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\AtBeginDocument{{\noindent\small {\em
Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 76, pp. 1--13.\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--2003/76\hfil Attractors of degenerate diffusion equations]
{Global attractors for a class of degenerate diffusion equations}

\author[Shingo Takeuchi \& Tomomi Yokota\hfil EJDE--2003/76\hfilneg]
{Shingo Takeuchi \& Tomomi Yokota}

\address{Shingo Takeuchi\hfill\break
Department of General Education, Kogakuin University \\
2665-1 Nakano-machi, Hachioji-shi, Tokyo 192-0015, Japan}
\email{shingo@cc.kogakuin.ac.jp}

\address{Tomomi Yokota\hfill\break
Department of Mathematics, Tokyo University of Science \\
26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-0827, Japan}
\email{yokota@rs.kagu.tus.ac.jp}


\date{}
\thanks{Submitted January 29, 2003. Revised May 8, 2003. Published July 11, 2003.}
\thanks{S. Takeuchi was supported by Grant-in-Aid for Young Scientists (B), No. 15740110.}
\subjclass[2000]{35K65, 37L30}
\keywords{Global attractors, $p$-Laplacian, degenerate diffusion}


\begin{abstract}
 In this paper we  give two existence results for a class of
 degenerate diffusion equations with $p$-Laplacian.
 One is on a unique global strong  solution, and the other is
 on a global attractor. It is also shown that the global attractor
 coincides with the unstable set of the set of all stationary solutions.
 As a by-product, an a-priori estimate for solutions
 of the corresponding elliptic equations is obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}

 \section{Introduction and Results} % sec. 1.

 Let $\Omega \subset \mathbb{R}^N$, $N \ge 1$,
 be a bounded domain of class $C^2$  with boundary $\partial \Omega$.
 We consider the  degenerate diffusion equation
 \begin{equation}  \label{P}
\begin{gathered}
    u_t=\lambda \Delta_p u +f(u),
    \quad  (x,t) \in \Omega \times (0,+\infty), \\
    u(x,t)=0, \quad (x,t) \in \partial \Omega \times (0,+\infty),
   \end{gathered}
 \end{equation}
 with the initial condition $u(x,0)=u_0(x)$
 in $\Omega$, where $\Delta_p u:=\operatorname{div}(|\nabla
 u|^{p-2}\nabla u)$ with $p>2$. We assume that
 $f \in C^1(\mathbb{R})$, $f(0)=0$, and one of the following two conditions
 is satisfied:
 \begin{enumerate}
  \item[(F1)] $p>N$ and
        $ \limsup_{|s| \to +\infty}
        \frac{f'(s)}{(p-1)|s|^{p-2}} < \lambda_1\lambda$\! ;
  \item[(F2)] $p \le N$ and
        $ \sup_{s \in \mathbb{R}} f'(s) < +\infty$,
 \end{enumerate}
 where $\lambda_1$ is the first eigenvalue of $-\Delta_p$ with the
 Dirichlet boundary condition and is characterized by
 $\lambda_1=\inf\{\|\nabla u\|_p^p/\|u\|_p^p;u \in W^{1,p}_0(\Omega)
 \setminus \{0\}\} \in (0,+\infty)$.
 For example of $f$ satisfying the above conditions,
 we can give
 $f(s)=s$;
 $f(s)=|s|^{q-2}s(1-|s|^r)$ with $q \ge 2$ and $r>0$;
 $f(s)=|s|^{q-2}s$ with $q \in (2,p)$ and $p>N$; and
 $f(s)=|s|^{p-2}s$ with $p>N$ when $\lambda>1/\lambda_1$.
 It is important that we assume only one sided boundedness on
 $f'$.

 For semilinear parabolic equations, i.e. $p=2$,
 there are many studies on the existence of global attractors
 and on the existence of solutions; see for example Temam \cite{Te}.
 A fundamental result in this field appeared on the paper \cite{Ma}
 by Marion. He assumes that $f$ has a polynomial growth nonlinearity
 and becomes negative for sufficiently large $u$,
 and that $f'$ is bounded from above. Under these conditions,
 he showed that a global attractor of \eqref{P} in $L^2(\Omega)$
 exists and is bounded in $L^\infty(\Omega)$.
 In fact, the boundedness
 can be proved even in $D(\Delta)=H^1_0(\Omega) \cap H^2(\Omega)$;
 see \cite{Te}. Moreover, it was also proved that
 the fractal and Hausdorff dimensions of the global attractor are both
 finite, which roughly means that solutions of
 \eqref{P} eventually behave with a finite number of
 ``degrees of freedom'' as $t \to +\infty$.
 The analysis for the dimensions relies on the method of linearization,
 which is very operative tool to investigate the time-local behavior
 of solutions.

 This article concerns the degenerate case; i.e., $p>2$.
 We start with the existence of solutions for \eqref{P}.
 The Galerkin method is well-known for constructing
 (weak) solutions of partial differential equations
 (see e.g., Tsutsumi \cite{T}). However,
 the method of monotone operators  gives a more straightforward proof of
 the existence of (strong) solutions than the Galerkin method,
 when available.
  Indeed,  \^Otani \cite{O} extended the abstract theory of
 monotone operators of Br\'ezis \cite{B} to nonlinear
 evolution equations with a difference term of subdifferentials,
 and then succeeded in obtaining better properties of solutions
 of $u_t=\Delta_p u+|u|^{q-2}u$
 than those had been known in \cite{T} by the Galerkin method.
 He also proved in \cite{O2} that the solution converges to
 the set of all stationary solutions (c.f.,
 Theorem \ref{thm:globalattractor} below).
 For our first result, we  use the method in
 \cite{O} to establish the existence of unique
 global strong solutions of \eqref{P}
 and give regularity properties. The definition of
 (global) strong solutions is given in Definition \ref{defn:stsol}, below.

 \begin{thm}
  \label{thm:existence}
  Let $N \ge 1$, $p>2$ and $f \in C^1(\mathbb{R})$ with
  $f(0)=0$. Assume that either (F1) or (F2) is satisfied.
  Let $u_0 \in L^2(\Omega)$. Then for any $T>0$
  there exists a unique strong solution
  $u \in C([0,T];L^2(\Omega))$ of \eqref{P} in $[0,T]$
  with $u(0)=u_0$, and $u$ can be extended to
  a global strong solution  which is denoted again by $u$.
  Moreover, $u$ satisfies
  \begin{gather}
   u \in C^{0,1}_{\rm loc}((0,+\infty);L^2(\Omega)) \cap
   C^{0,\frac{1}{p}}_{\rm loc}((0,+\infty);W^{1,p}_0(\Omega)),
   \label{eq:reg1}\\
   u \in C^{\alpha}(\overline{\Omega} \times [\delta,T]) \quad
   \mbox{for all $\alpha \in (0,1)$},
   \label{eq:reg6}\\
   \nabla u \in C^{\alpha}(\overline{\Omega} \times [\delta,T]) \quad
   \mbox{for some $\alpha \in (0,1)$},
   \label{eq:newreg}\\
   u_t \in L^2(\delta,+\infty;L^2(\Omega)),\
   u \in L^\infty(\delta,+\infty;W^{1,p}_0(\Omega)),
   \label{eq:reg4}\\
   t^{1-\frac{1}{\sigma}}u_t,\ t^{1-\frac{1}{\sigma}}\Delta_pu
    \in L^{\sigma}(0,T;L^2(\Omega)) \quad
    \mbox{for all $\sigma \in [2,+\infty]$},
   \label{eq:reg2}
  \end{gather}
  where $\delta$ and $T$ $(0<\delta<T<+\infty)$ are arbitrary.
 \end{thm}

 \begin{rem} \label{rmk1.1} \rm
  Under the assumption (F1),
  the uniqueness and local existence of strong solutions
  with $u(0)=u_0 \in W^{1,p}_0(\Omega)$  follow from
  Ishii \cite[Theorem 3.3]{I}, since $f$ is locally Lipschitz
  in $L^2(\Omega)$ with the domain $W^{1,p}_0(\Omega)$.
  However, it seems that his proof can not be applied
  in case of (F2).
  We give a unified proof for ``weak'' reactions
  (F1) and (F2), and obtain some
  regularity properties for initial data in $L^2(\Omega)$.
 \end{rem}

 Due to Theorem \ref{thm:existence},
 \eqref{P} produces a nonlinear semigroup on
 $L^2(\Omega)$ and it is significant to investigate the asymptotic
 behavior of solutions, which induces us to study a global attractor.
 Global attractors for degenerate diffusion equations with
 a Lipschitz perturbation have been discussed
 in \cite[Section III.5]{Te}.
 A few years ago,
 Carvalho, Cholewa and Dlotko \cite{CCD} proved the existence
 results of solutions and a global attractor for abstract
 evolution equations with a maximal monotone operator
 and a globally Lipschitz perturbation,
 which involve \cite{Te}.
 Our following theorem
 extends their results to non globally Lipschitz perturbation,
 and furthermore we give regularity results of the global attractor and
 its characterization by the set of all stationary solutions for \eqref{P}
 (though it is assumed that the diffusion
 term is represented by a subdifferential of functions).

 %%%%%%%%%%     Theorem 1.2     %%%%%%%%%%%

 \begin{thm}
  \label{thm:globalattractor}
  Suppose the same conditions as in Theorem \ref{thm:existence}.
  Then there exists a connected global attractor
  $\mathcal{A}_{\lambda}$ in $L^2(\Omega)$ of
  \eqref{P}. $\mathcal{A}_{\lambda}$ and
  $\{\Delta_p\phi; \phi \in \mathcal{A}_\lambda\}$
  are bounded in
  $C^{1,\alpha}(\overline{\Omega})$ for some $\alpha \in (0,1)$
  and in $L^2(\Omega)$, respectively.
  Moreover,
  $\mathcal{A}_{\lambda}=\mathcal{M}_{+}(\mathcal{E}_{\lambda})$,
  where $\mathcal{E}_{\lambda}$ consists of all solutions
  $\phi \in W^{1,p}_0(\Omega) \cap L^{\infty}(\Omega)$ of
   \begin{equation}
    \label{eq:stationary}
     \lambda \Delta_p \phi +f(\phi)=0,
      \quad x \in \Omega,
 \end{equation}
  and $\mathcal{M}_{+}(\mathcal{E}_{\lambda})$ is the unstable set of
  $\mathcal{E}_\lambda$. (For the definition of unstable sets,
  see Definition \ref{dfn2.3})
 \end{thm}

  Theorem \ref{thm:globalattractor} assures that, for instance,
  the equation $u_t=\lambda \Delta_p u+u$ has
  a global attractor for all $\lambda>0$. However,
  the equation $u_t=\lambda \Delta u+u$
  no longer has a global attractor for $\lambda \le 1/\lambda_1$.
  This is the simplest and most remarkable distinction
  on asymptotic behavior
  of solutions
  between two cases $p>2$
  and $p=2$ (see also \cite{CCD}).

  A self-evident fact that $\mathcal{A}_\lambda$
  contains $\mathcal{E}_\lambda$
  prompts us to observe as follows.
  It is known that
  $\mathcal{E}_\lambda$ is generally contained in $C^{1,\alpha}$
  and not to $C^2$ even if $\Omega$ and $f$ are
  in $C^\infty$. Indeed, we can explicitly
  compose such solutions (see Takeuchi and Yamada \cite[Remark 3.2]{TY}).
  Therefore we can not expect that $\mathcal{A}_\lambda$ is included
  in $C^2$, though $\Omega$ and $f$ are both very smooth.
  Next,
  if $\mathcal{E}_\lambda$ is discrete, fortunately,
  then the global attractor
  can be exactly represented by the union of the unstable sets
  associated to the functions
  in $\mathcal{E}_\lambda$, i.e., $\mathcal{A}_\lambda=
  \bigcup_{\phi \in \mathcal{E}_\lambda} \mathcal{M}_{+}(\phi)$
  (c.f., Temam \cite[Theorem VII.4.1]{Te}).
  However, since $\mathcal{E}_\lambda$ often includes some continua (c.f.,
  \cite[Theorems 3.1--3.3]{TY}), we have no conclusion about it from the
  abstract theory for dynamical systems.

  In addition, concerning the $p$-Laplacian,
  we note that there is no guarantee for the validity
  of linearization. This seems to be the reason why
  equations with the $p$-Laplacian are not extensively treated
  in terms of dynamical systems.

  \begin{rem} \label{rmk1.2} \rm
  Dung \cite{Du} has obtained the ultimately uniform
  boundedness of solutions and gradients for degenerate parabolic
  systems including \eqref{P} with bounded initial data,
  and shown the existence
  of a global attractor in the space of bounded continuous functions
  only under the Neumann boundary conditions.
  Note that we are not subject to the boundedness for initial data.
 \end{rem}

  \begin{rem} \label{rml1/3} \rm
  It is possible to relax the assumptions on $f$ if one pays no
  attention to the uniqueness of solutions.
  Even in this case, we may be able to show only the existence
  of global attractors, which is especially defined for
  multivalued semiflow (see Valero \cite{V}).
 \end{rem}

 As a by-product of Theorem \ref{thm:globalattractor},
 an a-priori (uniform) estimate for solutions of the elliptic equation
 \eqref{eq:stationary} is immediately deduced.


  \begin{cor}    \label{cor:elliptic}
   Suppose the same conditions as in Theorem \ref{thm:existence}.
   Then
   there exists a positive constant $M_\lambda$ such that
   $\|\phi\|_{C^{1,\alpha}(\overline{\Omega})} \le M_\lambda$
   for all $\phi \in \mathcal{E}_\lambda$.
  \end{cor}

 The contents of this paper are as follows. Section 2 is devoted to
 the preliminaries in which we define strong solutions and global
 attractors, and give some lemmas.
 We will prove Theorems \ref{thm:existence} and
 \ref{thm:globalattractor} in Sections 3 and 4, respectively.


 \section{Preliminaries} % sec. 2

 In this section we give some definitions and elementary lemmas.
 Throughout this paper,
 $L^p(\Omega)$ and $W^{1,p}_0(\Omega)$, $1 \le p \le \infty$,
 are the usual Lebesgue and Sobolev
 spaces with norms $\|\cdot\|_p$ and $\|\nabla \cdot\|_p$,
 respectively.
 The scalar product of $L^2(\Omega)$ is denoted by $(\cdot,\cdot)$.
 $C^{\alpha}(\overline{\Omega} \times [\delta,T])$, $0<\alpha<1$,
 is the H\"{o}lder space with norm
 \[
      [u]_{\alpha,\overline{\Omega} \times [\delta,T]}
      =\sup_{(x,t) \in \overline{\Omega} \times [\delta,T]}|u(x,t)|
      +\sup_{(x,t),(y,\tau) \in \overline{\Omega} \times [\delta,T]}
      \frac{|u(x,t)-u(y,\tau)|}{|x-y|^{\alpha}+|t-\tau|^{\alpha/p}}.
 \]
 Also, $C^{1,\alpha}(\overline{\Omega})$, $0<\alpha<1$,
 is the usual H\"{o}lder space.

 \begin{defn} \label{defn:stsol} \rm
  A function $u \in C([0,T];L^2(\Omega))$
  is called a \textit{strong solution of
  \eqref{P} in $[0,T]$ with $u(0)=u_0$}
  if $u$ is locally absolutely continuous on $(0,T)$,
  $u(t) \in W^{1,p}_0(\Omega),\ \Delta_p u(t) \in L^2(\Omega),\
  f(u(t)) \in L^2(\Omega)$
  for a.a. $t \in (0,T)$ and $u$ satisfies
  \begin{gather*}
   u_t=\lambda \Delta_p u+f(u)
   \quad  \mbox{a.e. in } \Omega \times (0,T), \\
   u(0)=u_0 \quad \mbox{a.e. in } \Omega.
  \end{gather*}
  Moreover, we say that a function $u \in C([0,+\infty);L^2(\Omega))$
  is a \textit{global strong solution of} \eqref{P} if
  $u$ is a strong solution of \eqref{P} in $[0,T]$ with
  $u(0)=u_0$ for every $T>0$.
 \end{defn}

 \begin{defn} \label{defn2.2} \rm
  Let $\{S(t)\}_{t \ge 0}$ be a semigroup on $L^2(\Omega)$.
  An \textit{attractor} for the semigroup $\{S(t)\}_{t \ge 0}$
  is a set $\mathcal{A} \subset L^2(\Omega)$ satisfying the
  following two properties:
  \begin{enumerate}
   \item $\mathcal{A}$ is an invariant set under $\{S(t)\}_{t \ge 0}$,
     i.e., $S(t)\mathcal{A}=\mathcal{A}$ for all $t \ge 0$, and
   \item $\mathcal{A}$ possesses an open neighborhood $\mathcal{U}$
     such that for every $u_0 \in \mathcal{U}$, $S(t)u_0$ converges
     to $\mathcal{A}$ as $t \to +\infty$:
     $$\inf_{y \in \mathcal{A}}\|S(t)u_0-y\|_2 \to 0 \quad
     \mbox{as $t \to +\infty$.}$$
  \end{enumerate}
  We say that $\mathcal{A} \subset L^2(\Omega)$ is a \textit{global
  attractor} for the semigroup $\{S(t)\}_{t \ge 0}$ if
  $\mathcal{A}$ is a compact attractor that attracts any bounded
  sets of $L^2(\Omega)$:
  $$\sup_{x \in S(t)B} \inf_{y \in \mathcal{A}}\|x-y\|_2 \to 0 \quad
  \mbox{as $t \to +\infty$}$$
  for any bounded set $B \subset L^2(\Omega)$.
 \end{defn}


 \begin{defn} \label{dfn2.3} \rm
  The \textit{unstable set} $\mathcal{M}_{+}(X)$ of $X \subset L^2(\Omega)$
  is the (possibly empty)
  set of points $u_*$ which belong to a complete orbit
  $\{u(t);t \in \mathbb{R}\}$ such that
  $$\inf_{y \in X}\|u(t)-y\|_2 \to 0
  \quad \mbox{as $t \to -\infty$.}$$
 \end{defn}


  \begin{lem}[Ghidaglia's inequality]
  \label{lem:ghidaglia}
  Let $y(\cdot)$ be a positive absolutely continuous function on
  $(0,+\infty)$ which satisfies
  $$y'+\gamma y^{\frac{p}{2}} \le \delta$$
  with $p>2,\ \gamma>0$ and $\delta \ge 0$. Then for $t >0$
  $$y(t) \le \left(\frac{\delta}{\gamma}\right)^\frac{2}{p}+
  \left(\frac{\gamma(p-2)}{2}t\right)^{-\frac{2}{p-2}}.$$
 \end{lem}

For the proof of this lemma, see Temam \cite[Lemma III.5.1]{Te}.


 Define a function
 $(u-M)_+:=\max\{u-M,0\}$ for a function $u$
 and a constant $M$, and $\chi[u > \alpha]$ denotes the
 characteristic function of the set $\{x \in \Omega; u(x)>\alpha\}$.


 \begin{lem}   \label{lem:-1}
  Let $\{k_n\}_{n=0}^\infty$ be a strictly increasing sequence
  of nonnegative numbers. Then for any $u \in L^2(\Omega)$
  \begin{equation}
   \label{eq:-1}
    \Big(\int_\Omega u(u-k_{n+1})_+dx\Big)^{1/2}
    \le \frac{\|(u-k_n)_+\|_2}
    {1-\frac{k_n}{k_{n+1}}}.
  \end{equation}
 \end{lem}

 \begin{proof}
  Easily we obtain estimates that yield
  \begin{align*}
   \big(1-\frac{k_n}{k_{n+1}}\big)^2 \int_\Omega u(u-k_{n+1})_+dx
   & \le \int_\Omega \big(u-k_n\frac{u}{k_{n+1}}\big)^2
   \cdot \chi[u>k_{n+1}]dx\\
   & \le \int_\Omega (u-k_n)^2 \cdot \chi[u>k_{n+1}]dx\\
   & \le \|(u-k_n)_+\|_2^2,
  \end{align*}
  which implies \eqref{eq:-1}.
 \end{proof}

 \begin{lem}   \label{lem:0}
  Let $\{t_n\}_{n=0}^\infty$ and $\{k_n\}_{n=0}^\infty$ be strictly
  increasing sequences of nonnegative numbers. Then for any $u \in
  L^\infty_{\rm loc}(0,+\infty;L^2(\Omega)) \cap
  L^p_{\rm loc}(0,+\infty;W^{1,p}_0(\Omega))$, the function
  \begin{equation}
   \label{eq:Y_n}
  Y_n(t)=\int_{t_n}^t \|(u-k_n)_+(s)\|_2^2ds,\quad t>t_n,
  \end{equation}
  satisfies
  \begin{multline}
   \label{eq:0}
    Y_{n+1}^{q/2} \leq \\
    \frac{C_0     \big(
     \operatornamewithlimits{ess\,sup}_{t_{n+1}<s<t}
     \|(u-k_{n+1})_+(s)\|_2^2 \big)^{p/N}
    \int_{t_{n+1}}^t\|\nabla((u-k_{n+1})_+)(s)\|^p_pds}
    {(k_{n+1}-k_n)^{q-2}}
    Y_n^\frac{q-2}{2}
  \end{multline}
  for all $t > t_{n+1}$ and some constant $C_0>0$, where $q=(N+2)p/N$.
 \end{lem}

 \begin{proof}
  By the H\"{o}lder and the Gagliardo-Nirenberg inequality
  \begin{align*}
   Y_{n+1}^{q/2}
   & = \Big(\int_{t_{n+1}}^t \int_\Omega (u-k_{n+1})_+^2 \cdot
   \chi[u>k_{n+1}]dxds\Big)^{q/2} \\
   & \le \int_{t_{n+1}}^t \|(u-k_{n+1})_+\|_q^qds \cdot
   |A_{n+1}|^\frac{q-2}{2}\\
   & \le C_0
   \Big( \operatornamewithlimits{ess\,sup}_{(t_{n+1},t)}
   \|(u-k_{n+1})_+\|_2^2 \Big)^{p/N}
   \int_{t_{n+1}}^t\|\nabla((u-k_{n+1})_+)\|_p^pds \cdot
   |A_{n+1}|^\frac{q-2}{2},
   \end{align*}
  where
  $|A_{n+1}|$ denotes the Lebesgue measure of
  $\{(x,s) \in \Omega \times [t_{n},t];u(x,s)>k_{n+1}\}$.
  Combining this with
  \[
   Y_n  \ge \int_{t_{n}}^t \int_\Omega (u-k_n)_+^2 \cdot
   \chi[u>k_{n+1}]\,dx\,ds
   \ge (k_{n+1}-k_n)^2|A_{n+1}|,
  \]
  we obtain \eqref{eq:0}.
 \end{proof}

 Finally we provide a simple, but nice bright lemma.

 \begin{lem}   \label{lem:2}
  Let $\{Y_n\}_{n=0}^\infty$ be a sequence of positive numbers, satisfying
  that there exist $a>0$, $b>1$ and $\theta>0$ such that
  \begin{equation}
   \label{eq:2}
    Y_{n+1} \le ab^nY_n^{1+\theta},\ n=0,1,2,\ldots.
  \end{equation}
  Then
  $Y_0 \le a^{-1/\theta}b^{-1/\theta^2}$
  implies that $Y_n \to 0$ as $n \to +\infty$.
 \end{lem}
 \begin{proof}
  The lemma is introduced in the book of DiBenedetto \cite[Lemma
  I.4.1]{Di} without its proof. Though it is proved easily,
  we show it here for confirmation.
  Using the recursive inequality \eqref{eq:2} repeatedly, we have
  $$Y_n \le a^\frac{(1+\theta)^n-1}{\theta}
  b^\frac{(1+\theta)^n-1-\theta n}{\theta^2}Y_0^{(1+\theta)^n}
  \le a^{-\frac{1}{\theta}}b^{-\frac{1+\theta n}{\theta^2}} \to 0$$
  as $n \to +\infty$.
 \end{proof}

 \section{Proof of Theorem \ref{thm:existence}} %sec. 3

  Let $\kappa$ be a positive constant such that
  \begin{align*}
  &\kappa < 1-\max\Big\{\limsup_{|s|\to+\infty}
  \frac{f'(s)}{\lambda_1\lambda(p-1)|s|^{p-2}},\ 0 \Big\}
  \quad \text{when (F1) is satisfied};\\
  &\kappa=1 \quad \text{when (F2) is satisfied}.
  \end{align*}
  Then there exists a constant $C_1>0$ such that
  $f'(s) \le (1-\kappa)\lambda_1\lambda(p-1)|s|^{p-2}+C_1$
  for all $s \in \mathbb{R}$.
  Putting
  $$g(s):=(1-\kappa)\lambda_1\lambda|s|^{p-2}s+C_1 s-f(s),
  $$
  we can see that $g \in C^1(\mathbb{R})$, $g(0)=0$,
  $g$ is nondecreasing on $\mathbb{R}$,
  and equation \eqref{P}
  can be represented by
  \begin{equation}
   \label{eq:monotone}
    u_t-\lambda \Delta_p u+g(u)=
    (1-\kappa)\lambda_1\lambda
    |u|^{p-2}u+C_1 u.
  \end{equation}
  Defining the following proper lower semi-continuous convex
  functions on $L^2(\Omega)$:
  \begin{equation*}
   \varphi_1(u):=
    \begin{cases}
     \frac{\lambda}{p} \|\nabla u\|_p^p, & u \in
     W^{1,p}_0(\Omega),\\
     +\infty, & \mbox{otherwise},
    \end{cases}
  \end{equation*}
  \begin{equation*}
   \varphi_2(u):=
    \begin{cases}
      \int_\Omega \int_0^u g(s)dsdx, & u \in
     L^2(\Omega) \quad \mbox{with $
     \int_0^ug(s)ds \in L^1(\Omega)$},\\
     +\infty, & \mbox{otherwise},
    \end{cases}
  \end{equation*}
  and
  \begin{equation*}
   \psi(u):=
    \begin{cases}
     \frac{(1-\kappa)\lambda_1\lambda}{p}\|u\|_p^p+\frac{C_1}{2}
     \|u\|_2^2, & u \in L^p(\Omega),\\
     +\infty, & \mbox{otherwise},
    \end{cases}
  \end{equation*}
  we rewrite \eqref{eq:monotone} as
  \begin{equation}
   \label{eq:abst}
    u_t+\partial \varphi_1(u)+\partial \varphi_2(u)
    =\partial \psi(u)
    \quad \mbox{in $(0,+\infty)$},
  \end{equation}
  where $\partial \varphi(u)$ denotes the subdifferential of $\varphi$
  at $u$.
  Since $\partial \varphi_1+\partial \varphi_2$ is $m$-accretive
  (maximal monotone) in $L^2(\Omega)$
  (see Br\'ezis,\ Crandall and Pazy \cite[Theorem 3.1]{BCP} and
  Okazawa \cite[Theorem 1]{Ok}), it follows that
  $\partial \varphi_1+\partial \varphi_2=
  \partial \varphi$, where $\varphi=\varphi_1+\varphi_2$.
  Hence \eqref{eq:abst} is rewritten as
  \begin{equation}
   \label{eq:abst2}
    u_t+\partial\varphi(u)=\partial \psi(u)
    \quad \mbox{in $(0,+\infty)$}.
  \end{equation}

 The next lemma holds the key
  to establishing the existence of global strong solutions
  of \eqref{eq:abst2}.

 \begin{lem}
  \label{lem:absorbing}
  Let $\kappa$, $C_1$, $\varphi$, $\varphi_1$ and $\psi$
  be as above. Then
  \begin{gather}
   \|\partial\psi(u)\|_2 \le
   C_1\|u\|_2+C_2(\varphi_1(u))^{1-\frac{1}{p}}
    \quad \mbox{for all $u \in W^{1,p}_0(\Omega)$,}
   \label{eq:otani2} \\
  \psi(u) \le (1-\kappa) \varphi_1(u)+\frac{C_1}{2} \|u\|_2^2
   \quad \mbox{for all $u \in W^{1,p}_0(\Omega)$},
   \label{eq:absorb} \\
   (\partial\psi(u),u)
    \le (1-\kappa) (\partial\varphi_1(u),u)+C_1\|u\|_2^2
   \quad \mbox{for all $u \in D(\partial \varphi_1)$}
   \label{eq:otani4}
   \end{gather}
   for some constant $C_2>0$.
 \end{lem}

 \begin{proof}[Proof of Lemma \ref{lem:absorbing}]
  It is clear that \eqref{eq:otani2} and \eqref{eq:absorb}
  follow from Sobolev's embedding theorem and
  Poincar\'e's inequality, respectively.
  Also, we can obtain by \eqref{eq:absorb}
  \[
     (\partial\psi(u),u)
   =p\Big(\psi(u)-\big(\frac{1}{2}-\frac{1}{p}\big)C_1
     \|u\|_2^2\Big) %\\
    \le (1-\kappa)
   p \varphi_1(u)+C_1\|u\|_2^2,
  \]
  which proves \eqref{eq:otani4}.
 \end{proof}

  The set
  $\{u \in L^2(\Omega);\ \varphi(u)+\|u\|_2 \le L\}$ is compact in
  $L^2(\Omega)$ for every $L<+\infty$ by Rellich's theorem.
  Therefore, by the same argument as in the proof of \^Otani
  \cite[Theorem 5.3]{O} (use \eqref{eq:absorb} and
  \eqref{eq:otani4} instead of (5.11) in \cite{O}), we see that
  for any $u_0 \in L^2(\Omega)$ and for any $T>0$
  there exists a strong solution $u \in C([0,T];L^2(\Omega))$ of
  \eqref{P} in $[0,T]$ with $u(0)=u_0$ such that
  $$\partial\varphi(u),\ \partial \psi(u) \in
  L^2(\delta,T;L^2(\Omega)) \quad
  \mbox{for all $\delta \in (0,T)$}.$$
%  The proof of the existence of strong solutions of \eqref{P} in $[0,T]$
%  is completed.

We need the following lemmas to prove
\eqref{eq:reg1}--\eqref{eq:reg2}
  and the uniqueness.

  \begin{lem}
   \label{lem:111}
   Take $T>0$ and $\delta \in (0,T]$.
   Let $u$ be a strong solution of \eqref{P} in $[0,T]$ obtained
   as above. Then there exist positive constants $C_3$ and $C_4$
   independent of $T$ such that
   \begin{gather}
    \label{eq:add1}
    \frac{1}{2}\|u(T)\|_2^2 + \kappa\int_0^T \varphi(u(t))dt
    \le C_3 T + \frac{1}{2}\|u_0\|_2^2,\\
    \label{eq:add2}
    \int_{\delta}^T \|u_t(t)\|_2^2dt + \frac{\kappa}{2}
    \varphi(u(T)) \le \varphi(u(\delta)) + C_4,\\
    %\quad \mbox{for all $\delta \in (0,T]$},\\
    \label{eq:add3}
    \int_0^T t\|u_t(t)\|_2^2dt  + \kappa T \varphi_1(u(T))
    \le c(T,\|u_0\|_2),\\ %\quad \mbox{for all $T>0$},\\
    \label{eq:add4}
    t^{1-\frac{1}{\sigma}} (\varphi_1 (u))^{1-\frac{1}{p}}
    \in L^{\sigma}(0,T), %\quad \mbox{for all $T>0$},
   \end{gather}
    where
    $c(T,\|u_0\|_2)=(C_1 T+1/\kappa)
    (C_3 T + \|u_0\|_2^2/2)$ and $\sigma \in [1,+\infty]$ is arbitrary.
   \end{lem}

   \begin{proof}[Proof of Lemma \ref{lem:111}]
    Taking the scalar product of \eqref{eq:abst2} in $L^2(\Omega)$ with
    $u$ and using \eqref{eq:otani4} and some inequalities with $p>2$,
    we have
    \begin{align*}
     \frac{1}{2}\frac{d}{dt}\|u\|_2^2+(\partial \varphi(u),u)
     & = (\partial \psi(u),u) \\
     & \le (1-\kappa) (\partial\varphi_1(u),u)+C_3+
     \frac{\kappa(p-1)\lambda}{p}\|\nabla u\|_p^p \\
     & =\big(1-\frac{\kappa}{p}\big)
     (\partial \varphi_1(u),u)+C_3;
    \end{align*}
    so that
    \begin{equation*}
     \frac{1}{2}\frac{d}{dt}\|u\|_2^2+
      \frac{\kappa}{p}
      (\partial \varphi_1(u),u)+(\partial \varphi_2(u),u)
      \le C_3.
    \end{equation*}
    Since $(\partial \varphi_1(u),u)=p\varphi_1(u)$
    and $(\partial \varphi_2(u),u) \ge \varphi_2(u)$,
    we obtain
    \begin{equation}
     \frac{1}{2}
      \frac{d}{dt}\|u(t)\|_2^2+\kappa
      \varphi(u(t)) \le C_3 \quad \textrm{for a.a. } t>0.
      \label{eq:absorbing}
    \end{equation}
    Integrating this inequality gives \eqref{eq:add1}.

    Next, setting
    $J(u(t)):=\varphi(u(t))-\psi(u(t))$, we see from \eqref{eq:abst2}
    that
    \begin{equation}\label{eq:j}
     \frac{d}{dt}J(u(t))
      =(\partial\varphi(u(t))-\partial\psi(u(t)),u_t(t))
      =-\|u_t(t)\|_2^2.
    \end{equation}
    Integrating it over $[\delta,T]$ ($0< \delta \le T$) and
    using \eqref{eq:absorb}, we have
    \begin{align*}
     \int_{\delta}^T \|u_t(t)\|_2^2dt
     + \varphi(u(T)) - \varphi(u(\delta))
     &= \psi(u(T)) - \psi(u(\delta)) \\
     &\le (1-\kappa)\varphi(u(T))+\frac{C_1}{2}\|u(T)\|_2^2 \\
     &\le (1-\kappa)\varphi(u(T))+\frac{\kappa}{2}\varphi(u(T))+C_4,
    \end{align*}
    which implies \eqref{eq:add2}.
    Here we used Sobolev's embedding theorem and Young's inequality
    in the last inequality.

    Multiplying \eqref{eq:j} by $t \ge 0$ and integrating it over $[0,\tau]$,
    we have
    $$\int_0^\tau t\|u_t(t)\|_2^2dt + \tau J(u(\tau))
    = \int_0^\tau J(u(t))dt. $$
    Since $\kappa \varphi_1(u)-C_1\|u\|_2^2/2 \le J(u) \; (\le \varphi(u))$
    by \eqref{eq:absorb}, it follows that
    $$\int_0^\tau t\|u_t(t)\|_2^2dt + \kappa \tau \varphi_1(u(\tau))
    \le \frac{C_1}{2}\tau \|u(\tau)\|_2^2 + \int_0^\tau \varphi(u(t))dt. $$
    Setting $\tau=T$ and applying \eqref{eq:add1} to the right-hand side,
    we obtain \eqref{eq:add3}, and
    $$\tau^{1-\frac{1}{\sigma}} (\varphi_1(u(\tau)))^{1-\frac{1}{p}}
    \le \left(\frac{c(T,\|u_0\|_2)}{\kappa}\right)^{1-\frac{1}{p}}
    \frac{1}{\tau^{\frac{1}{\sigma}-\frac{1}{p}}} \in L^{\sigma}(0,T).$$
    This is nothing but \eqref{eq:add4}.
   \end{proof}

   \begin{lem}
    \label{lem:3.3}
    Let $u$ and $c(\cdot,\cdot)$
    be as in Lemma \ref{lem:111}. Then for any $t \in (0,T]$
    there exists a constant
    $L(t)>0$ such that
     \begin{gather}
   \label{eq:difference2}
   \|u_t(t)\|_2 \le e^{L(t)}\|u_t(\delta)\|_2 \quad
      \mathrm{for\ a.a.}\
   t \ge \delta,\\
   \label{eq:difference3}
      t^2\|u_t(t)\|_2^2 \le 2 c(t,\|u_0\|_2)e^{2L(t)}.
  \end{gather}
   \end{lem}

   \begin{proof}[Proof of Lemma \ref{lem:3.3}]
    Let $0<h<1$. Then the monotonicity of $\partial\varphi$ implies that
    \begin{align*}
     & \quad \frac{1}{2}\frac{d}{dt}\|u(t+h)-u(t)\|_2^2 \\[5pt]
     &\le (\partial\psi(u(t+h))-\partial\psi(u(t)),u(t+h)-u(t)) \\
    \begin{split}
     &\le (1-\kappa)\lambda_1\lambda(p-1)\int_{\Omega}
     \max\{|u(t+h)|^{p-2},|u(t)|^{p-2}\}|u(t+h)-u(t)|^2\,dx \\
     &\quad +C_1 \|u(t+h)-u(t)\|_2^2
    \end{split}\\[5pt]
     &\le K(\|\nabla u(t+h)\|_p^{p-2},\|\nabla u(t)\|_p^{p-2})
     \|u(t+h)-u(t)\|_2^2,
    \end{align*}
    where $K(a,b):=(1-\kappa)\lambda_1\lambda(p-1)C_5\max\{a,b\}+C_1$.
    Note that $C_5$ is given by the Sobolev embedding
    $W^{1,p}_0(\Omega) \hookrightarrow L^{\infty}(\Omega)$ only if
    $p>N$, otherwise $\kappa=1$.
    Applying Gronwall's inequality to the preceding estimate yields that
    for all $\delta>0$
    \begin{equation}\label{eq:difference}
     \|u(t+h)-u(t)\|_2 \le e^{\int_{\delta}^t
      K(\|\nabla u(s+h)\|_p^{p-2},\|\nabla u(s)\|_p^{p-2})ds}
      \|u(\delta +h)-u(\delta)\|_2,
    \end{equation}
    where $\int_\delta^tKds$ is bounded with respect to $h$
    by \eqref{eq:add1} in Lemma \ref{lem:111}.
    Dividing \eqref{eq:difference} by $h$ and letting $h \to +0$,
    we obtain \eqref{eq:difference2}.

    Applying \eqref{eq:difference2}:
    $\|u_t(T)\|_2 \le e^{L(T)}\|u_t(t)\|_2$ $(0<t \le T)$ to the
    integrand of the first term on the left-hand side of
    \eqref{eq:add3},
    we obtain
    $$e^{-2L(T)}\|u_t(T)\|_2^2 \int_0^T tdt
    \le c(T,\|u_0\|_2),$$
    and hence \eqref{eq:difference3} follows.
   \end{proof}


   \begin{lem}
    \label{lem:222}
    Let $u$ be as in Lemma \ref{lem:111}.
    Then for any $T>0$ there exists a constant $k_T>0$
    such that $u(t) \in L^{\infty}(\Omega)$ and
    \begin{equation}
     \label{eq:bdd}
       \|u(t)\|_\infty \le k_T \quad
      \mbox{for all $ t \in [\frac{T}{2},T]$}.
    \end{equation}
   \end{lem}

   \begin{proof}[Proof of Lemma \ref{lem:222}]
  In case of (F1), the assertion is trivial by \eqref{eq:add2}
    with Sobolev's embedding theorem.
  We consider the case (F2). However, we note that
  the following proof does not need the condition $p \le N$ in (F2).

  The key to the proof of \eqref{eq:bdd}
  is to deduce a global iterative inequality
  (c.f., DiBenedetto \cite[Chapter V]{Di}).
  Take any $T>0$ and $k>0$.
  Define sequences $\{t_n\}_{n=0}^\infty$, $\{k_n\}_{n=0}^\infty$
  of nonnegative numbers and a sequence of functions
  $\{\zeta_n\}_{n=0}^\infty$ as follows:
  \begin{gather*}
   t_n  = \frac{T}{2}\big(1-\frac{1}{2^n}\big), \quad
   k_n = k \big(1-\frac{1}{2^n}\big) \quad \mbox{and} \\
   \zeta_n(t) =
   \begin{cases}
    0, & 0 \le t \le t_n,  \\
     \frac{t-t_n}{t_{n+1}-t_n},\quad & t_n < t < t_{n+1},\\
    1, & t_{n+1} \le t \le T.
   \end{cases}
  \end{gather*}
  Differentiating $\|(u-k_{n+1})_+(s)\|_2^2\zeta_n(s)$ with respect to
  $s$ and using
  \eqref{eq:monotone} with $\kappa=1$ and $(u-k_{n+1})_+g(u) \ge 0$,
  we obtain
  \begin{multline*}
   \frac{d}{ds}(\|(u-k_{n+1})_+\|_2^2\zeta_n)+
    2\lambda\|\nabla((u-k_{n+1})_+)\|_p^p\zeta_n \\
    \le \|(u-k_{n+1})_+\|_2^2\zeta_n'+2C_1\int_\Omega u(u-k_{n+1})_+\zeta_ndx.
  \end{multline*}
  Integrating this over $[t_n,t]$ with $t_{n+1} \le t \le T$
  and noting the properties
  of $\zeta_n$, we have
  \begin{align*}
   & \quad \|(u-k_{n+1})_+(t)\|_2^2+2\lambda
   \int_{t_{n+1}}^t\|\nabla((u-k_{n+1})_+)\|_p^pds\\
   & \le \frac{2^{n+2}}{T}\int_{t_n}^t\|(u-k_{n+1})_+\|_2^2ds
   +2C_1\int_{t_n}^t \int_\Omega u(u-k_{n+1})_+dxds\\
   & \le \frac{2^{n+2}}{T}\int_{t_n}^t\|(u-k_n)_+\|_2^2ds
   +2C_1(2^{n+1}-1)^2\int_{t_n}^t \|(u-k_n)_+\|_2^2ds,
  \end{align*}
  where we used an obvious inequality and Lemma \ref{lem:-1}
  in the second inequality. Thus
  \begin{multline}
   \label{eq:1}
   \sup_{[t_{n+1},T]}\|(u-k_{n+1})_+\|_2^2+2\lambda
   \int_{t_{n+1}}^T\|\nabla((u-k_{n+1})_+)\|_p^pds\\
   \le C_6\left(1+\frac{1}{T}\right)4^n
   \int_{t_n}^T \|(u-k_n)_+\|_2^2ds
  \end{multline}
  for some constant $C_6>0$. Now we put $Y_n$ as \eqref{eq:Y_n}
  with $t=T$ and it follows from \eqref{eq:1} and \eqref{eq:0}
  in Lemma \ref{lem:0}
  that \eqref{eq:2} in Lemma \ref{lem:2} is satisfied with
  \begin{gather*}
   a  = a_k=\frac{C_7}{k^{\frac{2}{q}(q-2)}}
    \big(1+\frac{1}{T}\big)^{\frac{2}{q}\left(1+\frac{p}{N}\right)},\\
   b = 4^{1+\frac{2p}{Nq}}\ (>1),\quad
   \theta = \frac{2p}{Nq},\quad
   q = \frac{(N+2)p}{N},
  \end{gather*}
  where $C_7$ is a positive constant. Since
  it is possible to take $k=k_T$ sufficiently large as
  \begin{equation}
   \label{eq:k}
  Y_0 \le \int_0^T\|u(s)\|_2^2ds
  \le a_k^{-\frac{1}{\theta}}b^{-\frac{1}{\theta^2}},
  \end{equation}
  Lemma \ref{lem:2}
  gives $Y_n \to 0$ as $n \to +\infty$. Hence
  $$\int_{\frac{T}{2}}^T\|(u-k_T)_+\|_2^2ds=0,$$
  which implies that
  $u \le k_T$ a.e. in $\Omega \times [T/2,T]$. The same argument holds
  true for $-u$ so that Lemma \ref{lem:222} is established.
  \end{proof}

  Now we are in a position to complete the proof of Theorem
  \ref{thm:existence}.
  By Lemma \ref{lem:222} we see that
  $f(u) \in L^{\infty}(\delta,T;L^{\infty}(\Omega))$,
  and hence DiBenedetto \cite[Theorems X.1.1 and X.1.2]{Di}
  (see also Chen and DiBenedetto \cite[Theorems 1 and 2]{CD})
  yields \eqref{eq:reg6}, \eqref{eq:newreg} and
  \begin{equation}
  \label{eq:newbdd}
    \|u(t)\|_{C^{1,\alpha}(\overline{\Omega})} \le
    \gamma(p,N,\delta,T) \quad
    \mbox{for all $t \in [\delta,T]$},
  \end{equation}
  where $\gamma(p,N,\delta,T)$ depends also on
  $\int_{\delta}^T \|\nabla u(t)\|_p^p dt$.

  The first claim of \eqref{eq:reg2} is proved by \eqref{eq:add3}
  in Lemma \ref{lem:111} and \eqref{eq:difference3} in Lemma
  \ref{lem:3.3} because $\|t^{1-1/\sigma}u_t\|_2^{\sigma}=\|\sqrt{t}u_t\|_2^2
  \|tu_t\|_2^{\sigma-2}$.
  Since $(\partial \varphi_1(u), \partial \varphi_2(u)) \ge 0$
  (see \cite[p.138, l.6]{BCP} and \cite[(5)]{Ok}), we have
  \begin{align}
   \label{eq:abst3}
   \|\partial \varphi_1(u)\|_2
   &\le \|\partial\varphi(u)\|_2 \notag \\
   &\le \|u_t\|_2+\|\partial \psi(u)\|_2 \quad \mbox{by
   \eqref{eq:abst2}} \notag \\
   &\le \|u_t\|_2+C_1\|u\|_2+C_2
   (\varphi_1(u))^{1-\frac{1}{p}}
   \quad \mbox{by \eqref{eq:otani2}}.
  \end{align}
  Multiplying \eqref{eq:abst3} by $t^{1-1/\sigma}$ and integrating it over
  $[0,T]$, we obtain the second claim of \eqref{eq:reg2}
  by virtue of the first one and \eqref{eq:add4}.

  In view of \eqref{eq:difference2} in Lemma \ref{lem:3.3} we have
    the first claim of \eqref{eq:reg1}.
  It follows from Tartar's inequality that
  \begin{align*}
   & \ \|\nabla u(t)-\nabla u(s)\|_p^p \\
   &\le 2^{p-2}\int_{\Omega}(|\nabla u(t)|^{p-2}\nabla u(t)-
   |\nabla u(s)|^{p-2}\nabla u(s))\!\cdot\!(\nabla u(t)-\nabla u(s))\,dx \\
   &\le 2^{p-2}(\|\Delta_p u(t)\|_2+\|\Delta_p u(s)\|_2)
   \|u(t)-u(s)\|_2.
  \end{align*}
  Noting that $\Delta_p u \in L^{\infty}(\delta,T;L^2(\Omega))$
  (see \eqref{eq:reg2} with $\sigma=+\infty$) and applying the first claim
  to the right-hand
  side, we obtain \eqref{eq:reg1}
  (c.f., Okazawa and Yokota \cite{OY} for \eqref{eq:reg1}
  in case (F2) is satisfied).

  The uniqueness of solutions of \eqref{P} in $[0,T]$ is proved as follows.
  Let $u$ and $v$ be
  strong solutions of \eqref{P} in $[0,T]$ with $u(0)=u_0 \in L^2(\Omega)$ and
  $v(0)=v_0 \in L^2(\Omega)$, respectively.
  As in the proof of \eqref{eq:difference}, we have
  $$\|u(t)-v(t)\|_2 \le e^{\int_0^t
  K(\|\nabla u(s)\|_p^{p-2},\|\nabla v(s)\|_p^{p-2})ds}
  \|u_0-v_0\|_2.$$
  This implies the uniqueness of solutions of \eqref{P} in $[0,T]$.

  Finally, since $T>0$ is arbitrary, we see that $u$ can be extended
  uniquely to a global strong solution of \eqref{P}.
  Noting that $C_4$ in \eqref{eq:add2} of Lemma \ref{lem:111}
  is independent of $T$, we obtain \eqref{eq:reg4}.

  \begin{rem} \label{rmk3.1} \rm
  To prove the first claim of \eqref{eq:reg1}, we have used
   \eqref{eq:difference2}. If \eqref{eq:difference3} is employed
   instead of \eqref{eq:difference2}, then we see that
   $$\|u(t)-u(s)\|_2 \le e^{L(T)}\sqrt{2c(T,\|u_0\|_2)}
   \cdot |\log{t}-\log{s}|,
   \quad   t,\ s \in (0,T].$$
  \end{rem}


\begin{rem} \label{rmk3.2} \rm
 Let $f(u)$ be replaced by the
 spatially inhomogeneous reaction $f(x,u)$.
 If $f \in C^1(\overline{\Omega}
 \times \mathbb{R})$, $f(x,0)=0$ for every
 $x \in \Omega$ and either (F1) or (F2) is satisfied
 uniformly with respect to $x \in \Omega$,
 then under some condition on $\nabla_x f$
 (see Okazawa \cite{Ok}), we can prove the unique existence of
 global strong solutions of \eqref{P} with $f(u)$ replaced
 by $f(x,u)$.
 \end{rem}


\section{Proof of Theorem \ref{thm:globalattractor}} %sec. 4

 Thanks to Theorem \ref{thm:existence},
 an operator $S(t):L^2(\Omega) \to L^2(\Omega)$ for each $t \ge 0$
 is well defined by $S(t)u_0=u(t;u_0)$.
 Then it is easy to verify that
 the family of operators
 $\{S(t)\}_{t \ge 0}$ enjoys the \textit{$C^0$-semigroup} properties on
 $L^2(\Omega)$, that is, $\{S(t)\}_{t \ge 0}$ is a semigroup
 and the mapping $(t,u_0) \mapsto S(t)u_0$ from $(0,+\infty) \times
 L^2(\Omega)$ into $L^2(\Omega)$ is continuous.

 \begin{proof}[Proof of Theorem \ref{thm:globalattractor}]
  Let $\kappa, C_1$ and $C_3$ be the same constants defined
  in the proof of Theorem \ref{thm:existence}.
  It is sufficient to show the existence of a compact absorbing set in
  $L^2(\Omega)$ for the semigroup $\{S(t)\}_{t \ge 0}$ (see, e.g.,
  Temam \cite[Theorem I.1.1]{Te}).

  From \eqref{eq:absorbing} in the proof of Lemma \ref{lem:111}, in particular
  \begin{equation*}
   \frac{1}{2}\frac{d}{dt}\|u(t)\|_2^2+C_8\|u(t)\|^p_2 \le C_3
  \end{equation*}
  for some $C_8>0$. Hence, Ghidaglia's inequality (Lemma
  \ref{lem:ghidaglia} with $y(t)=\|u(t)\|_2^2$)
  gives
  \begin{equation}
   \label{eq:l^2bounded}
   \|u(t)\|_2^2 \le \Big(\frac{C_3}{C_8}\Big)^\frac{2}{p}
   +(C_8(p-2)t)^{-\frac{2}{p-2}} \quad \mbox{for all $t>0$.}
  \end{equation}
  Next, it follows from \eqref{eq:j} that $J(u(t))$ is nonincreasing in
  $t>0$ and hence
  \begin{equation}
   \label{eq:absorbing1}
   J(u(t+1)) \le \int_t^{t+1}J(u(s))ds \le \int_t^{t+1}\varphi(u(s))ds
  \end{equation}
  when $t>0$. By \eqref{eq:absorb} in Lemma \ref{lem:absorbing},
  we obtain $J(u(t+1)) \ge \kappa\varphi_1(u(t+1))-
  C_1\|u(t+1)\|_2^2/2$.
  Moreover, integrating \eqref{eq:absorbing} over $[t,t+1]$
  gives $\kappa\int_t^{t+1}
  \varphi(u(s))ds \le C_3+\|u(t)\|_2^2/2$. Applying these two
  inequalities to \eqref{eq:absorbing1}, we have
  \begin{equation*}
   2\kappa^2\varphi_1(u(t+1)) \le 2C_3+\|u(t)\|_2^2
    +\kappa C_1 \|u(t+1)\|_2^2;
  \end{equation*}
  and hence \eqref{eq:l^2bounded} yields that there exist positive
  constants $C_9$ and $C_{10}$ such that
  \begin{equation}
   \label{eq:w}
   \varphi_1(u(t)) \le C_9+C_{10}((p-2)(t-1))^{-\frac{2}{p-2}} \quad
    \mbox{for all $t>1$.}
  \end{equation}
  Since $C_9$ and $C_{10}$ are independent of the solution,
  \eqref{eq:w} implies that there exists a number $t_0>1$ such that
  $S(t)B \subset B_{\rho_0}(0)$
  for any bounded set $B \subset L^2(\Omega)$ and $t \ge t_0$,
  where $B_{\rho_0}(0)=\{u \in W^{1,p}_0(\Omega);\|\nabla u\|_p \le \rho_0\}$
  and $\lambda \rho_0^p/p>C_7$. Therefore $B_{\rho_0}(0)$ is a
  compact absorbing set in $L^2(\Omega)$ and
  $\mathcal{A}_\lambda=\bigcap_{t \ge 0}
  \overline{\bigcup_{s \ge t}S(s)B_{\rho_0}(0)}$ is a connected
  global attractor
  in $L^2(\Omega)$.
  In addition, for all $\phi \in \mathcal{A}_\lambda$
  \begin{equation}
   \label{eq:size}
    \|\phi\|_2^2 \le \Big(\frac{C_3}{C_8}\Big)^{2/p}\quad
    \mbox{and} \quad
   \|\nabla \phi\|_p^p \le \frac{pC_9}{\lambda}<\rho_0^p.
  \end{equation}
  Indeed, by the invariance property of global attractor, for every
  $\phi \in \mathcal{A}_\lambda$
  there exists a $u_n \in \mathcal{A}_\lambda$ such that $S(n)u_n=
  u(n;u_n)=\phi$.
  Applying \eqref{eq:l^2bounded} and \eqref{eq:w}
  to $u(t)=S(t)u_n$, and setting
  $t=n \to +\infty$, we obtain these estimates.

  We will prove the boundedness of $\mathcal{A}_\lambda$ in
  $C^{1,\alpha}(\overline{\Omega})$ for some $\alpha \in (0,1)$.
  It follows from \eqref{eq:newbdd} and \eqref{eq:w} that
  \begin{equation}
  \label{eq:newbdd2}
      \|u(t)\|_{C^{1,\alpha}(\overline{\Omega})} \le
    \gamma(p,N) \quad
    \mbox{for all $t \in [1,2]$},
  \end{equation}
  where $\alpha$ and $\gamma(p,N)$ are independent of the solution.
  Now take any $\phi \in \mathcal{A}_{\lambda}$ and $u_2$ be as above.
  Applying \eqref{eq:newbdd2} to $S(t)u_2$, we see that
  $\|S(t)u_2\|_{C^{1,\alpha}(\overline{\Omega})} \le \gamma(p,N)$
  for all $t \in [1,2]$.
  Setting $t=2$, we obtain
  $$\|\phi\|_{C^{1,\alpha}(\overline{\Omega})} \le \gamma(p,N) \quad
  \mbox{for all $\phi \in \mathcal{A}_{\lambda}$;}$$
  that is, $\mathcal{A}_{\lambda}$
  is bounded in
  $C^{1,\alpha}(\overline{\Omega})$.

  The boundedness of $\{\Delta_p\phi;\phi \in \mathcal{A}_\lambda\}$
  in $L^2(\Omega)$
  is also shown in a similar way. The solution $u(t;u_2) \in
  \mathcal{A}_\lambda$ satisfies \eqref{eq:abst3}.
  Multiplying it by $t$ yields
  \begin{align*}
  \|t\partial \varphi_1(u)\|_2
   &\le \|tu_t\|_2+C_1t\|u\|_2+C_2t
   (\varphi_1(u))^{1-\frac{1}{p}}\\
   &\le \|tu_t\|_2+C_1t\|u\|_2+\frac{C_2t}{p}+
   \frac{C_2(p-1)}{p}t\varphi_1(u) \\
   &\le \tilde{c}(t,\|u_0\|_2),
  \end{align*}
   where $\tilde{c}(\cdot,\cdot)$ is a continuous function
   and increasing with respect to the first variable,
   determined by \eqref{eq:difference3},\ \eqref{eq:add1} and
   \eqref{eq:add3}. Hence,
  $\lambda \|\Delta_p u(t;u_2)\|_2 \le \tilde{c}(2,\|u_2\|_2)$
  for all $t \in [1,2]$. Since
  $u_2 \in \mathcal{A}_\lambda$ and \eqref{eq:size} is satisfied,
  there exists a constant $C_{11}>0$
  such that $\|\Delta_p u(t;u_2)\|_2 \le C_{11}$
  for all $t \in [1,2]$. Setting $t=2$,
  we conclude that
  $$\|\Delta_p \phi\|_2 \le C_{11} \quad
  \mbox{for all $\phi \in \mathcal{A}_\lambda$}.$$

  Finally we show that $\mathcal{A}_\lambda=\mathcal{M}_{+}(\mathcal{E}_\lambda)$.
  Since $\mathcal{A}_{\lambda}$ is relatively compact
  in $C^1(\overline{\Omega})$, the function $J(u)=\varphi(u)-\psi(u)$
  is continuous on $\mathcal{A}_{\lambda}$ with respect to
  the $L^2(\Omega)$-topology. This fact and \eqref{eq:j}
  mean that
  $J:\mathcal{A}_{\lambda} \to \mathbb{R}$ is a Lyapunov function of
  $S(\cdot)$.
  Therefore it follows from \cite[Theorem VII.4.1]{Te} that
  $\mathcal{A}_{\lambda}$ coincides with the unstable set of
  $\mathcal{E}_{\lambda}$.
 \end{proof}

 \begin{rem} \label{rmk4.1} \rm
  The absorbing time $t_0$ of the absorbing set $B_{\rho_0}(0)$
  is independent of the set of initial data $B$. Indeed,
  \eqref{eq:w} implies that
  all solutions belong to
  $B_{\rho_0}(0)$ uniformly with respect to the initial data
  when $t \ge t_0$, where
  $$t_0:=1+\frac{1}{p-2}\Big(\frac{pC_{10}}{\lambda \rho_0^p-pC_9}\Big)
  ^{(p-2)/2}.$$
 \end{rem}

\subsection*{Acknowledgment}
The authors would like to thank 
the anonymous referee for the careful reading of the manuscript. 

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\end{document}