\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 80, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/80\hfil Hyperbolic systems with  boundary differential inclusions]
{Solutions for a hyperbolic system with boundary differential
inclusion and nonlinear second-order boundary damping}

\author[Jong Yeoul Park \&  Sun Hye Park\hfil EJDE--2003/80\hfilneg]
{Jong Yeoul Park \&  Sun Hye Park}

\address{Department of Mathematics,  Busan  National University, \newline
 30  Changjeon-dong,  Keumjeong-ku,  Busan,  609-735, South Korea}
 \email[Jong Yeoul Park]{jyepark@pusan.ac.kr}

\date{}
\thanks{Submitted April 9, 2003. Published August 5, 2003.}
\thanks{Supported by grant R01-2002-000-00491-0 from the Korea Science
and Engineering Foundation}
\subjclass[2000]{35L85, 35B40}
\keywords{Differential inclusion, boundary damping, generalized solution}

\begin{abstract}
  In this paper we study the existence of generalized solutions for a
  hyperbolic system with a discontinuous multi-valued term and nonlinear
  second-order damping terms on the boundary.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}

\section{ Introduction }

The main purpose of this paper is to investigate the initial boundary value
problem for a hyperbolic system with differential inclusion on the boundary
\begin{equation} \label{e1.1}
\begin{gathered}
u''-\Delta u'-M(\|\nabla u\|^2)\Delta u=f \quad \text{in }
(x,t)\in Q=\Omega \times (0,T), \\
u(x,0)=u'(x,0)=0 \quad \text{in } x\in \Omega, \\
u=0 \quad \text{on } \Sigma_0=\Gamma_0 \times (0,T), \\
\frac{\partial u'}{\partial \nu}+M(\|\nabla u\|^2)\frac{\partial u}{\partial \nu}
+K(u)u''+|u'|^{\rho}u'+\Xi=0 \quad
\text{on } \Sigma_1=\Gamma_1 \times (0,T),     \\
\Xi(x,t)\in \varphi(u'(x,t)) \quad \text{a.e. } (x,t)\in
\Sigma_1=\Gamma_1 \times (0,T),
\end{gathered}
\end{equation}
where $\Omega$ is a bounded open set of $\mathbb{R}^n (n\geq 3)$
with sufficiently smooth boundary $\Gamma=\partial \Omega$ such that
$\Gamma=\Gamma_0 \cup \Gamma_1$, $\bar{\Gamma_0} \cap \bar{\Gamma_1}=\emptyset$
and $\Gamma_0, \Gamma_1$ have positive measures, $\rho \in (1,\infty),
M(s)$ is a $C^1$ class function such that $M(s)> m_0>0$ for some constant
$m_0$,  $K(s)$ is a continuously differentiable positive function,
$\Delta u = \sum_{i=1}^n \frac{\partial^2u }{\partial x_i^2},\|\nabla u\|^2=
\sum_{i=1}^n \int_{\Omega}|\frac{\partial u}{\partial x_i}|^2dx, \nu$
is the outward unit normal vector on $\Gamma$,
$\varphi$ is a discontinuous and nonlinear set valued mapping
and $T$ is a positive real number. The precise hypothesis on the above system
will be given in the next section.

The background of these problems is in physics, especially in solid mechanics,
where non-monotone and multi-valued constitutive laws lead to differential
inclusion. For a brief account of the works on such variational inequalities
we refer the reader to [3,4,5].
Doronin et al. [1] investigated the existence of generalized solutions
for the hyperbolic equation of the form
\begin{gather*}
u''-\Delta u=f \quad\text {in } (x,t)\in Q, \\
\frac{\partial u}{\partial \nu}+K(u)u''+|u'|^{\rho}u'=0 \quad\text{on }
\Sigma_1=\Gamma_1 \times (0,T), \\
u=0 \quad\text{on } \Sigma_0=\Gamma_0\times (0,T), \\
u(x,0)=u'(x,0)=0 \quad\text{on } \Omega.
\end{gather*}
Motivated the results of [1], in this paper we study the existence of solutions of
the variational inequalities (1.1).
It is important to observe that as far as we are concerned it has never been
considered differential inclusion acting on the boundary in the literature.
The plan of this paper is as follows. In section 2, the assumptions and the main
results are given. In section 3, the existence of a solution to problem (1.1)
is proved.

\section{Assumptions and Main results}

Throughout this paper we denote
\begin{gather*}
H_1(\Omega)=\{ u\in H^1(\Omega) : u=0 \text{ on } \Gamma_0 \},
\quad (u,v)=\int_{\Omega}u(x)v(x)dx, \\
(u,v)_{\Gamma_1}=\int_{\Gamma_1}u(x)v(x)d\Gamma, \quad
 \|u\|_{p,\Gamma_1}=(\int_{\Gamma_1}|u(x)|^pd\Gamma)^{1/p}.
\end{gather*}
For simplicity, we denote $\|\cdot\|_{L^2(\Omega)}$ and
$\|\cdot\|_{2,\Gamma_1}$ by $\|\cdot\|$ and $\|\cdot\|_{\Gamma_1}$, respectively.
We formulate the following assumptions:
\begin{itemize}
\item[(A1)] $K(s)$ is a continuous real function satisfying the conditions
\begin{gather}
0< K_0 \leq K(s) \leq K_1(1+|s|^{\rho}),       \\
|K'(s)|^{\frac{\rho}{\rho -1}}\leq K_2(1+K(s)),
\end{gather}
for some $K_0,K_1,K_2 >0$.

\item[(A2)] $b : \mathbb{R} \to \mathbb{R}$ is a locally bounded function
satisfying
\begin{equation}
|b(s)|\leq \mu_1(1+|s|), \enskip \forall s \in \mathbb{R},
\end{equation}
for some $\mu_1 >0$.
\end{itemize}


The multi-valued function $\varphi : \mathbb{R} \to \mathbb{R}$ is
obtained by filling
in jumps of a function $b : \mathbb{R}\to \mathbb{R}$ by means of the
functions $\underline{b}_{\epsilon},\overline{b}_{\epsilon},\underline{b},
\overline{b} : \mathbb{R}\to \mathbb{R}$ as follows:
\begin{gather*}
\underline{b}_{\epsilon}(t)=\mathop{\rm ess\,inf}{}_{|s-t|\leq \epsilon} b(s),
\quad
\overline{b}_{\epsilon}(t)=\mathop{\rm ess \, sup}{}_{|s-t|\leq \epsilon} b(s),         \\
\underline{b}(t)=\lim_{\epsilon \to 0^+}\underline{b}_{\epsilon}(t), \quad
\overline{b}(t)=\lim_{\epsilon \to 0^+}\overline{b}_{\epsilon}(t),     \\
\varphi(t)=[\underline{b}(t),\overline{b}(t)].
\end{gather*}
We shall use the regularization of $b$ defined by
\begin{equation*}
b^m(t)=m\int^{\infty}_{-\infty}b(t-\tau)\rho(m\tau)d\tau,
\end{equation*}
where $\rho \in C_0^{\infty}((-1,1)), \rho \geq 0$ and
$\int^1_{-1}\rho(\tau)d\tau=1$.

\begin{remark} \label{rmk2.1}\rm
It is easy to show that $b^m$ is continuous for all $m\in \mathbb{N}$ and
that $\underline{b}_{\epsilon}$, $\overline{b}_{\epsilon}$, $\underline{b}$,
$\overline{b}$, $b^m$ satisfy  condition (A2) with a possibly different
constant when $b$ satisfies (A2).
\end{remark}

\noindent {\bf Definition} A function $u(x,t)$ such that
\begin{gather*}
u\in L^{\infty}(0,T;H_1(\Omega)), \\
u'\in L^2(0,T;H_1(\Omega))\cap L^{\infty}(0,T;L^{\rho +2}(\Gamma_1)), \\
u''\in L^2(0,T;L^2(\Omega)\cap L^2(\Gamma_1)), \\
u(x,0)=u'(x,0)=0
\end{gather*}
is a generalized solution to (1.1) if there exists $\Xi \in L^2(0,T;L^2(\Gamma_1))$
and for any functions $v\in W=H_1(\Omega) \cap  L^{\rho +2}(\Gamma_1)$
and $\psi \in C^1(0,T)$ with $\psi(T)=0$ the relations hold:
\begin{gather} \label{e2.4}
\begin{aligned}
&\int^T_0\Big\{ (u'',v)+(\nabla u',\nabla v)+
M(\|\nabla u\|^2)(\nabla u,\nabla v)   \\
&+(|u'|^{\rho}u'-K'(u)(u')^2+\Xi,v)_{\Gamma_1} \Big\} \psi(t)dt
-\int^T_0(K(u)u',v)_{\Gamma_1}\psi'(t)dt  \\
&=\int^T_0(f,v)\psi(t)dt,
\end{aligned}\\
\Xi(x,t)\in \varphi(u'(x,t)) \quad \text{a.e. } (x,t)\in \Sigma_1.
\label{e2.5}
\end{gather}

Now we are in position to state our existence result.

\begin{theorem} \label{thm2.1}
Assume that (A1) and (A2) hold and $f\in L^2(0,T;L^2(\Omega))$. Then, for all $T>0$ there exists a generalized solution to the problem (1.1).
\end{theorem}

\section{Proof of main theorem}

In this section we are going to show the existence of solution for problem (1.1)
using the Faedo-Galerkin's approximation. For this end we represent by
$\{ w_j\}_{j\geq 1}$ a basis in
$W=H_1(\Omega)\cap L^{\rho+2}(\Gamma_1)$.
Let $W_m=\mathop{\rm span}\{ w_1,w_2,\dots,w_m\}$. Next we define the
approximations $u_m(t)=\sum_{j=1}^m g_{jm}(t)w_j$, where $g_{jm}(t)$ are
solutions to the Cauchy problem
\begin{gather} \label{e3.1}
\begin{aligned}
(u''_m,w_j)+(\nabla u'_m,\nabla w_j)+
M(\|\nabla u_m\|^2)(\nabla u_m,\nabla w_j)& \\
+(K(u_m)u''_m+|u'_m|^{\rho}u'_m+b^m(u'_m),w_j)_{\Gamma_1}&=(f,w_j), \end{aligned}\\
u_m(0)=u'_m(0)=0. \label{e3.2}
\end{gather}
By the same argument as in [1], the approximate system (3.1) and (3.2) has
solutions $u_m(t)$ in $[0,t_m)$. The extension of these solutions to the whole
interval $[0,T]$ is a consequence of the priori estimate which we are going to
prove below.


\noindent{\bf Step 1 : A priori estimate.}
Multiplying (3.1) by $g'_{jm}(t)$ and summing from $j=1$ to $j=m$, we get
\begin{equation} \label{e3.3}
\begin{aligned}
&\frac{1}{2} \frac{d}{dt} \Big \{ \|u'_m(t)\|^2
+\bar{M}(\|\nabla u_m(t)\|^2)+\int_{\Gamma_1}K(u_m(t))(u'_m(t))^2 d\Gamma \Big \}\\
&+(b^m(u'_m(t)),u'_m(t))_{\Gamma_1}
+\|\nabla u'_m(t)\|^2+\|u'_m(t)\|^{\rho+2}_{\rho +2, \Gamma_1}\\
&-\frac{1}{2} \int_{\Gamma_1}K'(u_m(t))(u'_m(t))^3 d\Gamma \\
&=(f(t),u'_m(t)),
\end{aligned}
\end{equation}
where $\bar{M}(s)= \int^s_0 M(r)dr$. By the condition (A2), we have
\begin{equation} \label{e3.4}
\begin{aligned}
\|b^m(u'_m(t))\|_{\Gamma_1}^2
&=\int_{\Gamma_1} \big ( b^m(u'_m(x,t))\big )^2d\Gamma\\
&\leq \int_{\Gamma_1} c_1 (1+|u'_m(x,t)|)^2d\Gamma \\
&\leq 2c_1 \int_{\Gamma_1}  (1+|u'_m(x,t)|^2)d\Gamma
=c_2+2c_1\|u'_m(t)\|_{\Gamma_1}^2,
\end{aligned}
\end{equation}
where $c_1, c_2$ are positive constants(dependent on the geometry of $\Gamma$
but independent of $m$).
In what follows $c_i (i\geq 3)$ denote generic constants independent of $m$. \\
Inequality (3.4) and H$\dot {o}$lder's inequality imply that
\begin{equation} \label{e3.5}
\begin{aligned}
&\big|\int^t_0(b^m(u'_m(s)),u'_m(s))_{\Gamma_1}ds\big|\\
&\leq (\int^t_0\|b^m(u'_m(s))\|_{\Gamma_1}^2ds)^{1/2}
(\int^t_0\|u'_m(s)\|_{\Gamma_1}^2ds)^{1/2} \\
&\leq \big (\int^t_0(c_2+2c_1\|u'_m(s)\|_{\Gamma_1}^2)ds\big)^{1/2}
\big(\int^t_0\|u'_m(s)\|_{\Gamma_1}^2 ds \big)^{1/2}  \nonumber \\
&\leq c_3(1+\int^t_0\|u'_m(s)\|_{\Gamma_1}^2ds).
\end{aligned}
\end{equation}
Note that, by Young's inequality,
\begin{equation} \label{e3.6}
\begin{aligned}
&\int^t_0\{ \|u'_m(s)\|^{\rho+2}_{\rho +2, \Gamma_1}
-\frac{1}{2}\int_{\Gamma_1}K'(u_m(s))(u'_m(s))^3d\Gamma \}ds \\
&\geq \int^t_0 \int_{\Gamma_1}|u'_m(s)|^2
\{ |u'_m(s)|^{\rho}-\epsilon |u'_m(s)|^{\rho}
-C(\epsilon)|K'(u_m(s))|^{\frac{\rho}{\rho-1}} \}d\Gamma ds,
\end{aligned}
\end{equation}
where $\epsilon$ is an arbitrary positive number.  Therefore, integrating (3.3) over
$(0,t)$ and taking $\epsilon=\frac{1}{2}$ in (3.6), from  (2.2), (3.5) and (3.6) we
obtain
\begin{equation} \label{e3.7}
\begin{aligned}
&\frac{1}{2}\big \{ \|u'_m(t)\|^2+\bar{M}(\|\nabla u_m(t)\|^2)+
\int_{\Gamma_1}K(u_m(t))(u'_m(t))^2d\Gamma \big \}     \\
&+\int^t_0\|\nabla u'_m(s)\|^2ds
+\frac{1}{2}\int^t_0\|u'_m(s)\|^{\rho+2}_{\rho +2,\Gamma_1}ds   \\
&\leq c_3(1+\int^t_0\|u'_m(s)\|_{\Gamma_1}^2ds)+\int^t_0\|f(s)\|^2ds+
\int^t_0\|u'_m(s)\|^2ds  \\
&\quad +c_4\int^t_0 \int_{\Gamma_1}|u'_m(s)|^2(1+K(u_m(s)))d\Gamma ds.
\end{aligned}
\end{equation}
On the other hand, note that $K(u)\geq C_0(1+K(u))$ where
$2C_0=\min\{ 1,K_0\}$. Thus, letting
$$
E_m(t)=\frac{1}{2}(\|u'_m(t)\|^2+\bar{M}(\|\nabla u_m(t)\|^2)+
C_0\int_{\Gamma_1}(1+K(u_m(t)))|u'_m(t)|^2d\Gamma),
$$
from (3.7) we have
$E_m(t)\leq c_5(1+\int^t_0 E_m(s)ds)$.
Thus, by Gronwall's lemma, we conclude that
\begin{equation} \label{e3.8}
E_m(t) \leq c_6, \enskip \forall t\in [0,T].
\end{equation}
This inequality and (3.7) imply  that for all $t\in (0,T)$
\begin{equation} \label{e3.9}
\int^t_0 \|\nabla u'_m(s)\|^2ds \leq c_7,  \quad
\int_{\Gamma_1}|u'_m(t)|^2d\Gamma \leq c_8.
\end{equation}
By imbedding theorem, from (3.9) we have
\begin{equation} \label{e3.10}
\int^t_0 \|u'_m(s)\|^2ds \leq c_9.
\end{equation}
Furthermore, from (3.4) and (3.9) we obtain
\begin{equation} \label{e3.11}
\int^t_0 \|b^m(u'_m(s))\|_{\Gamma_1}^2ds\leq c_{10}.
\end{equation}
Since $\bar{M}(\|\nabla u_m(t)\|^2)\geq m_0\|\nabla u_m(t)\|^2$, by (3.8)
\begin{equation} \label{e3.12}
\|\nabla u_m(t)\|^2\leq c_{11}.
\end{equation}
Next, multiplying (3.1) by $g''_{jm}(t)$ and summing from $j=1$ to $j=m$, we have
\begin{equation} \label{e3.13}
\begin{aligned}
&\|u''_m(t)\|^2+\frac{1}{2}\frac{d}{dt}\|\nabla u'_m(t)\|^2+
\ M(\|\nabla u_m(t)\|^2)\frac{d}{dt}(\nabla u_m(t),\nabla u'_m(t))   \\
& -M(\|\nabla u_m(t)\|^2) \|\nabla u'_m(t)\|^2
+(b^m(u'_m(t)),u''_m(t))_{\Gamma_1}\\
&+\int_{\Gamma_1}K(u_m(t))(u''_m(t))^2d\Gamma r
 +\frac{1}{\rho+2}\frac{d}{dt}\|u'_m(t)\|^{\rho+2}_{\rho+2,\Gamma_1}\\
&=(f(t),u''_m(t)).
\end{aligned}
\end{equation}
Integrating this inequality over $(0,t)$ and using (2.1) and Young's inequality,
we obtain
\begin{align}
&\int^t_0\|u''_m(s)\|^2ds+\frac{1}{2}\|\nabla u'_m(t)\|^2
+K_0\int^t_0\|u''_m(s)\|^2_{\Gamma_1}ds+
\frac{1}{\rho+2}\|u'_m(t)\|^{\rho+2}_{\rho+2,\Gamma_1} \nonumber\\
&\leq \int^t_0 M(\|\nabla u_m(s)\|^2)\|\nabla u'_m(s)\|^2ds
-M(\|\nabla u_m(t)\|^2)(\nabla u_m(t),\nabla u'_m(t))  \nonumber \\
&\quad +2M'(\|\nabla u_m(t)\|^2)(\nabla u_m(t),\nabla u'_m(t))^2
+\epsilon \int^t_0\|u''_m(s)\|_{\Gamma_1}^2ds  \label{e3.14} \\
&\quad +C(\epsilon)\int^t_0\|b^m(u'_m(s))\|_{\Gamma_1}^2ds
+\epsilon \int^t_0\|u''_m(s)\|^2ds+C(\epsilon)\int^t_0\|f(s)\|^2ds, \nonumber
\end{align}
where we have used $u_m(0)=u'_m(0)=0$.
Since $\epsilon$ is arbitrary and $M(s)$ is a $C^1$ function, from (3.8),(3.9),(3.11) and (3.12), we conclude that
\begin{equation} \label{e3.15}
\int^t_0\|u''_m(s)\|^2ds+\|\nabla u'_m(t)\|^2+\int^t_0\|u''_m(s)\|^2_{\Gamma_1}ds
+\|u'_m(t)\|^{\rho+2}_{\rho+2,\Gamma_1}\leq c_{12}.
\end{equation}
From (3.8)-(3.12), and (3.15), taking into consideration that
$u\big{|}_{\Gamma_0}=0$, we obtain
\begin{equation} \label{e3.16}
\begin{gathered}
(u_m) \text{ is bounded in } L^{\infty}(0,T;H_1(\Omega)), \\
(u'_m) \text{ is bounded in } L^{\infty}(0,T;H_1(\Omega))\cap L^{\infty}(0,T;L^{\rho+2}(\Gamma_1)), \\
(u''_m) \text{ is bounded in } L^2(0,T;L^2(\Omega)\cap L^2(\Gamma_1)), \\
(b^m(u'_m)) \text{ is bounded in } L^2(0,T;L^2(\Gamma_1)).
\end{gathered}
\end{equation}

\noindent{\bf Step 2 : Passage to the limit.}
Multiplying (3.1) by $\psi \in C^1(0,T)$ with $\psi(T)=0$ and integrating over
$(0,T)$, we obtain
\begin{equation} \label{e3.17}
\begin{aligned}
&\int^T_0 \Big \{ (u''_m(t),w_j)+(\nabla u'_m(t),\nabla w_j)
+M(\|\nabla u_m(t)\|^2)(\nabla u_m(t),\nabla w_j)  \\
&+(b^m(u'_m(t)),w_j)_{\Gamma_1}+(|u'_m(t)|^{\rho}u'_m(t)
-K'(u_m(t))(u'_m(t))^2,w_j)_{\Gamma_1} \Big \} \psi(t)dt \\
&-\int^T_0 (K(u_m(t))u'_m(t),w_j)_{\Gamma_1}\psi'(t)dt\\
&=\int^T_0 (f(t),w_j)\psi(t)dt.
\end{aligned}
\end{equation}
 From  (3.16), we have  subsequences
(in the sequel we denote subsequences by the same symbols as original sequences)
such that
\begin{gather}
u_m \to u \quad \text{ weakly star in } L^{\infty}(0,T;H_1(\Omega)), \\
u'_m \to u' \quad \text{ weakly star in }
L^{\infty}(0,T;H_1(\Omega))\cap L^{\infty}(0,T;L^{\rho+2}(\Gamma_1)), \\
u''_m \to u'' \quad \text{ weakly in } L^2(0,T;L^2(\Omega)\cap L^2(\Gamma_1)), \\
b^m(u'_m) \to \Xi \quad \text{ weakly in } L^2(0,T;L^2(\Gamma_1)).
\end{gather}
 From (3.18)--(3.21), considering that the imbedding $H_1(\Omega) \hookrightarrow L^2(\Gamma)$
is continuous and compact and using Aubin compactness theorem [2], we have
\begin{gather}
|u'_m|^{\rho}u'_m, K(u_m)u'_m, K'(u_m)(u'_m)^2 \in L^q(\Sigma_1),\quad
 q=\frac{\rho+2}{\rho+1} >1, \\
u_m\to u \text{ a.e. on } \Sigma_1 \quad \text{and}\quad
 u'_m\to u' \text{ a.e. on } \Sigma_1\,.
\end{gather}
Therefore,
\begin{equation}
\begin{gathered}
|u'_m|^{\rho}u'_m \to  |u'|^{\rho}u', \quad K(u_m)u'_m \to K(u)u',\\
 K'(u_m)(u'_m)^2  \to K'(u)(u')^2  \quad \text{ a.e. on } \Sigma_1.
\end{gathered}
\end{equation}

\noindent{\bf Step 3 : $(u,\Xi)$ is a solution of (1.1).}
Letting $m$ tend to infinity in (3.17) and using (3.18)-(3.24), we have
\begin{equation} \label{e3.25}
\begin{aligned}
&\int^T_0 \big \{ (u''(t),w_j)+(\nabla u'(t),\nabla w_j)
+M(\|\nabla u(t)\|^2)(\nabla u(t),\nabla w_j)  \\
&+(\Xi(t),w_j)_{\Gamma_1}+(|u'(t)|^{\rho}u'(t)
-K'(u(t))(u'(t))^2,w_j)_{\Gamma_1} \big \} \psi(t)dt  \\
&-\int^T_0 (K(u(t))u'(t),w_j)_{\Gamma_1}\psi'(t)dt \\
&=\int^T_0 (f(t),w_j)\psi(t)dt.
\end{aligned}
\end{equation}
Since $\{ w_j \}$ is dense in $H_1(\Omega)\cap L^{\rho+2}(\Gamma)$, we conclude that
(2.4) hold.
It remains to show that (2.5), i.e., $\Xi(x,t)\in \varphi(u'(x,t))$ a.e.
$(x,t)\in \Sigma_1$. By the Aubin-Lions compactness Lemma[2], we get from
(3.19)-(3.20) that
$$
u'_m \to u' \quad \text{strongly in } L^2(0,T;L^2(\Gamma_1)).
$$
This implies $u'_m(x,t) \to u'(x,t)$  a.e. in $\Sigma_1$.
Thus, for given $\eta >0$, using the theorems of Lusin and Egoroff, we can choose
a subset $\omega \subset \Sigma_1$ such that
$\mathop{\rm meas}(\omega)<\eta, u'\in L^{\infty}(\Sigma_1 \setminus \omega)$
and $u'_m\to u'$ uniformly on $\Sigma_1 \setminus \omega$. Thus, for each $\epsilon>0$, there is an $N>\frac{2}{\epsilon}$ such that
\[
|u'_m(x,t)-u'(x,t)|<\frac{\epsilon}{2}, \quad
 \forall (x,t)\in  \Sigma_1 \setminus \omega.
\]
Then, if $|u'_m(x,t)-s|<1/m$, we have $|u'(x,t)-s|<\epsilon $ for all
$m>N$ and $(x,t)\in \Sigma_1 \setminus \omega$. Therefore,
\[
\underline{b}_{\epsilon}(u'(x,t))\leq b^m(u'_m(x,t))\leq
\overline{b}_{\epsilon}(u'(x,t)),
\quad \forall m>N, (x,t)\in  \Sigma_1 \setminus \omega.
\]
Let $\phi \in L^{\infty}(\Sigma_1), \phi \geq 0$. Then
\begin{equation}
\begin{aligned}
\int_{\Sigma_1 \setminus \omega}\underline{b}_{\epsilon}(u'(x,t))\phi(x,t)
d \Gamma dt
&\leq \int_{\Sigma_1 \setminus \omega}b^m(u'_m(x,t))\phi(x,t) d \Gamma dt    \\
&\leq \int_{\Sigma_1 \setminus \omega}\overline{b}_{\epsilon}(u'(x,t))\phi(x,t)
d\Gamma dt.
\end{aligned}
\end{equation}
Letting $m$ approach $\infty$ in (3.26) and using (3.21), we obtain
\begin{equation}
\begin{aligned}
\int_{\Sigma_1 \setminus \omega}\underline{b}_{\epsilon}(u'(x,t))\phi(x,t) d\Gamma dt
&\leq \int_{\Sigma_1 \setminus \omega}\Xi(x,t) \phi(x,t) d\Gamma dt   \\
&\leq \int_{\Sigma_1 \setminus \omega}\overline{b}_{\epsilon}(u'(x,t))\phi(x,t)
d\Gamma dt.
\end{aligned}
\end{equation}
Letting $\epsilon \to 0^+$ in (3.27), we infer that
$$
\Xi(x,t)\in \varphi(u'(x,t)) \quad\mbox{a. e. in } \Sigma_1 \setminus \omega,
$$
and letting $\eta \to 0^+$ we get
$$
\Xi(x,t)\in \varphi(u'(x,t))\quad \text{a.e. in } \Sigma_1.
$$
This completes the proof.

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