\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 63, pp. 1--11.\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--2003/63\hfil Weak solutions]
{Weak solutions for a viscous $p$-Laplacian equation}

\author[Changchun Liu\hfil EJDE--2003/63\hfilneg]
{Changchun Liu}

\address{Changchun Liu \hfill\break
Department of Mathematics, Nanjing Normal University \\
 Nanjing 210097, China \hfill\break
Department of Mathematics, Jilin University, Changchun 130012, China}
\email{mathlcc@21cn.com}


\date{}
\thanks{Submitted August 5, 2002. Published June 10, 2003.}
\subjclass[2000]{35G25, 35Q99, 35K55, 35K70}
\keywords{Pseudo-parabolic equations, existence, uniqueness}


\begin{abstract}
  In this paper,  we consider the pseudo-parabolic equation
  $ u_t-k\Delta u_t=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$.
  By using the time-discrete method, we establish the existence
  of weak solutions, and also discuss the uniqueness.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

This paper concerns the study of the viscous $p$-Laplacian
equation
$$
\frac{\partial u}{\partial t} -k\frac{\partial\Delta u}{\partial t}
=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u),\quad x\in\Omega,\; p>2, \eqno(1.1)
$$
with boundary condition
$$
u\big|_{\partial\Omega}=0, \eqno(1.2)
$$
and initial condition
$$
u(x, 0)=u_0(x),\quad x\in\Omega. \eqno(1.3)
$$
Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ and $k>0$ is the viscosity
coefficient.
The term $k\frac{\partial\Delta u}{\partial t}$ in (1.1) is interpreted as due to viscous relaxation
effects, or viscosity; hence, the equation (1.1) is called
``viscous $p$-Laplacian equations''.
The well-known $p$-Laplacian equation is obtained by setting $k=0$.

Equation (1.1) arises as a regularization of the pseudo-parabolic equation
$$
\frac{\partial u}{\partial t}-k\frac{\partial\Delta u}{\partial t}=\Delta u,
\eqno(1.4)
$$
which arises in various physical phenomena. (1.4) can be assumed as a model
for diffusion of fluids in fractured porous media \cite{b1,d1,c2}, or as a model for heat
conduction involving a thermodynamic temperature $\theta=u-k\Delta u$ and a
conductive temperature $u$ \cite{t1,c1}.
Equation (1.4) has been extensively studied, and
there are many outstanding results concerning  existence, uniqueness, regularity,
and special properties of solutions, see for example 
\cite{c2,d1,d2,n1,r1,s1,t2}.

To derive (1.4), B. D. Coleman, R. J. Duffin and V.
J. Mizel considered a special kinematical situation, of
nonsteady simple shearing flow \cite{c2}. In fact, when the influence of
many factors, such as the molecular and ion effects, are considered,
one has the nonlinear relation $\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$
in stead of $\Delta u$ in right-hand side of (1.4). Hence, we obtain (1.1).

Equation (1.1) is something like the $p$-Laplacian equation,
but many methods which are useful for the $p$-Laplacian equation are no
longer valid for this equation.
Because of the degeneracy, problem (1.1)-(1.3) does not admit classical
solutions in general. So, we study weak solutions in the sense of following

\noindent{\bf Definition} A function  $u$  is said to be a weak
solution of (1.1)-(1.3), if the following conditions are satisfied:
\begin{enumerate}
\item $u\in L^\infty(0, T;W^{1, p}_0(\Omega))\cap C(0,T;H^1(\Omega))$,
$\frac{\partial u}{\partial t}\in L^\infty(0, T;W^{-1, p'}(\Omega))$, where $p'$ is conjugate
exponent of $p$.

\item For $\varphi\in C_0^\infty(Q_T)$ and $Q_T=\Omega\times(0,T)$,
$$
\iint_{Q_T}u\frac{\partial\varphi}{\partial t} dx\,dt+k\iint_{Q_T}\nabla u
\frac{\partial \nabla \varphi}{\partial t} dx\,dt
-\iint_{Q_T}|\nabla u|^{p-2}\nabla u\nabla \varphi dx\,dt=0\,.
$$
\item $u(x,0)=u_0(x)$.
\end{enumerate}

In this paper, we discuss first the existence of weak solutions.
Most proofs of existence for (1.4) are based on the Yoshida approximations
\cite{d2}, but these methods do not apply to (1.1).
Our method for proving the existence of weak solutions is based on a
time discrete method that constructs approximate solutions.
Later on, we discuss the uniqueness of a solution.
For simplicity we set $k=1$ in this paper.

\section{Existence of weak solutions}

\begin{theorem} \label{thm2.1}
If $u_0\in W_0^{1, p}(\Omega)$ with $p>2$,
then problem (1.1)-(1.3) has  at least one solution.
\end{theorem}

We use the a discrete method for constructing  an approximate solution.
First, divide the interval $(0,T)$ in $N$ equal segments and set $h=\frac T N$.
Then  consider the  problem
\begin{gather*}
\frac1h(u_{k+1}-u_k)-\frac1h(\Delta u_{k+1}-\Delta
u_k)=\mathop{\rm div}(|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}),
\tag{2.1}\\
u_{k+1}|_{\partial\Omega}=0, \quad k=0, 1, \dots, N-1,
\tag{2.2}
\end{gather*}
where $u_0$ is the initial value.

\begin{lemma} \label{lm2.1}
For a fixed $k$, if $u_k\in
H^1_0(\Omega)$,  problem (2.1)-(2.2) admits a weak solution
$u_{k+1}\in W_0^{1, p}(\Omega)$, such that for any
$\varphi\in C_0^\infty(\Omega)$, have
$$
\frac1h\int_\Omega(u_{k+1}-u_k)\varphi dx
+\frac1h\int_\Omega(\nabla u_{k+1}-\nabla u_k)\nabla\varphi dx
+\int_\Omega|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}\nabla \varphi dx=0.
\eqno(2.3)
$$
\end{lemma}

\begin{proof} On the space $W^{1,p}_0(\Omega)$, we consider the functionals
\begin{gather*}
\Phi_1[u]=\frac1p\int_\Omega|\nabla u|^p dx, \\
\Phi_2[u]=\frac12\int_\Omega|u|^2 dx, \\
\Phi_3[u]=\frac12\int_\Omega|\nabla u|^2 dx, \\
\Psi[u]=\Phi_1[u]+\frac1h\Phi_2[u]+\frac1h\Phi_3[u]-\int_\Omega fu dx,
\end{gather*}
where $f\in H^{-1}(\Omega)$ is a known function. Using Young's inequality,
there exist constants $C_1, C_2>0$, such that
\begin{align*}
\Psi[u]&=\frac1p\int_\Omega|\nabla u|^p dx+\frac1{2h}\int_\Omega|u|^2 dx
+\frac1{2h}\int_\Omega|\nabla u|^2 dx-\int_\Omega f u \,dx\\
&\geq C_1\int_\Omega|\nabla u|^p dx-C_2\|f\|_{-1};
\end{align*}
hence $\Psi[u]\to\infty$, as $\|u\|_{1, p}\to+\infty$. Here
$\|u\|_{1,p}$ denotes  the norm of $u$ in $W^{1,p}_0(\Omega)$.

Since the norm is lower semi-continuous and $\int_\Omega fudx$ is a
continuous functional,
$\Psi[u]$ is weakly lower semi-continuous on $W_0^{1, p}(\Omega)$ and
satisfying the coercive condition. From \cite{c3} we conclude that there exists
$u_*\in W^{1,p}_0(\Omega)$, such that
$$
\Psi[u_*]=\inf\Psi[u],
$$
and $u_*$ is the weak solutions of the Euler equation corresponding to
$\Psi[u]$,
$$
\frac1h u-\frac1h\Delta u-\mathop{\rm div}(|\nabla u|^{p-2}\nabla
u)=f.
$$
Taking $f=(u_k-\Delta u_k)/h$, we obtain a weak solutions $u_{k+1}$
of (2.1)--(2.2). The proof is complete.
\end{proof}

Now, we need to establish  a priori estimates, for the weak solutions
$u_{k+1}$ of  (2.1)--(2.2). First, we define the weak solutions of
(1.1)--(1.3) as follows:
\begin{gather*}
u^h(x, t)=u_k(x), \quad kh<t\leq (k+1)h, \; k=0, 1, \dots, N-1, \\
u^h(x, 0)=u_0(x).
\end{gather*}

\begin{lemma} \label{lm 2.2}
 The weak solutions $u_k$ of (2.1)--(2.2) satisfy
\begin{gather}
h\sum_{k=1}^N\int_\Omega|\nabla u_k|^p dx\leq C, \tag{2.4} \\
\sup_{0<t<T}\int_\Omega|\nabla u^h(x, t)|^pdx\leq C, \tag{2.5}
\end{gather}
where $C$ is a constant independent of $h$ and $k$.
\end{lemma}

\begin{proof} i) We take $\varphi= u_{k+1}$ in the integral equality (2.3)
(we can easily prove that for
$\varphi\in W^{1,p}_0(\Omega)$, (2.3) also holds).
\begin{align*}
\frac1h\int_\Omega(u_{k+1}-u_k)u_{k+1}dx
+\frac1h\int_\Omega(\nabla u_{k+1}-\nabla u_k)\nabla u_{k+1}dx&\\
+\int_\Omega|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}\nabla u_{k+1}dx&=0,
\end{align*}
i.e.,
\begin{align*}
\frac1h\int_\Omega|u_{k+1}|^2dx+\frac1h\int_\Omega|\nabla u_{k+1}|^2dx
-\frac1h\int_\Omega u_ku_{k+1}dx &\\
-\frac1h\int_\Omega\nabla u_{k+1}\nabla u_kdx
+\int_\Omega|\nabla u_{k+1}|^pdx&=0.
\end{align*}
Thus
\begin{align*}
&\frac1h\int_\Omega|u_{k+1}|^2dx+\frac1h\int_\Omega|\nabla u_{k+1}|^2dx
+\int_\Omega|\nabla u_{k+1}|^pdx\\
&=\frac1h\int_\Omega
u_ku_{k+1}dx+\frac1h\int_\Omega\nabla u_{k+1}\nabla u_k dx.
\end{align*}
By Young's inequality,
\begin{align*}
&\frac1h\int_\Omega|u_{k+1}|^2dx+\frac1h\int_\Omega|\nabla u_{k+1}|^2dx
+\int_\Omega|\nabla u_{k+1}|^pdx \\
&\leq\frac1{2h}\int_\Omega|u_k|^2dx+\frac1{2h}\int_\Omega|u_{k+1}|^2dx+
\frac1{2h}\int_\Omega|\nabla u_k|^2dx+\frac1{2h}\int_\Omega|\nabla
u_{k+1}|^2dx;
\end{align*}
that is,
\[
\begin{aligned}
&\frac12\int_\Omega|u_{k+1}|^2dx+\frac12\int_\Omega|\nabla u_{k+1}|^2dx
+h\int_\Omega|\nabla u_{k+1}|^pdx \\
&\leq\frac12\int_\Omega|u_k|^2dx+
\frac12\int_\Omega|\nabla u_k|^2dx.
\end{aligned}
\eqno(2.6)
\]
Adding these inequalities for $k$ from $0$ to $N-1$, we have
$$
h\sum_{k=1}^N\int_\Omega|\nabla u_k|^pdx\leq\frac12\int_\Omega|u_0|^2dx
+\frac12\int_\Omega|\nabla u_0|^2dx.
$$
Therefore, (2.4) holds.

\noindent ii) We take $\varphi=u_{k+1}-u_k$ in the integral equality
(2.3) and have
\begin{align*}
\frac1h\int_\Omega(u_{k+1}-u_k)(u_{k+1}-u_k)dx+\frac1h\int_\Omega(\nabla u_{k+1}
-\nabla u_k)\nabla(u_{k+1}-u_k)dx &\\
+\int_\Omega|\nabla u_{k+1}|^{p-2}\nabla
u_{k+1}\nabla(u_{k+1}-u_k)dx &=0\,.
\end{align*}
Since the first term and the second term of the left hand side of the above
equality is nonnegative, it follows that
\begin{align*}
\int_\Omega|\nabla u_{k+1}|^pdx
&\leq\int_\Omega|\nabla u_{k+1}|^{p-2} \nabla u_{k+1}\nabla u_k \,dx\\
&\leq\frac{p-1}p\int_\Omega|\nabla u_{k+1}|^pdx
+\frac1p\int_\Omega|\nabla u_k|^pdx;
\end{align*}
thus,
$$
\int_\Omega|\nabla u_{k+1}|^pdx\leq\int_\Omega|\nabla u_k|^pdx.
$$
For any $m$, with $1\leq m\leq N-1$, adding the above inequality for $k$
from $0$ to $m-1$, we have
$$
\int_\Omega|\nabla u_m|^pdx\leq\int_\Omega|\nabla u_0|^pdx.
$$
Therefore, (2.5) holds.
\end{proof}

\begin{lemma} \label{lm 2.3}
For a weak solutions $u_{k+1}$ of  (2.1)--(2.2), we have
$$
-Ch\leq\int_\Omega|u_{k+1}|^2dx+\int_\Omega|\nabla
u_{k+1}|^2dx-\int_\Omega|u_k|^2dx-\int_\Omega|\nabla u_k|^2dx\leq0,
\eqno(2.7)
$$
where $C$ is a constant independently of $h$.
\end{lemma}

\begin{proof} The second inequality in (2.7) is an immediate consequence of
(2.6). To prove the first inequality, we choose $\varphi= u_k$
in (2.3) and obtain
\begin{align*}
\frac1h\int_\Omega(u_{k+1}-u_k)u_kdx+\frac1h\int_\Omega(\nabla u_{k+1}
-\nabla u_k)\nabla u_k\,dx&\\
+\int_\Omega|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}\nabla u_k\,dx&=0\,.
\end{align*}
Therefore,
\begin{align*}
&\int_\Omega|u_k|^2dx+\int_\Omega|\nabla u_k|^2dx-\int_\Omega u_{k+1}u_kdx-\int_\Omega\nabla
u_{k+1}\nabla u_kdx \\
&=h\int_\Omega|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}\nabla u_kdx \\
&\leq h\Big(\int_\Omega|\nabla u_{k+1}|^pdx\Big)^{(p-1)/p}
\Big(\int_\Omega|\nabla u_k|^pdx\Big)^{1/p}.
\end{align*}
Here we have used H\"{o}lder inequality. By (2.5) again, we obtain
$$
\int_\Omega|u_k|^2dx+\int_\Omega|\nabla u_k|^2dx-\int_\Omega u_{k+1}u_kdx-\int_\Omega\nabla
u_{k+1}\nabla u_kdx\leq Ch.
$$
Therefore,
\begin{align*}
&\int_\Omega|u_k|^2dx+\int_\Omega|\nabla u_k|^2dx\\
&\leq Ch+\int_\Omega u_{k+1}u_kdx+\int_\Omega\nabla u_{k+1}\nabla u_kdx\\
&\leq Ch+\frac12\int_\Omega|u_{k+1}|^2dx+\frac12\int_\Omega|u_k|^2dx
+\frac12\int_\Omega
|\nabla u_{k+1}|^2dx+\frac12\int_\Omega|\nabla u_k|^2dx.
\end{align*}
i.e.,
$$
\int_\Omega|u_k|^2dx+\int_\Omega|\nabla u_k|^2dx-\int_\Omega
|u_{k+1}|^2dx-\int_\Omega|\nabla u_{k+1}|^2dx\leq Ch
$$
which completes the proof. \end{proof}

\begin{lemma} \label{lm2.4}
$$
\sup_{0<t<T}\Big(\int_\Omega|u^h|^2dx+\int_\Omega|\nabla
u^h|^2dx\Big)\leq\int_\Omega|u_0|^2dx+\int_\Omega|\nabla u_0|^2dx. \eqno(2.8)
$$
\end{lemma}

The proof follows by adding (2.4), for $m$ with $1\leq m\leq N-1$, for $k$ from $0$
to $m-1$.

\begin{proof}[Proof of Theorem 2.1]
First, we define the operator $A^t$, $A^t(\nabla u^h)=|\nabla u_k|^{p-2}\nabla u_k$,
$\Delta^hu^h=u_{k+1}-u_k$,
where $kh<t\leq(k+1)h, k=0,1,\dots,N-1$. By the dispersion equation (2.1) and
the (2.4) in Lemma 2.2, we know that
$$
\frac1h(u_{k+1}-u_k)\quad
\hbox{in }L^\infty(0, T;W^{-1, p'}(\Omega))\quad\hbox{is bounded.}
\eqno(2.9)
$$
By (2.5), (2.7), (2.9) and (2.4) we known that exists a subsequence of $\{u^h\}$
(which we denote as the original sequence) such that
\begin{gather*}
u^h\to u\quad\hbox{in}\;L^\infty(0, T;W^{1, p}(\Omega))\quad\hbox{weak-}\star,\\
\nabla u^h\to \nabla u\quad\hbox{in }
L^\infty(0,T;L^2(\Omega))\quad\hbox{weak-}\star, \\
\frac1h(u_{k+1}-u_k)\to\frac{\partial u}{\partial t}
\quad\hbox{in }L^\infty(0, T;W^{-1, p'}(\Omega)\quad\hbox{weak-}\star, \\
|\nabla u^h|^{p-2}\nabla u^h\to w\quad\hbox{in }
L^\infty(0, T;L^{p'}(\Omega))\quad\hbox{weak-}\star,
\end{gather*}
where $p'$ is conjugate exponent of $p$. From
(2.3), we known, for any $\varphi\in C^\infty_0(Q_T)$,
$$
\iint_{Q_T}(\frac1h\Delta^hu^h\varphi+\frac1h\Delta^h\nabla u^h\nabla\varphi+|\nabla u^h|^{p-2}\nabla
u^h\nabla\varphi)dx\,dt=0,
$$
i.e.,
$$
\iint_{Q_T}(\frac1h\Delta^hu^h\varphi-\frac1h\Delta^hu^h\Delta\varphi+|\nabla
u^h|^{p-2}\nabla u^h\nabla\varphi) dx\,dt=0.
$$
Letting $h\to0$, we obtain, in the sense of distributions,
$$
\frac{\partial u}{\partial t}-\frac{\partial\Delta u}{\partial t}-\mathop{\rm div}(w)=0. \eqno(2.10)
$$
Next, we prove that $w=|\nabla u|^{p-2}\nabla u$ a.e.  in $Q_T$.
Define
\begin{align*}
f_h(t)&=\frac{t-kh}{2h}\left(\int_\Omega|u_{k+1}|^2dx+\int_\Omega|\nabla
u_{k+1}|^2dx-\int_\Omega|u_k|^2dx-\int_\Omega|\nabla
u_k|^2dx\right)\\
&\quad +\frac12\int_\Omega|u_k|^2dx+\frac12\int_\Omega|\nabla u_k|^2dx,
\end{align*}
where $kh< t\leq (k+1)h$. by (2.7) we have
\begin{gather*}
\frac12\int_\Omega|\nabla u_k|^2dx+\frac12\int_\Omega|u_k|^2dx-Ch\leq f_h(t)\leq
\frac12\int_\Omega|u_k|^2dx+\frac12\int_\Omega|\nabla u_k|^2dx, \\
-C\leq f'_h(t)\leq 0.
\end{gather*}
By Ascoli--Arzela theorem, there exists a function
$f(t)\in C([0,T])$, such that
$$
\lim_{h\to0}f_h(t)=f(t)\quad\hbox{for $t\in[0, T]$ uniformly}.
$$
Using (2.7), we  have
$$
\lim_{h\to0}\Big(\frac12\int_\Omega|\nabla u^h|^2dx
+\frac12\int_\Omega|u^h|^2dx\Big)=f(t)
\quad\hbox{for $t\in[0, T]$ uniformly}.
\eqno(2.11)
$$
By (2.6) again, we obtain
\[
\frac12\int_\Omega|u_N|^2dx+\frac12\int_\Omega|\nabla u_N|^2dx+\int\hskip-2mm
\int_{Q_T}|\nabla u^h|^pdx\,dt\\
\leq \frac12\int_\Omega|u_0|^2dx +\frac12\int_\Omega|\nabla u_0|^2dx.
\]
In the above inequality letting $h\to 0$, and using (2.10) we have
\begin{align*}
\lim_{h\to0}\iint_{Q_T}|\nabla u^h|^pdx\,dt
&\leq f(0)-f(T)\\
&=\lim_{\varepsilon\to0}\frac1\varepsilon\int_0^{T-\varepsilon}(f(t)-f(t+\varepsilon))dt
\\
&=\lim_{\varepsilon\to0}\lim_{h\to0}
\Big[\frac1{2\varepsilon}\int_0^{T-\varepsilon}\int_\Omega(
|u^h(x, t)|^2-|u^h(x, t+\varepsilon)|^2)dx\,dt \\
&\quad+\frac1{2\varepsilon}\int_0^{T-\varepsilon}\int_\Omega(|\nabla u^h(x, t)|^2
-|\nabla u^h(x, t+\varepsilon)|^2)dx\,dt\Big].
\end{align*}
Since $\Phi_2[u]=\frac12\int_\Omega|u|^2dx$ and
$\Phi_3[u]=\frac12\int_\Omega|\nabla u|^2dx$ are convex functionals,
and
$$
\frac{\delta\Phi_2[u]}{\delta u}=u,\quad
\frac{\delta\Phi_3[u]}{\delta u}=-\Delta u,
$$
we have
\begin{align*}
&\frac12\int_\Omega|u^h(x, t)|^2dx-\frac12\int_\Omega|u^h(x, t+\varepsilon)|^2dx\\
&+\frac12\int_\Omega |\nabla u^h(x, t)|^2dx-\frac12\int_\Omega|\nabla u^h(x, t
+\varepsilon)|^2dx \\
&\leq \int_\Omega u^h(x, t)(u^h(x, t)-u^h(x, t+\varepsilon))dx\\
&\quad +\int_\Omega\nabla u^h(x, t)(\nabla u^h(x, t)
-\nabla u^h(x, t+\varepsilon))dx.
\end{align*}
Therefore,
\begin{align*}
&\lim_{h\to0}\frac1{2\varepsilon}\Big[\int_0^{T-\varepsilon}\int_\Omega|u^h(x, t)|^2-
|u^h(x, t+\varepsilon)|^2)dx\,dt\\
&+\int_0^{T-\varepsilon}\int_\Omega
(|\nabla u^h(x, t)|^2-|\nabla u^h(x, t+\varepsilon)|^2)dx\,dt\Big] \\
&\leq\frac1\varepsilon\int_0^{T-\varepsilon}\int_\Omega(u(x, t)
-u(x, t+\varepsilon))u\,dx\,dt \\
&\quad +\frac1\varepsilon\int_0^{T-\varepsilon}
\int_\Omega(\nabla u(x, t)-\nabla u(x, t+\varepsilon))\nabla u\,dx\,dt\,.
\end{align*}
Hence, we obtain
$$
\lim_{h\to0}\iint_{Q_T}|\nabla u^h|^pdx\,dt\leq-\int_0^T\langle\frac{\partial u}{\partial t},
u\rangle dt
+\int_0^T\langle\frac{\partial u}{\partial t}, \Delta u\rangle dt,
$$
where $\langle,\rangle$ denotes the inner product. Form
(2.10), we obtain
$$
\lim_{h\to0}\iint_{Q_T}|\nabla u^h|^pdx\,dt\leq\iint_{Q_T}w\nabla u dx\,dt\,.
\eqno(2.12)
$$
Again by $\frac{\delta \Phi_1[u]}{\delta u}=-\mathop{\rm
div}(|\nabla u|^{p-2}\nabla u)$ and the convexity of $\Phi_1[u]$, for
any $g\in L^\infty(0,T;W_0^{1,p}(\Omega))$ we have
\begin{align*}
&-\frac1p\iint_{Q_T}|\nabla
g|^pdx\,dt+\frac1p\iint_{Q_T}|\nabla u^h|^pdx\,dt\\
&\leq\iint_{Q_T}-\mathop{\rm div}(|\nabla
u^h|^{p-2}\nabla u^h)(u^h-g)dx\,dt,
\end{align*}
that is
\begin{align*}
\frac1p\iint_{Q_T}|\nabla g|^pdx\,dt-\frac1p\iint_{Q_T}|\nabla
u^h|^pdx\,dt
&\geq \iint_{Q_T}\mathop{\rm div}(|\nabla
u^h|^{p-2}\nabla u^h)(u^h-g)dx\,dt \\
&=\iint_{Q_T}(|\nabla u^h|^{p-2}\nabla u^h)\nabla(g-u^h)dx\,dt.
\end{align*}
By (2.11) and $F(u)$ is weakly lower semicontinuous, in above equality letting
$h\to0$, we obtain
$$
\frac1p\iint_{Q_T}|\nabla g|^pdx\,dt-\frac1p\iint_{Q_T}|\nabla u|^pdx\,dt
\leq\iint_{Q_T}w\nabla(g-u)dx\,dt.
\eqno(2.13)
$$
In (2.13), we take $g=\varepsilon g+u$ to obtain
$$
\frac1\varepsilon\Big[\frac1p\iint_{Q_T}|\nabla(\varepsilon g+u)|^pdx\,dt
-\frac1p\iint_{Q_T}
|\nabla u|^pdx\,dt\Big]\geq\iint_{Q_T}w\nabla g\,dx\,dt.
$$
Letting $\varepsilon\to 0$,
$$
\iint_{Q_T}\frac{\delta\Phi_1[u]}{\delta u}g\,dx\,dt=\iint_{Q_T}
|\nabla u|^{p-2}\nabla u\nabla g\,dx\,dt\geq\iint_{Q_T}w\nabla g\,dx\,dt\,.
$$
Since $g$ is arbitrary, taking $g=-g$, we get the opposite inequality above;
hence
$$
w=|\nabla u|^{p-2}\nabla u.
$$
The strong convergence of $u^h$ in $C(0,T; H^1(\Omega))$ and the fact that
$u^h(x,0)=u_0(x)$ completes the proof.
\end{proof}

\section{Uniqueness of solutions}

In this section, we  prove that the weak solution is unique.
To this end we need the following lemma.

\begin{lemma} \label{lm3.1}
For $\varphi\in L^\infty(t_1,t_2;W^{1,p}_0(\Omega))$ with
$\varphi_t\in L^2(t_1,t_2; H^1(\Omega))$, the weak solutions $u$ of the
problem (1.1)-(1.3) on $Q_T$ satisfies
\begin{align*}
&\int_\Omega u(x,t_1)\varphi(x,t_1)dx+\int_\Omega \nabla u(x,t_1)\nabla\varphi(x,t_1)dx
\\
&+\int_{t_1}^{t_2}\int_\Omega\left(u\frac{\partial \varphi}{\partial t}
+\nabla u\frac{\partial\nabla\varphi}{\partial t}-|\nabla u|^{p-2}\nabla u\nabla\varphi\right)dx\,dt \\
&=\int_\Omega u(x,t_2)\varphi(x,t_2)dx+\int_\Omega\nabla u(x,t_2)
\nabla\varphi(x,t_2)dx.
\end{align*}
In particular, for $\varphi\in W_0^{1,p}(\Omega)$, we have
\[\begin{aligned}
\int_\Omega(u(x,t_1)-u(x,t_2))\varphi dx+\int_\Omega\nabla(u(x,t_1)-u(x,t_2))
\nabla\varphi dx& \\
-\int_{t_1}^{t_2}\int_\Omega|\nabla u|^{p-2}\nabla u\nabla\varphi dx\,dt
&=0\,.
\end{aligned}
\tag{3.1}
\]
\end{lemma}
\begin{proof} From $\varphi\in L^\infty(t_1,t_2;W^{1,p}_0(\Omega))$
and $\varphi_t\in L^2(t_1,t_2; H^1(\Omega))$,
it follows that there exists a sequence of functions $\{\varphi_k\}$,
for fixed $t\in(t_1,t_2), \varphi_k(\cdot,t)\in C_0^\infty(\Omega)$,
and as $k\to\infty$
$$
\|\varphi_{kt}-\varphi_t\|_{L^2(t_1,t_2; H^1(\Omega))}\to0,\quad
\|\varphi_k-\varphi\|_{L^\infty(t_1,t_2;W_0^{1,p}(\Omega))}\to0.
$$
Choose a function $j(s)\in C_0^\infty(R)$ such that
$j(s)\geq0$, for $s\in R$; $j(s)=0$, for $\forall |s|>1$; $\int_Rj(s)ds=1$.
For $h>0$, define
$j_h(s)=\frac1hj(\frac s h)$ and
$$
\eta_h(t)=\int_{t-t_2+2h}^{t-t_1-2h}j_h(s)ds.
$$
Clearly $\eta_h(t)\in
C_0^\infty(t_1,t_2)$, $\lim_{h\to0^+}\eta_h(t)=1$, for all
$t\in(t_1,t_2)$.
In the definition of weak solutions, choose $\varphi=\varphi_k(x,t)\eta_h(t)$.
We have
\begin{align*}
\int_{t_1}^{t_2}\int_\Omega u\varphi_k
j_h(t-t_1-2h)dx\,dt-\int_{t_1}^{t_2}\int_\Omega u\varphi_k j_h(t-t_2+2h)dx\,dt&
\\
+\int_{t_1}^{t_2}\int_\Omega \nabla u\nabla\varphi_k
j_h(t-t_1-2h)dx\,dt-\int_{t_1}^{t_2}\int_\Omega\nabla u\nabla\varphi_k j_h(t-t_2+2h)\,
dx\,dt& \\
+\int_{t_1}^{t_2}\int_\Omega u\varphi_{kt}\eta_h dx\,dt+
\int_{t_1}^{t_2}\int_\Omega \nabla u\nabla\varphi_{kt}\eta_h \,dx\,dt&\\
-\int_{t_1}^{t_2}\int_\Omega |\nabla u|^{p-2}
\nabla u\nabla\varphi_k\eta_h \,dx\,dt&=0.
\end{align*}
Observe that
\begin{align*}
&\big|\int_{t_1}^{t_2}\int_\Omega u\varphi_k
j_h(t-t_1-2h)dx\,dt-\int_\Omega(u\varphi_k)|_{t=t_1}dx\big|
\\
&=\big|\int_{t_1+h}^{t_1+3h}\int_\Omega u\varphi_k
j_h(t-t_1-2h)dx\,dt
-\int_{t_1+h}^{t_1+3h}\int_\Omega (u\varphi_k)|_{t=t_2}
j_h(t-t_1-2h)dx\,dt\big|
\\
&\leq \sup_{t_1+h<t<t_1+3h}\int_\Omega\big|(u\varphi_k)|_t-(u\varphi_k)
|_{t_1}\big|dx,
\end{align*}
and $u\in C(0,T;L^2(\Omega))$. We see that the right hand side tends to zero
as $h\to0$. Similarly,
\begin{gather*}
\Big|\int_{t_1}^{t_2}\int_\Omega u\varphi_k
j_h(t-t_2+2h)dx\,dt-\int_\Omega(u\varphi_k)|_{t=t_2}dx\Big|\to0,
\quad \hbox{as } h\to 0,\\
\Big|\int_{t_1}^{t_2}\int_\Omega \nabla u\nabla\varphi_k
j_h(t-t_1-2h)dx\,dt-\int_\Omega(\nabla u\nabla\varphi_k)|_{t=t_1}dx\Big|\to0,
\quad \hbox{as } h\to 0, \\
\Big|\int_{t_1}^{t_2}\int_\Omega \nabla u\nabla\varphi_k
j_h(t-t_2+2h)dx\,dt-\int_\Omega(\nabla u\nabla\varphi_k)|_{t=t_2}dx\Big|\to0,
\quad \hbox{as } h\to 0.
\end{gather*}
Letting $h\to0$ and $k\to\infty$, we obtain
\begin{align*}
&\int_\Omega u(x,t_1)\varphi(x,t_1)dx+\int_\Omega \nabla u(x,t_1)
\nabla\varphi(x,t_1)dx& \\
&+\int_{t_1}^{t_2}\int_\Omega\Big(u\frac{\partial \varphi}{\partial t}
+\nabla u\frac{\partial \nabla\varphi}{\partial t}
-|\nabla u|^{p-2}\nabla
 u\nabla\varphi\Big)dx\,dt \\
&=\int_\Omega u(x,t_2)\varphi(x,t_2)dx+\int_\Omega \nabla u(x,t_2)\nabla\varphi(x,t_2)dx.
\end{align*}
In particular for $\varphi\in W_0^{1,p}(\Omega)$, we have
\begin{align*}
\int_\Omega(u(x,t_1)-u(x,t_2))\varphi dx+\int_\Omega
(\nabla u(x,t_1)-\nabla u(x,t_2))\nabla\varphi dx&\\
-\int_{t_1}^{t_2}\int_\Omega|\nabla u|^{p-2}
\nabla u\nabla\varphi\, dx\,dt&=0
\end{align*}
which completes the proof.
\end{proof}

For a fixed $\tau\in(0,T)$, set $h$ satisfying $0<\tau<\tau+h<T$.
Letting $t_1=\tau$, $t_2=\tau+h$, then multiply (3.1) by $\frac1h$, for
$\varphi\in W^{1,p}_0(\Omega)$,  we obtain
$$
\int_\Omega(u_h(x,\tau))_\tau\varphi(x)dx+\int_\Omega((\nabla u)_h(x,\tau))_\tau\varphi(x)dx
+\int_\Omega(|\nabla u|^{p-2}\nabla u)_h(x,\tau)\nabla\varphi dx=0,
\eqno(3.2)
$$
where
$$
u_h(x,t)=\begin{cases}
\frac1h\int_t^{t+h}u(\cdot,\tau)d\tau, &t\in(0,T-h),\\
0,& t>T-h.
\end{cases}
$$

\begin{theorem} \label{thm3.1}
Problem (1.1)-(1.3) admits  only one weak solution.
\end{theorem}

\begin{proof}
 Suppose $u_1,u_2$ are two solutions of (1.1)-(1.3), then
\begin{align*}
\int_\Omega(u_1(x,\tau)-u_2(x,\tau))_{h\tau}\varphi(x)dx
+\int_\Omega((\nabla u_1-\nabla u_2)_h(x,\tau))_\tau\varphi(x)dx&
\\
+\int_\Omega(|\nabla u_1|^{p-2}\nabla u_1
-|\nabla u_2|^{p-2}\nabla u_2)_h(x,\tau)\nabla\varphi dx&=0.
\end{align*}
For a fixed $\tau$, we take $\varphi(x)=[u_1-u_2]_h\in W^{1,p}_0(\Omega)$,
and hence
\begin{align*}
&\int_\Omega(u_1(x,\tau)-u_2(x,\tau))_{h\tau}(u_1-u_2)_h dx\\
&+\int_\Omega\nabla(u_1(x,\tau)-u_2(x,\tau))_{h\tau}\nabla(u_1-u_2)_h dx
\\
&=-\int_\Omega[(|\nabla u_1|^{p-2}\nabla u_1-|\nabla u_2|^{p-2}\nabla u_2)_h](x,\tau)
\nabla(u_1-u_2)_hdx,
\end{align*}
i.e.,
\begin{align*}
&\int_\Omega(u_1(x,\tau)-u_2(x,\tau))_{h\tau}(u_1-u_2)_h dx\\
&+\int_\Omega\nabla(u_1(x,\tau)-u_2(x,\tau))_{h\tau}\nabla(u_1-u_2)_h dx\\
&=-\int_\Omega[(|\nabla u_1|^{p-2}\nabla u_1
-|\nabla u_2|^{p-2}\nabla u_2)_h](x,\tau)\nabla(u_1-u_2)_hdx.
\end{align*}
Integrating the above equality with respect to $\tau$ over $(0,t)$,
$$
\int_\Omega|(u_1-u_2)_h|^2(x,t)dx+\int_\Omega|\nabla(u_1-u_2)_h|^2(x,t)dx\leq0,
$$
 we have $\int_\Omega|(u_1-u_2)_h|^2dx=0$;
therefore, $u_1=u_2$.
\end{proof}

\subsection*{Acknowledgment}
The author would like to thank referee for his/her valuable suggestions and
for providing the references E. Di Benedtto \& M. Pierre \cite{d1}
 and E. Di Benedetto \& R. E. Showalter \cite{d2}.

\begin{thebibliography}{00}

\bibitem{b1} G. I. Barwnblatt, Iv. P. Zheltov, and I. N. Kochina,
\textit{Basic concepts in the theory of seepage of homogeneous
liquids in fissured rocks}, J. Appl. Math. Mech.,
24(1960), 1286-1303.

\bibitem{c3} Kungching Chang, \textit{Oritical point theory and its applications},
Shanghai Sci. Tech. Press, Shanghai, 1986.

\bibitem{c1} P. J. Chen and M. E. Gurtin, \textit{On a theory of heat conduction
involving two temperatures}, Z. Angew. Math. Phys., 19(1968), 614-627.

\bibitem{c2} B. D. Coleman, R. J. Duffin, and V. J. Mizel,
\textit{Instability, uniqueness and
non-existence theorems for the equations, $u_t=u_{xx}-u_{xtx}$ on a strip},
Arch. Rat. Mech. Anal., 19(1965), 100-116.

\bibitem{d1} E. DiBenedtto and M. Pierre,
\textit{On the maximum principle for pseudoparabolic equations},
Indiana Univ. Math. J., 30(6)(1981), 821-854.

\bibitem{d2} E. DiBenedetto and R. E. Showalter,
\textit{Implicit decenerate evolution equations and
applications}, SIAM J. Math. Anal., 12(5)(1981), 731-751.

\bibitem{n1} A. Novick-Cohen and R. L. Pego,  \textit{Stable patterns in a viscous
diffusion equation}, Trans. Amer. Math. Soc., 324(1991), 331--351.

\bibitem{r1} V. R. G. Rao and T. W. Ting, \textit{Solutions of pseudo-heat equation
in whole space}, Arch. Rat. Mech. Anal., 49(1972), 57-78.

\bibitem{s1} R. E. Showalter and T. W. Ting, \textit{Pseudo-parabolic partial differential
equations}, SIAM J. Math. Anal., 1(1970), 1-26.

\bibitem{t1} T. W. Ting, \textit{A cooling process according to two-temperature
theory of heat conduction}, J. Math. Anal. Appl., 45(1974), 23-31.

\bibitem{t2} T. W. Ting, \textit{Parabolic and pseudoparabolic partial differential
equations}, J. Math. Soc. Japan, 21(1969), 440-453.

\end{thebibliography}
\end{document}