}}(1-x)^{2}G(1-x)\,dxdt+\frac{C}{\gamma{\delta_{0}^{2}}}\int_{1-\kappa}^{1-\delta_{0}}(1-x)(1-\varepsilon_{0})^{2-\alpha}\int_{0}^{T}\int_{x}^{1}C_{\alpha,\varepsilon_{0}}\|u(t,\cdot)\|_{H^{1}_{\alpha}}^{2}\,dsdt\,dx$
$\displaystyle\leq\frac{C}{\gamma{\delta_{0}^{2}}}G(\delta_{0})\int_{1-\kappa}^{1-\delta_{0}}(1-x)^{2}\,dx+\frac{C_{\alpha,\varepsilon_{0}}}{\gamma{\delta_{0}^{2}}}\sup_{[0,T]}\|u(t,\cdot)\|_{H^{1}_{\alpha}}^{2}\int_{1-\kappa}^{1-\delta_{0}}(1-x)^{2}\,dx$
$\displaystyle\leq C\frac{9}{4}G(\delta_{0})+CN_{0}.$

Therefore, substituting $I_{1}$ and $I_{2}$ into (3.10), we finally get

$\int_{0}^{T}\int_{1-\kappa}^{1}x^{\alpha}|u_{x}||u||\rho_{\kappa}^{\prime\prime}|\,dxdt\leq\left(\frac{9}{4}G(\delta_{0})^{2}+CN_{0}G(\delta_{0})\right)^{\frac{1}{2}}\leq\frac{3}{2}G(\delta_{0})+CN_{0},$

as required in (3.9). ∎

###### Proof of Theorem 1.3.

Let us prove that Theorem 1.3 as a consequence of Theorem 1.4. Indeed, from
(3.7)

$\displaystyle\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|u(t,x)|^{2}\,dxdt$
$\displaystyle=\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\frac{1-x}{x^{\alpha}}\int_{x}^{1}u_{x}^{2}(t,s)s^{\alpha}\,ds\,dxdt$
$\displaystyle\leq\frac{1}{(1-\varepsilon_{0})^{\alpha}\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}(1-x)\int_{x}^{1}u_{x}^{2}(t,s)s^{\alpha}\,ds\,dxdt$
$\displaystyle\leq\frac{1}{(1-\varepsilon_{0})^{\alpha}\varepsilon^{2}}\int_{1-\varepsilon}^{1}(1-x)\frac{1}{\varepsilon}\int_{0}^{T}\int_{1-\varepsilon}^{1}u_{x}^{2}(t,s)s^{\alpha}\,dsdt\,dx$
$\displaystyle\leq\frac{CN_{0}}{(1-\varepsilon_{0})^{\alpha}\varepsilon^{2}}\int_{1-\varepsilon}^{1}(1-x)\,dx$
$\displaystyle\leq CN_{0}.$

∎

##  4 Regularity for very weak solutions

We start this section recalling that the concept of weak solutions for the
wave equations, with initial data $(f,u_{0},u_{1})$ in
$L^{1}(0,T;L^{2}(0,1))\times H^{1}_{\alpha}\times L^{2}(0,1)$, is placed in
Definition 1.2. So, Theorems 1.3 and 1.4 bring regularity results for this
first type of solution. On the other hand, Definition 1.5 is about very weak
solutions for wave equations, when the initial data belongs to
$L^{1}(0,T;L^{2}(0,1))\times L^{2}(0,1)\times H^{-1}_{\alpha}$. This current
section is devoted to the proof of Theorem 1.6, which provides some
information about the regularity of this second class of solutions.

Before achieving the main goal of this part, let us describe a useful
procedure which allows us to get some additional regularity for solutions of
(1.3). Take a very weak solution $z$ of (1.3) related to the initial data
$(g,z^{0},z^{1})\in L^{1}(0,T;L^{2}(0,1))\times L^{2}(0,1)\times
H^{-1}_{\alpha}$ and let $\psi^{0}$ be the solution for the elliptic problem

$(x^{\alpha}\psi^{0}_{x})_{x}=z^{1},\ \ \psi^{0}\in H^{1}_{\alpha}.$

We can see that

$\psi(t,x)=\int_{0}^{t}z(s,x)\,ds+\psi^{0}(x)$

is a weak solution of (1.3) with initial data $(G,\psi^{0},z^{0})\in
W^{1,1}(0,T;L^{2}(0,1))\times H^{1}_{\alpha}(0,1)\times L^{2}(0,1)$, where
$G(t,x)=\int_{0}^{t}g(s,x)\,ds$. In this case, Proposition 2.8 assures that

$\displaystyle\psi\in C^{0}([0,T];H^{1}_{\alpha})\cap
C^{1}([0,T];L^{2}(0,1)).$

As a consequence, from Proposition 2.9, $\psi_{x}(\cdot,1)\in L^{2}(0,T)$,
whence

$z_{x}(\cdot,1)=\psi_{xt}(\cdot,1)\in H^{-1}(0,T).$

We notice that, given $a>0$, $\psi\in C^{0}([0,T];H^{1}(a,1))\hookrightarrow
L^{2}(0,T;H^{1}(a,1))\equiv H^{1}(a,1;L^{2}(0,T))$ and so that

$\displaystyle z=\psi_{t}\in H^{1}(a,1;H^{-1}(0,T))\hookrightarrow
C^{0}([a,1];H^{-1}(0,T)).$

Next, we prove the main result desired here.

###### Proof of Theorem 1.6.

As described at the beginning of this section, starting from the very weak
solutions $\varphi_{\varepsilon},\varphi$ of (1.3), we can obtain the related
weak solutions

$\Phi_{\varepsilon},\Phi\in C^{0}([0,T];H_{\alpha}^{1})\cap
C^{1}([0,T];L^{2}(0,1)).$

satisfying

$\begin{cases}(x^{\alpha}\Phi_{\varepsilon
x}^{0})_{x}=\varphi_{\varepsilon}^{1},\text{ where }\Phi_{\varepsilon}^{0}\in
H_{\alpha}^{1},\\\
\displaystyle\Phi_{\varepsilon}(t,x)=\int_{0}^{t}\varphi_{\varepsilon}(s,x)ds+\Phi_{\varepsilon}^{0}(x),\end{cases}$

and

$\begin{cases}(x^{\alpha}\Phi_{x}^{0})_{x}=\varphi^{1}$, where $\Phi^{0}\in
H_{\alpha}^{1},\\\
\displaystyle\Phi(t,x)=\int_{0}^{t}\varphi(s,x)ds+\Phi^{0}(x),\end{cases}$

respectively. To be more precise, $\Phi_{\varepsilon}$ comes from the initial
data $(H_{\varepsilon},\Phi_{\varepsilon}^{0},\varphi_{\varepsilon}^{0})$,
where $H_{\varepsilon}(t,x)=\int_{0}^{t}h_{\varepsilon}(s,x)\,ds$, while
$\Phi$ is associated to $(H_{,}\Phi^{0},\varphi^{0})$, with
$H(t,x)=\int_{0}^{t}h(s,x)\,ds$.

Additionally,

$\Phi_{\varepsilon x}(\cdot,1),\Phi_{x}(\cdot,1)\in L^{2}(0,T)\text{ and
}\Phi_{\varepsilon x}(\cdot,1)\rightharpoonup\Phi_{x}(\cdot,1)\text{ in
}L^{2}(0,T),$

following

$\varphi_{\varepsilon x}(\cdot,1),\varphi_{x}(\cdot,1)\in H^{-1}(0,T)\text{
and }\varphi_{\varepsilon x}(\cdot,1)\rightharpoonup\varphi_{x}(\cdot,1)\text{
in }H^{-1}(0,T).$

In order to get $\varphi_{x}(\cdot,1)\in L^{2}(0,T)$, it suffices to prove the
next claim.

Claim: There exists $C>0$ such that

$|\langle\varphi_{x}(\cdot,1),u\rangle|\leq C\|u\|_{L^{2}(0,T)}\emph{ for any
}u\in\mathcal{D}(0,T).$ (4.1)

In fact, let us take $u\in\mathcal{D}(0,T)$ and consider $w(t,x)=(1-x)u(t)$.
We notice that $w(t,1)=0$, $w_{x}(t,1)=-u(t)$ and

$\displaystyle\mbox{supp }w(\cdot,x)=\mbox{supp}(u)$

for each $x\in(1-\varepsilon,1)$. In this case,

$\displaystyle\displaystyle\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|w(t,x)|^{2}dxdt$
$\displaystyle=\frac{1}{\varepsilon^{3}}\int_{0}^{T}|u(t)|^{2}\int_{1-\varepsilon}^{1}|x-1|^{2}dxdt$
$\displaystyle=\frac{1}{3}\int_{0}^{T}|u(t)|^{2}dt,$

which means that

$\bigg{(}\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|w(t,x)|^{2}dxdt\bigg{)}^{1/2}=\frac{1}{\sqrt{3}}\|u\|_{L^{2}(0,T)}.$
(4.2)

Since

$\displaystyle\Phi_{\varepsilon x}(t,x)=\int_{0}^{t}\varphi_{\varepsilon
x}(s,x)ds+\Phi_{\varepsilon x}^{0}(x),$

we have $\Phi_{\varepsilon xt}(t,1)=\varphi_{\varepsilon x}(t,1)$ and so that

$\displaystyle\displaystyle\langle\varphi_{\varepsilon x}(\cdot,1),u\rangle$
$\displaystyle=\langle\Phi_{\varepsilon
xt}(\cdot,1),u\rangle=-\langle\Phi_{\varepsilon
x}(\cdot,1),u^{\prime}\rangle=-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt$
$\displaystyle=\begin{multlined}\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt\\\
-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt\end{multlined}\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt\\\
-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt$
$\displaystyle=\begin{multlined}-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\varphi_{\varepsilon}(t,x)w(t,x)dxdt\\\
-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt\end{multlined}-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\varphi_{\varepsilon}(t,x)w(t,x)dxdt\\\
-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt$ $\displaystyle=:A_{\varepsilon}+B_{\varepsilon},$

where

$A_{\varepsilon}=-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\varphi_{\varepsilon}(t,x)w(t,x)dxdt$

and

$B_{\varepsilon}=-\frac{3}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)w_{t}(t,x)dxdt-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt$

From (1.4) and (4.2), it is very clear that

$\displaystyle|A_{\varepsilon}|\leq
3\bigg{(}\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|\varphi_{\varepsilon}(t,x)|^{2}dxdt\bigg{)}^{1/2}\bigg{(}\frac{1}{\varepsilon^{3}}\int_{0}^{T}\int_{1-\varepsilon}^{1}|w(t,x)|^{2}dxdt\bigg{)}^{1/2}\leq(3C)^{1/2}\|u\|_{L^{2}(0,T)},$
(4.3)

where $C>0$ does not depend on $\varepsilon$. The remainder of this proof is
devoted to check that $B_{\varepsilon}\to 0$ as $\varepsilon\to 0$. In fact,
using

  * 1.

$\Phi_{\varepsilon}(t,x)w_{t}(t,x)=\Phi_{\varepsilon}(t,x)(1-x)u^{\prime}(t)$;

  * 2.

$\displaystyle\int_{x}^{1}\Phi_{\varepsilon
x}(t,\xi)d\xi=\Phi_{\varepsilon}(t,1)-\Phi_{\varepsilon}(t,x)=-\Phi_{\varepsilon}(t,x)$;

  * 3.

$\displaystyle\Phi_{\varepsilon
x}(t,1)=\frac{3}{\varepsilon^{3}}\int_{1-\varepsilon}^{1}(1-x)\int_{x}^{1}\Phi_{\varepsilon
x}(t,1)d\xi dx$,

we certainly get

$\displaystyle\displaystyle B_{\varepsilon}$
$\displaystyle=-\frac{3}{\varepsilon^{3}}\int_{0}^{T}u^{\prime}(t)\int_{1-\varepsilon}^{1}\Phi_{\varepsilon}(t,x)(1-x)dxdt-\int_{0}^{T}\Phi_{\varepsilon
x}(t,1)u^{\prime}(t)dt$
$\displaystyle=-\frac{3}{\varepsilon^{3}}\int_{0}^{T}u^{\prime}(t)\int_{1-\varepsilon}^{1}\bigg{[}\Phi_{\varepsilon}(t,x)(1-x)+(1-x)\int_{x}^{1}\Phi_{\varepsilon
x}(t,1)d\xi\bigg{]}dxdt$
$\displaystyle=-\frac{3}{\varepsilon^{3}}\int_{0}^{T}u^{\prime}(t)\int_{1-\varepsilon}^{1}(1-x)^{2}\bigg{[}\frac{1}{1-x}\int_{x}^{1}-\Phi_{\varepsilon
x}(t,\xi)d\xi+\Phi_{\varepsilon x}(t,1)\bigg{]}dxdt$
$\displaystyle=\frac{3}{\varepsilon^{3}}\int_{1-\varepsilon}^{1}(1-x)^{2}\bigg{[}\frac{1}{1-x}\int_{x}^{1}\langle\Phi_{\varepsilon
x}(\cdot,\xi)-\Phi_{\varepsilon
x}(\cdot,1),u^{\prime}\rangle_{H^{-1}(0,T),H_{0}^{1}(0,T)}d\xi\bigg{]}dx.$

It suffices to prove that $\|\Phi_{\varepsilon x}(\cdot,\xi)\|_{H^{-1}(0,T)}$
is uniformly continuous. Indeed, let $\Psi_{\varepsilon}$ be the weak solution
of (1.3), with initial data
$(\mathbf{H}_{\varepsilon},\Psi_{\varepsilon}^{0},\Phi_{\varepsilon}^{0})$,
where

$\begin{cases}(x^{\alpha}\Psi_{\varepsilon
x}^{0})_{x}=\varphi_{\varepsilon}^{0},\text{ where }\Psi_{\varepsilon}^{0}\in
H_{\alpha}^{1}\cap H^{2}_{\alpha},\\\
\displaystyle\Psi_{\varepsilon}(t,x)=\int_{0}^{t}\Phi_{\varepsilon}(s,x)ds+\Psi_{\varepsilon}^{0}(x),\end{cases}$

and $\mathbf{H}_{\varepsilon}(t,x)=\int_{0}^{t}h_{\varepsilon}(s,x)\,ds$.
Hence, $\Psi_{\varepsilon}\in L^{2}(0,T;H^{1}_{\alpha}\cap H^{2}_{\alpha})$
and, in particular, $\Psi_{\varepsilon}\in L^{2}(0,T;H^{2}(1-\varepsilon,1))$.
As a result, $\Psi_{\varepsilon x}\in L^{2}(0,T;H^{1}(1-\varepsilon,1))\equiv
H^{1}(1-\varepsilon,1;L^{2}(0,T))$, whence

$\Phi_{\varepsilon x}=\Psi_{\varepsilon xt}\in
H^{1}(1-\varepsilon,1;H^{-1}(0,T))\hookrightarrow
C^{0}([1-\varepsilon,1];H^{-1}(0,T)),$

following the required uniform continuity.

Finally, (1.5) is an immediate consequence of $\langle\varphi_{\varepsilon
x}(\cdot,1),u\rangle=A_{\varepsilon}+B_{\varepsilon},$ (4.3) and
$B_{\varepsilon}\to 0$. ∎

##  Appendix A Appendix

The aim of this section is to obtain the inequality (A.3), that was
fundamental in the proof of Theorem 1.4. Besides that, we end up proving an
embedding for the Sobolev space $H_{\alpha}^{1}$ in a Hölder continuous space.

Given $a\in(0,1)$, let us denote by $H^{1}(a,1)$ and $C^{0,1/2}([a,1])$
classical Sobolev and Hölder spaces.

######  Lemma A.11.

$H^{1}_{\alpha}\hookrightarrow H^{1}(a,1),\ \forall a\in(0,1)$ and

$\|u\|_{{H^{1}(a,1)}}\leq C_{\alpha,a}\|u\|_{H^{1}_{\alpha}},$

where $C_{\alpha,a}=\sqrt{\max\\{1,1/a^{\alpha}\\}}$.

###### Proof.

Given $a\in(0,1)$, if $u\in H^{1}_{\alpha}$, then $u\in L^{2}(a,1)$. Besides
that, we have

$\|u^{\prime}\|_{{L^{2}(a,1)}}^{2}=\int_{a}^{1}x^{-\alpha}x^{\alpha}|u^{\prime}|^{2}\,dx\leq\frac{1}{a^{\alpha}}|u^{\prime}|^{2}\,dx=\frac{1}{a^{\alpha}}\|x^{\alpha/2}u^{\prime}\|_{{L^{2}(0,1)}}^{2}.$
(A.1)

Therefore,

$\|u\|_{{H^{1}(0,1)}}^{2}\leq\max\\{1,1/a^{\alpha}\\}\|u\|_{H^{1}_{\alpha}}^{2}.$

∎

Classical Sobolev Imbedding, together with Lemma A.11, give us that

$H^{1}_{\alpha}\hookrightarrow H^{1}(a,1)\hookrightarrow
C^{0,1/2}([a,1]),\text{ for any }a\in(0,1).$ (A.2)

Moreover, we have that any $u\in H^{1}_{\alpha}$ satisfies

$u(x)-u(y)=\int_{y}^{x}u^{\prime}(\xi)\,d\xi,\ \forall x,y\in(0,1).$

However, let us give the proof of (A.2) in order to see how the constant of
embedding depends on $a,\alpha$.

######  Lemma A.12.

$H^{1}_{\alpha}\hookrightarrow C^{0,1/2}([a,1])$, for any $a\in(0,1)$, and

$\|u\|_{{C^{0,1/2}([a,1])}}\leq C_{\alpha,a}\|u\|_{H^{1}_{\alpha}},\ \forall
u\in H^{1}_{\alpha},$ (A.3)

where
$C_{\alpha,a}:=\max\left\\{(1-a)^{-1/2},\frac{(1-a)^{1/2}}{a^{\alpha/2}}\right\\}$.

###### Proof.

Given $x,y\in[a,1]$, from Cauchy-Schwartz inequality and (A.1), we have that

$\begin{split}|u(x)-u(y)|&\leq\int_{y}^{x}|u^{\prime}|\,d\xi\leq\left(\int_{y}^{x}\,d\xi\right)^{1/2}\left(\int_{y}^{x}|u^{\prime}|^{2}\
d\xi\right)^{1/2}\\\ &=|x-y|^{1/2}\left(\int_{a}^{1}|u^{\prime}|^{2}\
d\xi\right)^{1/2}\leq|x-y|^{1/2}\frac{1}{a^{\alpha/2}}\|u\|_{H^{1}_{\alpha}}.\end{split}$

Hence,

$\left[u\right]_{C^{0,1/2}([a,1])}\leq\frac{1}{a^{\alpha/2}}\|u\|_{H^{1}_{\alpha}}.$

Besides that,

$\begin{split}|u(y)|&\leq|u(x)|+|u(x)-u(y)|\leq|u(x)|+|x-y|^{1/2}\frac{1}{a^{\alpha/2}}\|u\|_{H^{1}_{\alpha}}\\\
&\leq|u(x)|+\frac{(1-a)^{1/2}}{a^{\alpha/2}}\|u\|_{H^{1}_{\alpha}},\ \forall
x,y\in[a,1].\end{split}$

Integrating with respect to $x$ over $[a,1]$, we get

$(1-a)|u(y)|\leq(1-a)^{1/2}\|u\|_{{L^{2}(0,1)}}+\frac{(1-a)^{3/2}}{a^{\alpha/2}}\|u\|_{H^{1}_{\alpha}},$

we obtain

$\displaystyle|u(y)|$
$\displaystyle\leq(1-a)^{-1/2}\|u\|_{{L^{2}(0,1)}}+\frac{(1-a)^{1/2}}{a^{\alpha/2}}\|u\|_{H^{1}_{\alpha}}$
$\displaystyle\leq\max\left\\{(1-a)^{-1/2},\frac{(1-a)^{1/2}}{a^{\alpha/2}}\right\\}\|u\|_{H^{1}_{\alpha}},\
\forall y\in[a,1],$

which gives us the desire inequality. ∎

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