\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 27, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/27\hfil Regularity of vibrations]
{Regularity result for the problem of vibrations of a
nonlinear beam}

\author[M. F. M'Bengue,  M. Shillor\hfil EJDE-2008/27\hfilneg]
{M'Bagne F. M'Bengue, Meir Shillor}

\address{M'Bagne F. M'Bengue \newline
Department of Mathematics and Statistics \\
Oakland University, Rochester, MI 48309, USA}
\email{mfmbengu@oakland.edu}

\address{Meir Shillor \newline
Department of Mathematics and Statistics \\
Oakland University, Rochester, MI 48309, USA}
\email{shillor@oakland.edu}

\thanks{Submitted January 25, 2008. Published February 28, 2008.}
\subjclass[2000]{74H30, 35L75, 74K10, 74D10, 74H45}
\keywords{Nonlinear beam; existence and uniqueness; energy balance;
\hfill\break\indent  pseudomonotone operators; regularity}

\begin{abstract}
 A model for the dynamics of the Gao nonlinear beam, which allows for
 buckling,  is studied. Existence and uniqueness of the local weak solution
 was established in Andrews {\it et al.} (2008). In this work  the further
 regularity in time of the weak solution is shown using recent results for
 evolution problems. Moreover, the weak solution is shown to be global, 
 existing on each finite time interval.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction} \label{sec:int}

A model for the vibrations of a nonlinear beam that takes into
account the beam's thickness which, however, is one-dimensional, was
derived by Gao in \cite{Gao96,Gao00}. The existence of the unique
local weak solution for the problem was established recently in
\cite{ADuMBS08}. In this work we show that the weak solution has
additional regularity in time, when the problem data is smoother.
This allows us to establish that the weak solution is also a
global solution existing on each finite time interval.

The motivation for the introduction of various models for beams
was  to capture more fully the nonlinearities that beams exhibit,
in particular buckling, which is associated with a double-well in
the energy function of the beam. Among the models derived in
\cite{Gao00}, the one-dimensional model studied in
\cite{ADuMBS08} and here is the simplest. However, it is
nonlinear, it allows for  buckling, and has interest in and of
itself. The literature on the beam includes also   \cite{RW02} and
the references therein.

This work is the continuation of  \cite{ADuMBS08} where, in
addition  to the analysis, a finite difference scheme for the beam
was introduced based on the Newmark time discretization and
Hermite cubic finite elements, and the results of numerical
simulations depicted.  Here, we prove that if the problem data is
more regular, then the solution has additional time regularity.
The proof is based on the problem for the time derivative of the
solution. We first study the truncated problem, and use results
for variational problems for pseudomonotone operators of
\cite{Ken86, KSpsm}. Then, we use a continuity argument to show
that there exists a unique weak solution such that for some time
the truncation is inactive. Using the energy balance and a priori
estimates derived from it, we also show that for a sufficiently
large truncation ceiling, the truncation is not active on each
preassigned time interval, and so the solution is global.

The rest of the paper is organized as follows. In Section 2 we
present  shortly the classical formulation of the model, following
\cite{Gao96, Gao00, ADuMBS08}.  In Section \ref{sec:Problem} the
weak formulation and the statement of the existence and uniqueness
result in \cite{ADuMBS08} is given, the weak or abstract
formulation of the problem for the time derivative presented and
the statements of the existence and uniqueness results for the
truncated problem and the full problem stated.  Our main
regularity result is states in Theorem \ref{thm34}. The proof is
provided in Section \ref{sec:proof}, and is based on the results
for the truncated problem in Section \ref{sec:proofR}. In Section
\ref{sec:E} we present the energy balance equation for the
original problem, derive a priori estimates on
$\|w_x\|_{L^\infty(0, T; L^\infty(0,1))}$, and based on this
estimate we conclude that the solution of the problem is global in
time.

\section{The Model}\label{sec:Problem}

The derivation of the model was done in Gao in \cite{Gao96, Gao00}, and a
more detailed description can be found in \cite{ADuMBS08}. Here,
we just present it with very few comments. The beam's  centerline
is, in dimensionless variables, $[0, 1]$ and its thickness $2h$,
and we denote by  $w(x,t)$ the transverse displacement of its
central axis, for $0\leq x\leq 1$ and $0\leq t \leq T$, for 
$0<T$.

\begin{figure}[ht]
\texttt{\setlength{\unitlength}{1pt}
\begin{picture}(218,90)(128,-20)
\thicklines
        \put(340,10){$x$}
        \put(314,30){$p(t)$}
        \put(300,-4){$1$}
         \put(322,25){\vector(-1,0){20}}
             \put(212,26){$w$}
                      \put(120,20){\vector(1,0){230}}
                  \put(120,-20){\vector(0,1){90}}
                  \put(120,30){\line(1,0){180}}
                  \put(120,10){\line(1,0){180}}
                    \put(300,10){\line(0,1){20}}
                   \put(210,20){\vector(0,1){27}}
{\color[rgb]{1,0,1} %pink
 \qbezier(120,30)(210, 80)(300,30)}
{\color[rgb]{1,0,1}
 \qbezier(115,10)(202, 60)(295,10)}
 \put(120,20){\line(1,0){40}}
{\color[rgb]{0,0,1}
 \qbezier(115,20)(202, 70)(295,20)}
\end{picture}}
\caption{The beam; $w$ is the displacement of the central axis.}
\end{figure}

The beam is subjected to a laterally distributed load $f=f(x,t)$,
and a  horizontal traction  which acts at the end $x=1$. The
simplest finite deformations nonlinear model derived in
\cite{Gao96, Gao00}  for the beam consists of the evolution equation
\begin{equation}
\label{g1}
w_{tt} +  k w_{xxxx}+  (\nu p-a w^{2}_{x}) w_{xx}={f},
\end{equation}
in $\Omega_T=(0,1)\times(0, T)$, where all the variables are in
dimensionless form, and $k, \nu$,  and $a$ are system parameters,
assumed to be positive constants.  The term with $p=p(t)$
describes the effect of the horizontal force (compression/tension)
acting on the right end. When $0<p$ the beam is being compressed,
and when $p<0$ it is under tension.


We assume that the beam is clamped at both ends and initially the
displacement is $w_0$ and the velocity is $v_0$. Also, for the
sake of generality, we add a viscosity term $\gamma w_{txxxx}$,
with viscosity coefficient   $\gamma >0$, assumed to be small.

The classical formulation of the {\it dynamic model for a beam
with finite deformations} with viscosity, is as follows.

\subsection*{Problem $P_{cl}$}
  Find the displacement field
$w=w(x, t)$ for $x\in (0, 1)$ and $t\in (0, T)$, such that
\begin{gather}
   w_{tt} + k w_{xxxx}+ \gamma w_{txxxx}
-(aw_x^2 - \nu  p) w_{xx}= f, \label{p1}\\
 w(0,t)=w_x(0,t) =0,    \label{p3}\\
 w(1,t)=w_x(1,t) =0,    \label{p4}\\
 w(x, 0)=w_0(x),\quad w_t(x,0)=v_0(x). \label{p5}
\end{gather}

The system is nonlinear and  the existence and uniqueness of  the
local weak solution to the problem has been established in
\cite{ADuMBS08}. We first present the weak formulation of the
problem and then show that additional assumptions on the problem
data lead to an improved regularity, in time, of the solution.


\section{Weak formulation and results}\label{sec:weak}

We first describe the weak formulation of Problem $P_{cl}$, the
assumptions  on the problem data, and state the existence and
uniqueness result for local weak solutions. We follow
\cite{ADuMBS08}. Then, we describe the problem obtained by
differentiating Problem $P_{cl}$ with respect to time.

We denote by $(\cdot, \cdot)$ the  inner product in $H=L^2(0, 1)$,
and let
\[
W=H^2_0(0,1)=\{u\in H^2(0, 1): u=u_x=0\;\mbox{at }\; x=0, 1\},
\]
be a Hilbert space endowed with the inner product $(w, u)_W =
\int_0^1 w_{xx} u_{xx}\,dx,$ and the associated norm
$\|w\|_W^2=(w,w)_W$ which,  in view of the boundary conditions and
the Poincare theorem,  is equivalent to the  usual $H^2(0,1)$ norm
on $W$. The dual of $W$ is denoted by $W'$, and since
$W\subseteq H \subseteq W'$, by identifying  $H'=H$, it follows that
$(W, H, W')$ is a Gelfand triple. Next,  let $\mathcal{H}=L^2(0, T; H)$,
and $\mathcal{W}=L^{2}(0,T; W)$ with inner product
$(\cdot, \cdot)_\mathcal{W}$ and duality pairing
$\langle \cdot, \cdot \rangle _\mathcal{W}$ between $\mathcal{W}$
and its dual $\mathcal{W'}$,  which we write as
$\langle \cdot, \cdot \rangle$.
Again, we have
\[
\mathcal{W} \subseteq \mathcal{H}= \mathcal{H'}\subseteq \mathcal{W'} .
\]

Next, proceeding as usual, we obtain the following
{\it variational formulation of  the problem of vibrations of a nonlinear
beam}.

\subsection*{Problem $P_V$}
 Find the displacement field $w : [0, T] \to  W$ and the
velocity $v=w_t$, such that for a.a. $t \in [0, T]$ and $ \psi \in W$,
\begin{gather}
\begin{gathered}
 \langle    v_{t}(t), \psi \rangle_W + k (w_{xx}(t),  \psi_{xx})
+ \gamma (v_{xx}(t),  \psi_{xx})\\
 +\frac13 a  (w_x^3(t),  \psi_x)
 - \nu p(t)( w_{x}(t),  \psi_{x})= ( f(t) ,  \psi),
\end{gathered} \label{v1}\\
w(0)=w_0,\quad w_t(0) =v_0.  \label{v3}
\end{gather}


We make the following assumptions on the problem data:
\begin{gather}
     w_{0}, \; v_{0} \in W, \quad \|w_{0xx}\|_{L^2(0, 1)},
    \|w_{0x}\|_{L^{\infty}(0, 1)}\leq R^*,  \label{w01}\\
     p\in C^1([0,T]),\quad |p|, |p'| \leq p^*,  \label{p}\\
     f \in \mathcal{H}.  \label{f0}
\end{gather}
Here, $R^*$ and $ p^*$ are two positive constants.

The main existence and uniqueness result in \cite{ADuMBS08} is the following.

\begin{theorem} \label{thm-main}
Assume that \eqref{w01}--\eqref{f0} hold. Then there exists $T^*>0$ and a
unique solution $w$ to  Problem $P_{V}$ on the time interval $[0, T^*)$
 such that
\begin{equation}
\label{36}
w, v \in L^{\infty}(0, T^*; W),\quad v' \in  L^2(0, T^*; { W}').
\end{equation}
\end{theorem}

To establish additional regularity of $w$, we study the problem for $v=w'$,
where here and below we denote by a prime the (weak) time derivative.
We differentiate  equation (\ref{v1}) with
respect to t, set $z=w''=v'$, and, for $\psi \in W$, we obtain
\begin{equation}
\begin{gathered}
 \langle   z'(t), \psi \rangle_W + k (v_{xx}(t), \psi _{xx})
+ \gamma (z_{xx}(t), \psi_{xx})
 + a  (w_x^2(t)v_x(t), \psi_x)  \\
 -\nu  p'(t)(w_{x}(t), \psi_{x})-\nu  p(t)(v_{x}(t),
\psi_{x})= ( f'(t) , \psi).
\end{gathered}\label{reg1}
\end{equation}
To obtain the initial condition for $z$, we formally  set $t=0$ in
(\ref{p1}) and obtain condition (\ref{reg3}) below.

We have the following problem for the triple $\{w, v, z \}$.

\subsection*{Problem $P_{Vz}$}
 Find the displacement field $w: [0,T]\to W$, the velocity field
$v:[ 0,T]\to W$, and  the acceleration $z:[0,T] \to W$ such that
for a.a. $t\in[0,T]$
and every $\psi \in W$ the variational equation (\ref{reg1}) holds,
together with
\begin{gather}
w(t)=w_0+\int_{0}^{t} v(\tau)\, d\tau, \quad v(t)
=v_0+\int_{0}^{t} z(\tau)\,d\tau,
\label{reg2} \\
 z(0)= - k w_{0xxxx}- \gamma v_{0xxxx}
+aw^2_{0x} w_{0xx} - \nu  p(0) w_{0xx}+ f(0). \label{reg3}
\end{gather}
Problem \eqref{reg1}--\eqref{reg3}  makes sense only if we assume,
in addition to (\ref{w01})--(\ref{f0}), that
\begin{equation}
  f, f' \in {\mathcal{W}'}, \quad f(0)\in H, \label{ff}
\end{equation}
and to ensure that $z(0)\in L^{2}( 0,1)$ we assume that
\begin{equation}
w_{0}, \; v_{0} \in H^4(0,1), \quad
w_{0} \in {H_0^2(0, 1)}, \quad
\|w_{0x}\|_{L^{\infty}(0, 1)}\leq R^*.  \label{reginit}
\end{equation}
We note that  the term $a w^2_{0x} w_{0xx}$ is  well defined.


To deal with the term with $w_x^2v_x$ we introduce the truncation
\begin{equation}
\label{etast}
\Psi_R(r)=\begin{cases}
  R   & \text{for } R \leq r,\\
  r    & \text{for } |r| \leq R, \\
 -R   & \text{for } r \leq - R,
\end{cases}
\end{equation}
where  $R$ is a large positive number, and we replace $w_x^2$ with
$\Psi^2_R(w_x)$.  Eventually, we show that when  $R$ is sufficiently
large, the truncation is inactive.

To proceed with the abstract formulation of the truncated problem,
we define the operators $ B,  K: \mathcal{W}\to {\mathcal{W'}}$, and
$ K_{RN} : \mathcal{W}\times  \mathcal{W} \to {\mathcal{W'}}$ by
\begin{gather}
   \langle {B}(w), \psi \rangle
=\int^{T}_{0}\int^{1}_{0} w_x \psi_{x}\, dx\,dt,    \label{B1}\\
\langle K( w), \psi \rangle =\int^{T}_{0}\int^{1}_{0}
w_{xx}\psi_{xx}\, dx\,dt,   \label{K1} \\
 \langle K_{NR}(w, v),\psi\rangle  =\int_{0}^{T}\int_{0}^{1}
\Psi_R^2(w_{x})v_{x}\psi_{x}\,dx\,dt.  \label{K3}
\end{gather}
We introduce the function space
\[
\mathcal{Y} =\mathcal{W}\times  \mathcal{W}\times\mathcal{W},
\]
and denote its dual by $\mathcal{Y}'$.

The abstract formulation of the truncated version of Problem $P_{Vz}$ is:

\subsection*{Problem $P_{Vz R}$}
 Find $(w,v,z)\in \mathcal{Y} $, with $z' \in \mathcal{W'}$ such that:
\begin{gather}
  v=w',   \quad  z = v', \label{318}\\
 z' + k K(v)+ \gamma K(z )  + aK_{NR}(w, v)
 -\nu  p' B(w)-\nu  p B(v)=f', \label{319}
\end{gather}
in $\mathcal{W'}$, together with (\ref{reg2}) and  (\ref{reginit}).


The abstract formulation of  Problem $P_{Vz}$ is obtained by
reinstating $w_x^2$ in place of $\Psi^2_R(w_x)$ in $K_{NR}$.

Next, we rewrite (\ref{318}) in an equivalent form,
\[
 K(v) =K(w'),   \quad  K(z) = K(v').
\]
The equivalence follows from the boundary conditions.  This allows us to
show the coercivity of the operator $A_{\rm reg}$.

The operator  $A: \mathcal{Y}\to\mathcal{Y}' $, for
$y=(w, v, \varphi )\in \mathcal{Y} $, is defined by
\[
A(w,v, \varphi) =k K(v) +\gamma K(\varphi)
+a K_{NR}(w, v)-\nu  p' B(w) -\nu  p B(v).
\]
The operator
$A_{\rm reg}:\mathcal{Y}  \to \mathcal{Y} ' $ is defined,
for $y=(w, v, \varphi )\in \mathcal{Y} $, by
\[
A_{\rm reg}(y)=\left( - K(v), - K(\varphi),  A(w, v, \varphi) \right).
\]

We let $G = W\times W\times H$ with dual $G'$, we define
$\mathcal{G}=\mathcal{W}\times \mathcal{W}\times \mathcal{H}$,
the operator $D:\mathcal{G} \to \mathcal{G}' $ is defined,
for $y=(w, v, \varphi )\in \mathcal{G}$, by
\[
D(y)=\left( K(w),  K(v), \varphi \right),
\]
and  the functional $F:\mathcal{Y} \to\mathbb{R}^3$ as
\[
 \langle F, y \rangle  =(0, 0, \int_0^T \int_{0}^{1} f' \varphi\,dx\,dt).
\]

Problem $P_{Vz R}$ can now be written in the following
abstract form.

\subsection*{Problem $P_{AR}$}
 Find $y=(w,v,z)\in \mathcal{Y} $ such that
\begin{gather*}
(Dy)' + A_{\rm reg}(y)= F, \quad \mbox{in } \mathcal{Y} ',\\
Dy(0)=Dy_{0}, \quad  \mbox{in } G',
\end{gather*}
where $w$ and $v$ are given in (\ref{reginit}),  $y_0=(w_0, v_0, z(0))$,
and $z(0)$ in (\ref{reg3}).

\begin{theorem} \label{thm32}
Assume that \eqref{w01}--\eqref{f0},  \eqref{ff} and \eqref{reginit}
hold. Then Problem $P_{AR}=P_{Vz R}$ has a unique weak solution.
\end{theorem}

The proof of the theorem is given in the next section.
The main step to the main result of this paper is  the following.

\begin{theorem} \label{thm33}
Assume that \eqref{w01}--\eqref{f0}, \eqref{ff} and \eqref{reginit}
hold. Then there exists $0<T^*$ such that Problem $P_{Vz}$ has a
unique weak solution on the time interval $[0, T^*)$.
\end{theorem}

The proof is provided in Section \ref{sec:proof}, and is based on the
observation that if $R$ is sufficiently large and $w_{0x}$ is bounded
in $L^\infty$, then, by the continuity of the solution,
the $L^\infty$ norm of $w_x$ is bounded by $R$ on a time interval
 $[0, T^*)$, hence the truncation $\Psi_R(w_x)$ is inactive, and
on this time interval the problem with or
without truncation has the same solution.

Then, the argument presented in the next section, which provides an a
priori estimate based on an energy balance, shows that the unique
weak solution is actually global in time,
and exists on each time interval $[0, T]$.

The main result of this paper shows that the solution of Problem $P_V$
has additional regularity in time.

\begin{theorem} \label{thm34}
Assume that \eqref{w01}--\eqref{f0}, \eqref{ff}--\eqref{reginit}
hold. Then, for each $0<T$,  Problem $P_{V}$, \eqref{v1} and
\eqref{v3}, has a unique weak  solution  on the time interval $[0, T]$,
and the solution satisfies
\begin{equation}
\label{321}
w, v, v' \in L^{\infty}(0, T; W),\quad v'' \in  L^2(0, T, { W}').
\end{equation}
\end{theorem}

We conclude that the acceleration $v'$ is a function and only its time
derivative may be a distribution, and moreover,
\begin{equation}\label{319a}
w\in C^1([0, T]); W), \quad v \in C([0, T]; W), \quad v' \in C([0, T]; H).
\end{equation}

In the next section we show that the solution is global and exists
on each finite time interval $[0, T]$.


\section{Energy estimate and global solution} \label{sec:E}

In this section we use an energy balance to obtain an a priori estimate
that allows us to conclude that the solution of Problem $P_V$ is global.

Proceeding formally, the following energy balance was obtained in
\cite{ADuMBS08} for
Problem $P_V$ on $[0, T^*)$:
\begin{equation}
\begin{aligned}
E(t) &\equiv \frac12 \| v(t)\|_H^2 + \frac k2 \|w_{xx}(t)\|_H^2
+ \frac a{12}\|w_x^2(t)\|_H^2-\frac 12 { \nu} p(t) \|w_x(t)\|_H^2
 \\
&=E(0) -\gamma \int_0^t \|v_{xx}(\tau)\|_H^2\,d\tau
- \frac 12 { \nu} \int_0^t p'(\tau)  \|w_x(\tau )\|_H^2\,d\tau \\
&\quad  + \int_0^t (f(\tau), v(\tau))\,d\tau.
\label{E}
\end{aligned}
\end{equation}
The initial energy is
\[
E(0)=\frac 12 \| v_0\|_H^2 + \frac k2 \|w_{0xx}\|_H^2
+ \frac a{12}\|w_{0x}^2\|_H^2
-\frac 12 { \nu} p(0) \|w_{0x}\|_H^2.
\]
The first integral on the right-hand side in (\ref{E})  is the viscous
dissipation, the second one is related to the work done by   $p$,
while the third one is the work of the applied force $f$.

 It follows from the assumptions (\ref{w01})--(\ref{f0}) on the problem
data that $|E(0)| = E_0 <\infty$,
 and also that the energy balance is actual, as the regularity
 of the solution, (\ref{321}) and (\ref{319a}),  justifies the manipulations that lead
to (\ref{E}).

Rearranging the terms, using the fact that
$\|w_{x}(t)\|_H^2 \leq \|w_{xx}(t)\|_H^2 $,
and manipulations that include the Cauchy inequality, we obtain for
$0\leq t < T^*$,
\begin{align*}
&\frac12 \| v(t)\|_H^2 + \frac k2 \|w_{xx}(t)\|_H^2
 +\gamma \int_0^t \|v_{xx}(\tau)\|_H^2\,d\tau  \\
& \leq E_0 +\frac 12 \int_0^t \|f(\tau)\|^2_H\,d\tau +
\frac 12 { \nu} p^* \|w_x(t)\|_H^2 - \frac a{12}\|w_x^2(t)\|_H^2\\
&\quad +C\int_0^t \left( \| v(\tau)\|_H^2 +
\|w_{xx}(\tau)\|_H^2 \right)\, d\tau.
\end{align*}
Here, $C=C(k, \nu, p^*)$ is a positive constant.
Next, we study $J=J(t)$ which is given by
\[
J(t)\equiv \frac 12 { \nu} p^* \|w_x(t)\|_H^2
- \frac a{12}\|w_x^2(t)\|_H^2 = \frac a {12}\int_0^1
\left( \frac{6 \nu p^*}{a} - w_x^2(x, t) \right)w_x^2(x, t)\,dx.
\]
Let $\chi_+(x, t)$ be the subset of $[0, 1]$ where
$w_x^2(x,t) \leq 6 \nu p^*/a$ and let
$\chi_- (x, t)$ be the complement where $w_x^2(x, t) > 6 \nu p^*/a$. Then
\[
J(t) \leq \frac a {12}\int_{\chi_+(x, t)}
\left( \frac{6 \nu p^*}{a} - w_x^2(x, t) \right)w_x^2(x, t)\,dx\leq 3 \nu^2 (p^*)^2 \equiv c_{*\nu}.
\]
Using now the Gronwall inequality yields
\begin{equation}
\label{42}
 \| v(t)\|_H^2 +  \|w_{xx}(t)\|_H^2 \leq \exp(C T^*)
\left((E_0 + c_{*\nu})T^* +\|f\|^2_{L^2(0, T^*, H)}
 \right),
\end{equation}
for all $0\leq t <T^*$. Thus, $ \|w_{xx}(t)\|_H\leq C^*$, where $C^*$
depends only on the data and on $T^*$.
Finally, we note that the H\"older inequality yields
\[
|w_x(x,t)|  \leq \|w_x(t)\|_{L^{\infty}(0,1)} \leq \int_0^1 |w_{xx}(r,t)|\,dr\leq \|w_{xx}(t)\|_{L^2(0,1)}
\leq C^*,
\]
for  $0\leq t \leq T^*$. But this means that if we choose the truncation
ceiling $R$ such that $C^* << R$, then by a continuity argument
the truncation $\Psi_R$
(used in \cite{ADuMBS08}) is inactive on $0\leq t \leq T^*+ \varepsilon$,
where $\varepsilon$ depends only on $C^*$,  and therefore, the
solution exists on any finite time interval $[0, T]$.

\section{Proof of Theorem 3.2}\label{sec:proofR}

The proof is based on the existence results for evolution problems
established in  Kuttler \cite{Ken86}  and  \cite{KSpsm}.
To use these results we need to show that
 the operator $A_{\rm reg}: \mathcal{Y} \to \mathcal{Y} '$ is bounded,
coercive, and pseudomonotone,
 that is,  for $y=(w, v, \varphi) \in \mathcal{Y} $,  it satisfies
the following conditions:
\begin{enumerate}
\item
 $\|A_{\rm reg}(y)\|_{\mathcal{Y} '}\leq C(1+\|y\|_{\mathcal{Y} })$.\vskip3pt
\item
$\displaystyle \lim_{\|y\|_{\mathcal{Y} }\to \infty}\frac{\langle \lambda
Dy
+A_{\rm reg}(y), \,
y \rangle}{\|y \|_{\mathcal{Y} }} = \infty$,   for $\lambda$ sufficiently
large.  \vskip3pt
\item
$y \to A_{\rm reg}(y)$  is a pseudomonotone map from  $\mathcal{Y} $ to
 $\mathcal{Y} '$.
 \end{enumerate}
 These conditions guarantee the existence of a weak solution to
the truncated problem.

 Actually, to obtain the coercivity (2)  one has to use an exponential
shift in which the new dependent variable $\widetilde{y}$ is given by
$\widetilde{y}= y e^{-\lambda t}$.
Then in all the linear terms we find that  $\lambda D+A_{\rm reg}$
replaces $A_{\rm reg}$ and the nonlinear term $K_{NR}$ changes with
an exponential  $e^{\lambda t}$.
Since the problem is only considered on a finite interval, to save on
notation and simplify the presentation, we ignore this minor  technical
detail and note that it suffices to show that (1)--(3) hold for
$y=(w,v,\varphi)\in \mathcal{Y}$.

These steps are summarized in the following lemmas. The proofs
parallel  the ones presented in \cite{ADuMBS08}, so we only
present the modifications which deal with the different  nonlinear
term.  Indeed, the operators $B$ and $K$ are linear, so we need
only to study $K_{NR}$.

 Below, we let $C$ denote a positive constant which depends on the
problem data and whose  value may change from place to place.

 \begin{lemma} \label{lemma1}
 The operator $K_{NR}$ is bounded.
\end{lemma}

\begin{proof}
Since $|\Psi_{R}^2(w_{\varphi\, x})|\leq R^2$, by using the H\"older
inequality we obtain
\begin{align*}
|\langle K_{NR}(w, v), \psi  \rangle|
&\leq \int_{0}^{T}\int_{0}^{1}|\Psi_{R}^2(w_x)| |v_x| |\psi_{x}|
\,dx\,dt \\
&\leq R^2\int_{0}^{T}\int_{0}^{1} |v_{x}| |\psi_{x}|\,dx\,dt \\
&\leq R^2\|v \|_{L^2(0,T;H^1)} \|\psi \|_{L^2(0,T;H^1)} \\
&\leq C R^2 \| v \|_{\mathcal{W}} \|\psi\|_{\mathcal{W}}.
\end{align*}
Therefore,
\[
\| K_{NR}(w, v)\|_{\mathcal{W}'} \leq CR^2 \| v \|_{\mathcal{W}}
\leq CR^2 \| y \|_{\mathcal{Y}}.
\]
The rest of the estimates leading to the boundedness of $A$ are
straightforward (see  \cite[Lemma 3.3]{ADuMBS08}) and from these
we obtain  the boundedness of  $A_{\rm reg}$.
\end{proof}


\begin{lemma} \label{lemma1b}
 The operator $\lambda D+A_{\rm reg}$ is coercive for all $\lambda $
sufficiently large.
\end{lemma}

\begin{proof}
We have
\[
\langle \lambda Dy +A_{\rm reg}(y), y \rangle=\langle \lambda Dy, y 
\rangle + \langle A_{\rm reg}(y), y \rangle.
\]
Then,
 \begin{align*}
\langle \lambda Dy, y \rangle
&=(\lambda Dy, y ) =
\lambda ( \| w_{xx} \|^2_{\mathcal{H}} +\| v_{xx} \|^2_{\mathcal{H}}
+ \| \varphi \|^2_{\mathcal{H}}) \\
&=\lambda ( \| w \|^2_{\mathcal{W}} +\| v \|^2_{\mathcal{W}} + \| \varphi \|^2_{\mathcal{H}}).
 \end{align*}
Here, we used $\| w_{xx} \|_{\mathcal{H}}$ as the norm on $\mathcal{W}$.
Next,
 \begin{align*}
 \langle A_{\rm reg}(y), y \rangle
&\geq - |\langle K v, w \rangle| - |\langle K\varphi, v \rangle|
 + \langle A(w, v, \varphi), \varphi \rangle| \\
&\geq - \| v \|_{\mathcal{W}}\| w \|_{\mathcal{W}}
 - \| \varphi\|_{\mathcal{W}}\| v \|_{\mathcal{W}}
 + \langle A(w, v, \varphi), \varphi \rangle.
\end{align*}
Using the Cauchy inequality with $\epsilon$ leads to
 \[
 \| v \|_{\mathcal{W}}\| w \|_{\mathcal{W}}
 +\| \varphi\|_{\mathcal{W}}\| v \|_{\mathcal{W}}
 \geq - {C}(\| v \|^2_{\mathcal{W}}+\| w \|^2_{\mathcal{W}})
 -\frac 14 \gamma  \| \varphi\|^2_{\mathcal{W}}.
 \]
Also,
 \begin{align*}
 \langle A(w, v, \varphi), \varphi \rangle
&=k ( v_{xx}, \varphi_{xx})_{\mathcal{H}}
 +\gamma \|  \varphi\|^2_{\mathcal{W}}
-\nu ( p' w_{x}, \varphi_x)_{\mathcal{H}} \\
&\quad -\nu (p v _{x}, \varphi_x)_{\mathcal{H}} +
  a(  \Psi_R^2(w_{x})v_{x}, \varphi_{x})_{\mathcal{H}}.
 \end{align*}
Therefore, using the Cauchy inequality with $\epsilon$, again, yields
 \begin{align*}
 \langle A(w, v, \varphi),   \varphi \rangle
&\geq \gamma \|  \varphi\|^2_{\mathcal{W}}
 -k\|v_{xx}\|_{\mathcal{H}} \|\varphi_{xx}\|_{\mathcal{H}}
-\nu p^* \|w_{x}\|_{\mathcal{H}} \|\varphi_{x}\|_{\mathcal{H}} \\
&-\nu p^* \|v_{x}\|_{\mathcal{H}} \|\varphi_{x}\|_{\mathcal{H}}
 -aR^2 \|v_{x}\|_{\mathcal{H}} \|\varphi_{x}\|_{\mathcal{H}} \\
&\geq \frac 12 \gamma \|  \varphi\|^2_{\mathcal{W}}
 - C\left(\| w \|^2_{\mathcal{W}} +  \| v \|^2_{\mathcal{W}} \right).
 \end{align*}
Collecting these estimates and rearranging the constants we obtain
 \[
 \langle A_{\rm reg}( y),   y \rangle  \geq \frac14 \gamma \|  \varphi\|^2_{\mathcal{W}}
 -  C\left(\| w \|^2_{\mathcal{W}} +  \| v \|^2_{\mathcal{W}} \right).
\]
Thus,
\begin{align*}
&\langle \lambda Dy +A_{\rm reg}( y),  y \rangle\\
&\geq \lambda ( \| w \|^2_{\mathcal{W}} +\| v \|^2_{\mathcal{W}})
 + (\lambda \| \varphi \|^2_{\mathcal{H}}
 + \frac14 \gamma \|  \varphi\|^2_{\mathcal{W}})
 - C\left(\| w \|^2_{\mathcal{W}} +  \| v \|^2_{\mathcal{W}} \right).
\end{align*}
It follows that when $\lambda$ is sufficiently large,
\[
 \lim_{\| y \|_{\mathcal{Y} }\to \infty}\frac{\langle (\lambda Dy
 +A_{\rm reg}) ( y),  y \rangle}  {\|y\|_{\mathcal{Y} }}=\infty.
\]
Hence, the operator $\lambda D+A_{\rm reg}$ is coercive.
 \end{proof}

We prove, next,  that the operator $A_{\rm reg}$ is pseudomonotone on a
smaller space, which is sufficient for our purposes. Let
\[
{\mathcal{Z}}= \{y \in \mathcal{Y};\;  y' \in \mathcal{Y}'\}
\]
be a Banach space with the product norm
$\|y \|_{\mathcal{Z}}=\| y \|_{\mathcal{Y}}+\| y' \|_{\mathcal{Y}'}$.
Then, we consider the operator as  $A_{\rm reg}:\mathcal{Z} \to \mathcal{Z} '$.
We have the following result.

 \begin{lemma} \label{lemma1c}
 The operator $A_{\rm reg}:\mathcal{Z} \to \mathcal{Z} '$ is
pseudomonotone. \end{lemma}

\begin{proof}
We have
\[
A(w,v, \varphi) =k K(v) +\gamma K(\varphi)
+a K_{NR}(w, v)-\nu  p' B(w) -\nu  p B(v).
\]
Since ${\mathcal{Z}}$ is a reflexive Banach space,
 it suffices to show that
\[
K_{NR}:{\mathcal{Z}}\to{\mathcal{Z}}'
\]
is a weak to norm continuous, where $K_{NR}$ is trivially restricted to 
to  ${\mathcal{Z}}$. We observe that the sum of the other
operators in $A_{\rm reg}$ is linear, bounded and monotone, thus
pseudomonotone.


Let $\{y_m\}$ be a sequence which converges weakly  in
${{\mathcal{Z}}}$ to $y$.
Then, the sequence $\{y_m=(w_{m}, v_{m}, \varphi_{m})\}$  converges weakly
in ${{\mathcal{Y}}}$ to $y=(w, v, \varphi)$.

Since $\{y_{m}\}$  is  bounded in $\mathcal{Z}$,  it follows from the
theorems of Simon \cite{Sim87} or Aubin \cite{Aubin}, that each one
of the sequences $\{\varphi_{m}\}, \{v_{m}\}$ and $\{w_{m}\}$
is relatively compact in $L^2(0,T; H^{1}(0,1))$. Thus, there exist
subsequences $\{\varphi_{m_{j}}\}, \{v_{m_{j}}\}$
and $\{w_{m_{j}}\}$  which converge strongly  in  $L^2(0,T;H^{1}(0,1))$
and almost everywhere in
$[0,1]\times[0, T]$ to $\varphi, v$ and $ w$, respectively.
Then,
\[
\Psi_{R}^{2}(w_{m_{j}x}){v}_{m_{j}x} \to
\Psi_{R}^{2}(w_{ x})v_{x},
\]
strongly in $\mathcal{H}=L^2(0, T; L^2(0,1)$ and a.e.
in $[0,1]\times[0, T]$.
Thus, for each $\psi \in\mathcal{Y}$, we have
\begin{align*}
&|\langle K_{NR}(w_{m_{j}}, v_{m_{j}})-K_{NR}(w, v), \psi \rangle| \\
&\leq C \|\Psi_{R}^{2}(w_{m_{j}x}) v_{m_{j} x}
- \Psi_{R}^{2}(w_{x}) v_{x} \|_\mathcal{H} \|\psi\|_\mathcal{Y},
\end{align*}
and, as $m_{j} \to \infty$,
\[
\| K_{NR}(w_{m_{j}}, v_{m_{j}})-K_{NR}(w, v)\|_{\mathcal{Y'}}\leq
 C\|\Psi_{R}^{2}(w_{m_{j}x}) v_{m_{j} x}
- \Psi_{R}^{2}(w_{x}) v_{x} \|_\mathcal{H} \to 0.
\]
Thus, $K_{NR}$ is weak to norm continuous and, since the sum
of the other terms is linear, bounded and monotone,  we conclude
that $A_{\rm reg}$ is pseudomonotone.
 \end{proof}

Since the conditions (1), (2) and (3) are satisfied, 
the abstract existence theorems in  \cite{Ken86, KSpsm}
provide  a solution to Problem $P_{VzR}$ for each 
sufficiently large $R$.  This completes the proof
of the existence part  of Theorem \ref{thm32}.
The uniqueness is shown by using the Gronwall inequality. However,
we skip it here since it will be used below in a similar manner.

\section{Proof of Theorem 3.3} \label{sec:proof}

We prove the theorem by showing the following result, and then prove
the uniqueness of the solution.

  \begin{proposition}  \label{propR}
 Let $R$ be sufficiently large.  Then, there exists $T^*>0$ such that
the solution $(w_R, v_R, z_R)$  of  Problem $P_{Vz R}$  satisfies
$\Psi_R(w_x)=w_x$ a.e. on $[0, 1]\times [0, T^*)$.
  \end{proposition}

\begin{proof}
 It follows from Theorem \ref{thm32} that $w=w_R, v=w_R'\in L^2(0, T: W)$,
 then  \cite[Lemma 1.2]{Lio69} asserts that
$w \in C([0, T]; W)$.
Thus,  the mappings  $w_x:[0,T]\to H_0^1(0,1)$, and
$w_{xx}:[0,T]\to L^2(0,1) $
are continuous. Now,  $w_x(x,t)=\int_0^x w_{xx}(r,t)\,dr$, and since
$w_x(t) \in H^1_0(0,1)$,
the H\"older inequality yields
\[
|w_x(x,t)|  \leq \|w_x(t)\|_{L^{\infty}(0,1)}
\leq \int_0^1 |w_{xx}(r,t)|\,dr\leq \|w_{xx}(t)\|_{L^2(0,1)},
\]
for  $0\leq t \leq T$. Let $h(t)=\|w_{xx}(t)\|_{L^2(0,1)}$
then $h:[0,T]\to \mathbb{R}$ is  continuous on a compact set,
so it is bounded. Since $h(0)=\|w_{0xx}\|_{L^2(0,1)} \leq R^* < R$,
there exists $T^*\leq T$ such that $h(t) \leq R$ for all $t\in [0,T^*)$.
It follows that
\begin{equation} \label{51}
\|w_{x}(t)\|_{L^{\infty}(0,1)}  \leq   R,
\end{equation}
and, therefore, the truncation is inactive on the time interval $[0, T^*)$,
i.e.,  $\Psi_{R}^2(w_{x}) = w^2_{x}$, and so the solution
$(w_R, v_R, z_R)=(w, v, z)$ is a solution of the Problem  $P_{Vz}$
on $[0, T^*)$.
\end{proof}

This completes the proof of the existence of a local solution of Problem
 $P_{Vz}$.

We note in passing that it follows from the proof  that if $R_1 <R_2$
then the associated times satisfy  $T^*_1 <T^*_2$.

We next prove the uniqueness of the solution.

\begin{proposition}  \label{propU}
 The solution $(w,v,z)\in \mathcal{Y} $ of Problem  $P_{Vz}$ on
$[0, T^*)$  is unique.
  \end{proposition}

\begin{proof}
Let $y_i=(w_i,v_i,z_i )$, for $i=1,2$, be two solutions of the problem.
We subtract  (\ref{reg1}) for $y_1$ from (\ref{reg1}) for $y_2$, use
$\psi=z_2(t)-z_1(t)$ as a test function, and for a.a. $t\in (0, T)$,
 obtain
\begin{align*} 
&\langle   z_2'(t)-z_1'(t), z_2(t)-z_1(t) \rangle_W
+ k (v_{2 xx}(t)-v_{1 xx}(t), z_{2xx}(t) -z_{1xx}(t)) \\
&+ \gamma \|z_{2xx}(t)- z_{1xx}(t)\|^2_H
 + \nu  p'(t)(w_{2xx}(t)-w_{1xx}(t),  z_{2}(t) -z_{1}(t)) \\
& +\nu  p(t)(v_{2xx}(t)-v_{1xx}(t),  \ z_{2}(t) -z_{1}(t)) \\
&= - a  (w_{2x}^2(t)v_{2x}(t)- w_{1x}^2(t)v_{1x}(t),  z_{2x}(t)
-z_{1x}(t)),
\end{align*}
where we used the facts that in view of the boundary conditions
\[
(v_{2x}(t)-v_{1x}(t),  \ z_{2x}(t) -z_{1x}(t))
=- (v_{2xx}(t)-v_{1xx}(t),  z_{2}(t) -z_{1}(t)),
\]
 and similarly for the expression with $w$.
We may write it as
\begin{align*}
&\frac{d}{dt}\| z(t)\|_{H}^2   + k \frac{d}{dt}\|v_{xx}(t)\|_H^2
    +2 \gamma \|z_{xx}(t)\|_H^2+ 2 {\nu}  p(t)(v_{xx}(t), z(t)) \\
&- 2 {\nu}  p'(t)(w_{xx}(t), z(t))  \\
&=- 2a(w^2_{2x}(t)(v_{2x}(t)-v_{1x}(t)),z_x(t))- 2a((w^2_{2x}(t)
-w^2_{1x}(t))v_{1x}(t), z_x(t)),
\end{align*}
where $z(t)=z_{2}(t) -z_{1}(t)$, $v=v_2(t)-v_1(t)$ and $w=w_2(t)-w_1(t)$.

Integrating over $0\leq \tau \leq t$ and using the H\"older inequality,
(\ref{p}) and the boundedness of $w_{x}$ and $v_{x}$, yields
\begin{equation} \label{ineq41}
\begin{aligned}
&\| z(t)\|_H^2 +k\|v_{xx}(t)\|_H^2
 + 2\gamma\int_{0}^{t}\|z_{xx}(\tau)\|_H^2\,d\tau\\
&\leq 2  {\nu}p^* \int_0^t   \|w_{xx}(\tau)\|_H\|z(\tau)\|_H\,d\tau
  + 2  {\nu}p^* \int_0^t   \|v_{xx}(\tau)\|_H\|z(\tau)\|_H\,d\tau\\
&\quad +  {C}\int_{0}^{t}\|v_{x}(\tau)\|_H\|z_{x}(\tau)\|_H\,d\tau
  + {C}\int_{0}^{t}\|w_{x}(\tau)\|_H\  \|z_x(\tau)\|_H\,d\tau,
\end{aligned}
\end{equation}
where $ C$ is a positive number which is independent of $z, v$ or $w$,
and we used the fact that $w(0)=v(0)=0$, and also (\ref{51}).
Using  the Cauchy's inequality with $\epsilon$ on the right-hand side
leads to 
\begin{align*}
& \leq C\int_0^t  \left ( \|w_{xx}(\tau)\|^2_H  +\|v_{xx}(\tau)\|^2_H
+\|z(\tau)\|^2_H \right)\,d\tau \\
& + \frac C{4 \epsilon}  \int_0^t  \left ( \|w_{x}(\tau)\|^2_H
+  \|v_{x}(\tau)\|^2_H  \right)\,d\tau
+\epsilon  C \int_0^t  \left ( \|z_{x}(\tau)\|^2_H  \right)\,d\tau,
\end{align*}
then, using the estimates on $w_{x}$, $w_{xx}$, $v_{x}$, and $z_{x}$
in terms of $v_{xx}$ and $z_{xx}$, we find that the
above expression is less than or equal to
\[
  C_\epsilon \int_0^t  \left( \|z(\tau)\|^2_H + \|v_{xx}(\tau)\|^2_H\right)
 \,d\tau
  + \epsilon   {C} \int_0^t   \|z_{x x}(\tau)\|^2_H  \,d\tau.
\]
Now, we choose $\epsilon$ sufficiently small, say 
 $\epsilon  =  \gamma/C$, and obtain
\begin{equation}
 \| z(t)\|_H^2 +k\|v_{xx}(t)\|_H^2 \leq C\int_0^t  ( \|z(\tau)\|^2_H
 + \|v_{xx}(\tau)\|^2_H )\,d\tau.    \label{ineq42}
\end{equation}

It follows from  the Gronwall inequality  that $\| z(t)\|_H^2 =0$
and $\|v_{xx}(t)\|_H^2=0$.   Since $\|v(t)\|_W \leq {C}\|v_{xx}(t)\|_H$ and
$\|w_{xx}(t)\|_H \leq C \|v_{xx}(t)\|_H$,  we conclude that the solution
is unique.
\end{proof}

This concludes the proof of Theorem \ref{thm33}, and then
Theorem \ref{thm34} follows from it
and the estimate in Section  \ref{sec:E}.

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee for the comments
which led to an improved manuscript.

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\end{document}