\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 77, pp. 1--10.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/77\hfil Strong global attractor]
{Strong global attractor for a quasilinear nonlocal
wave equation on $\mathbb{R}^N$}
\author[P. G. Papadopoulos, N. M. Stavrakakis\hfil EJDE-2006/77\hfilneg]
{Perikles G. Papadopoulos, Nikolaos M. Stavrakakis}

\address{Perikles G. Papadopoulos \newline
Department of Mathematics, National Technical University,
Zografou Campus, 157 80 Athens, Greece}
\email{perispap@math.ntua.gr}

\address{Nikolaos M. Stavrakakis \newline
Department of Mathematics, National Technical University,
Zografou Campus, 157 80 Athens, Greece}
\email{nikolas@central.ntua.gr}

\date{}
\thanks{Submitted May 10, 2006. Published Juy 12, 2006.}
\subjclass[2000]{35A07, 35B30, 35B40, 35B45, 35L15, 35L70, 35L80}
\keywords{Quasilinear hyperbolic equations; Kirchhoff strings;
  global attractor; \hfill\break\indent
unbounded domains; generalized Sobolev spaces;
 weighted $L^p$ spaces}


\begin{abstract}
 We study the long time behavior of solutions to the nonlocal quasilinear
 dissipative wave equation
 $$
 u_{tt}-\phi (x)\|\nabla u(t)\|^{2}\Delta u+\delta u_{t}+|u|^{a}u=0,
 $$
 in $\mathbb{R}^N$, $t \geq 0$,  with
 initial conditions  $ u(x,0) = u_0 (x)$  and  $u_t(x,0) = u_1(x)$.
 We consider  the case  $N \geq 3$, $\delta> 0$,  and
 $(\phi (x))^{-1}$ a positive function in
 $L^{N/2}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N )$.
 The existence of a global attractor is proved in the strong topology of the
 space  $\mathcal{D}^{1,2}(\mathbb{R}^N) \times L^{2}_{g}(\mathbb{R}^N)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

Our aim in this work is to study the  quasilinear
hyperbolic initial-value problem
\begin{gather} \label{e2.4}
u_{tt} - \phi(x) \|\nabla u(t)\|^{2} \Delta u +
\delta u_{t} +|u|^{a}u=0, \quad  x \in {\mathbb{R}^N}, \; t \geq
0,\\
\label{e2.5} u(x,0)= u_{0}(x), \quad
u_{t}(x,0)= u_{1}(x), \quad x \in {\mathbb{R}^N},
\end{gather}
with  initial conditions   $u_{0}$, $u_{1}$  in appropriate
function spaces, $N \geq 3$,  and $\delta > 0$. Throughout the
paper we assume that the functions  $\phi,g: {\mathbb{R}^N}
\to {\mathbb{R}}$  satisfy the  condition
\begin{itemize}
\item[(G1)]
$\phi(x)>0$,  for all $x \in {\mathbb{R}^N}$ and
$(\phi(x))^{-1}:= g(x) \in L^{N/2}(\mathbb{R}^N)
\cap L^{\infty}(\mathbb{R}^N)$.

\end{itemize}
For the modelling process we refer the reader to some of our
earlier papers \cite{pap-stav,pap-stav2} or to the original paper
by Kirchhoff  in 1883  \cite{Kir83}. There he proposed the
so called  Kirchhoff string model in the study of
oscillations of stretched strings and plates.

In bounded domains there is a vast literature concerning the
attractors of semilinear waves equations. We refer the reader
to the monographs \cite{hale88,tem88}. Also in the paper
\cite{hale-stav}, the existence of global attractor in a weak
topology is discussed for a general dissipative wave equation.
Ono \cite{ono97}, for  $\delta \geq 0$,  has proved global
existence, decay estimates, asymptotic stability and blow up
results for a degenerate non-linear wave equation of Kirchhoff
type with a strong dissipation.
On the other hand, it seems that very few results are achieved for
the unbounded domain case. In our previous work \cite{pap-stav},
we proved global existence and blow-up results
for an equation of Kirchhoff type in all of  $\mathbb{R}^N$. Also, in
\cite{pap-stav2} we proved the existence of compact invariant sets
for the same equation. Recently, in \cite{pap-stav2005} we studied
the stability of the origin for the generalized equation of
Kirchhoff strings on  $\mathbb{R}^N$,  using central manifold theory.
Also,  Karahalios and  Stavrakakis \cite{stav97},
\cite{stav98} proved existence of global attractors and estimated
their dimension for a semilinear dissipative wave equation on
$\mathbb{R}^N$.

The presentation of this paper is follows: In Section 2, we
discuss the space setting of the problem and the necessary
embeddings for constructing the evolution triple. In Section 3, we
prove existence of an absorbing set for our problem in the energy
space $\mathcal{X}_{0}$. Finally in Section 4, we prove that there
exists a global attractor  $\mathcal{A}$  in the strong
topology of the energy space $\mathcal{X}_{1}:=\mathcal{D}^{1,2}(\mathbb{R}^N)
\times L^{2}_{g}(\mathbb{R}^N)$, so extending some earlier results of us
on the asymptotic behavior of the problem (see \cite{pap-stav2}).

\subsection*{Notation} We denote by  $B_{R}$  the open ball
of $\mathbb{R}^N$  with center  $0$  and radius  $R$.  Sometimes for
simplicity we use the symbols
$C_{0}^{\infty}$,   ${\mathcal{D}^{1,2}}$,  $L^{p}$,
$1\leq p \leq {\infty}$,  for the spaces
$C_{0}^{\infty}(\mathbb{R}^N)$, $\mathcal{D}^{1,2}(\mathbb{R}^N)$,
$L^{p}(\mathbb{R}^N)$,
respectively; $\|\cdot\|_{p}$  for the norm
$\|\cdot\|_{L^{p}(\mathbb{R}^N)}$, where in case of  $p=2$  we may omit
the index. The symbol  $:=$  is used for definitions.

\section{Space Setting. Formulation of the Problem}

 As it is already shown  in the paper \cite{pap-stav},
the space setting for the initial conditions and the solutions of
problem \eqref{e2.4}-\eqref{e2.5}  is the product space
$$
\mathcal{X}_{0}:=D(A) \times \mathcal{D}^{1,2}(\mathbb{R}^N), \quad   N \geq
3.
$$
Also the  space
$\mathcal{X}_{1}:=\mathcal{D}^{1,2}(\mathbb{R}^N)
 \times L^{2}_{g}(\mathbb{R}^N)$, with the associated norm
$e_{1}(u(t)):= \|u\|^{2}_{\mathcal{D}^{1,2}} +\|u_{t}\|^{2}_{L^{2}_{g}}$
is introduced,  where the space
$L^{2}_{g}(\mathbb{R}^N)$  is defined to be the closure of
$C^{\infty}_{0}(\mathbb{R}^N)$  functions with respect to the inner
product
 \begin{equation}  \label{inner1}
(u,v)_{L_{g}^{2}(\mathbb{R}^N)} := \int_{\mathbb{R}^N} guv dx.
 \end{equation}
It is clear that  $L^{2}_{g}(\mathbb{R}^N)$  is a separable Hilbert
space and the embedding  $\mathcal{X}_{0}\subset \mathcal{X}_{1}$  is
compact. The homogeneous Sobolev space  $\mathcal{D}^{1,2}(\mathbb{R}^N)$
is defined, as the closure of  $C^{\infty}_{0}(\mathbb{R}^N)$  functions
with respect to the following {\sf energy norm} $\|u\|^{2}_{\mathcal{D}^{1,2}}:=\int_ {\mathbb{R}^N}{|\nabla u|}^2dx $.  It is  known that
$$
\mathcal{D}^{1,2}(\mathbb{R}^N)
=\big\{u \in L^{\frac{2N}{N-2}}(\mathbb{R}^N): \nabla u
\in(L^2(\mathbb{R}^N))^N \big\}
$$
and  $\mathcal{D}^{1,2}(\mathbb{R}^N)$  is embedded continuously in
$L^{\frac{2N}{N-2}}(\mathbb{R}^N)$,  that is, there exists  $k>0$  such
that
\begin{equation} \label{e2.1}
\|u\|_{\frac{2N}{N-2}} \leq k \|u\|_{\mathcal{D}^{1,2}}.
\end{equation}
The  space  $D(A)$  is going
to be introduced and studied later in this section. The following
 generalized version of Poincar\'e's inequality is going to
be frequently used
 \begin{equation}  \label{e2.2}
\int_{\mathbb{R}^N} |\nabla u|^{2} dx \geq \alpha \int_{\mathbb{R}^N} g
u^{2} dx,
\end{equation}
%
for all  $u \in C^{\infty}_{0}$  and  $g\in L^{N/2}$, where
$\alpha:= k^{-2}\|g\|^{-1}_{N/2}$ (see \cite[Lemma 2.1]{bro94}).
It  is  shown that  $\mathcal{D}^{1,2}(\mathbb{R}^N)$  is a separable
Hilbert space. Moreover, the following compact embedding is
useful.

\begin{lemma} \label{L.0}
 Let  $g\in L^{N/2}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$. Then
the embedding  $\mathcal{D}^{1,2} \subset L^{2}_{g}$  is compact.
Also, let  $g\in L^{\frac{2N}{2N-pN+2p}}(\mathbb{R}^N)$. Then the
following continuous embedding  $\mathcal{D}^{1,2}(\mathbb{R}^N)\subset
L^{p}_{g}(\mathbb{R}^N)$ is valid,  for all  $1\leq p\leq 2N/(N-2)$.
\end{lemma}

For the proof of the above lemma, we refer to \cite[Lemma 2.1]{stav99}.
To study the properties of the operator  $-\phi\Delta$, we consider the
equation
 \begin{equation}  \label{e2.3nn}
 -\phi (x)\Delta u(x)= \eta(x), \quad   x\in\mathbb{R}^N,
 \end{equation}
without boundary conditions. Since for every  $u,  v \in
C^{\infty}_{0}(\mathbb{R}^N)$ we have
 \begin{equation} \label{e2.4nn}
(-\phi\Delta u, v)_{L^{2}_{g}}=\int_{\mathbb{R}^N}\nabla u\nabla v \,dx,
 \end{equation}
we may consider (\ref{e2.3nn}) as an operator equation of
the form
 \begin{equation}
 \label{e2.5nn}
 A_{0}u=\eta, \quad  A_{0}:D(A_{0})\subseteq
L^{2}_{g}(\mathbb{R}^N) \to L^{2}_{g}(\mathbb{R}^N), \quad
 \eta \in L^{2}_{g}(\mathbb{R}^N).
 \end{equation}
The operator  $A_{0}=-\phi\Delta$  is a symmetric, strongly
monotone operator on  $L^{2}_{g}(\mathbb{R}^N)$. Hence, the theorem of
Friedrichs is applicable. The  energy scalar product given by
(\ref{e2.4nn}) is
$$
(u, v)_{E}=\int_{\mathbb{R}^N}\nabla u\nabla v dx
$$
and the  energy space  $X_{E}$  is the completion of
$D(A_{0})$  with respect to  $(u, v)_{E}$.  It is obvious that
the energy space is the homogeneous Sobolev space
$\mathcal{D}^{1,2}(\mathbb{R}^N)$. The  energy extension
  $A_{E}=-\phi\Delta$  of  $A_{0}$,
\begin{equation}  \label{energ}
 -\phi\Delta : \mathcal{D}^{1,2}(\mathbb{R}^N)  \to
 \mathcal{D}^{-1,2}(\mathbb{R}^N),
 \end{equation}
  is defined to be the duality mapping of  $\mathcal{D}^{1,2}(\mathbb{R}^N)$.
We define  $D(A)$  to be the set of all solutions of equation
(\ref{e2.3nn}), for arbitrary  $\eta \in L^{2}_{g}(\mathbb{R}^N)$. Using
the theorem of Friedrichs we have that the extension  $A$  of
$A_{0}$  is the restriction of the energy extension  $A_{E}$
to the set  $D(A)$.  The operator  $A=-\phi\Delta$  is
self-adjoint and therefore graph-closed. Its domain  $D(A)$,  is
a Hilbert space with respect to the graph scalar product
$$
(u, v)_{D(A)}=(u, v)_{L^{2}_{g}}+(Au, Av)_{L^{2}_{g}}, \quad
\mbox{for all }  u,  v \in D(A).
$$
The norm induced by the scalar product is
$$
\|u\|_{D(A)}=\Big\{\int_{\mathbb{R}^N}g|u|^{2}\,dx
+\int_{\mathbb{R}^N}\phi|\Delta u|^{2}\,dx\Big\}^{1/2},
$$
which is equivalent to the norm
$$
\|Au\|_{L^{2}_{g}}=\big\{\int_{\mathbb{R}^N}\phi|\Delta u|^{2}\,dx
\big\}^{1/2}.
$$
So we have established the  evolution quartet
\begin{equation} \label{evol}
D(A) \subset {\mathcal{D}^{1,2}}(\mathbb{R}^N) \subset
L^{2}_{g}(\mathbb{R}^N) \subset {\mathcal{D}^{-1,2}}(\mathbb{R}^N),
\end{equation}
where all the embeddings are dense and compact.  Finally, the
definition of  weak solutions for the problem
\eqref{e2.4}--\eqref{e2.5} is given.


\begin{definition} \label{D.1pb}  \rm
A \emph{weak solution} of \eqref{e2.4}-\eqref{e2.5}  is a function  $u$
such that the following three conditions are satisfied:
\begin{itemize}
\item[(i)]   $u \in L^{2}[0,T; D(A)]$,
$u_{t} \in L^{2}[0,T; \mathcal{D}^{1,2}(\mathbb{R}^N)]$,
$u_{tt} \in L^{2}[0,T; L^{2}_{g}(\mathbb{R}^N)]$,
\item[(ii)]   for all  $ v \in C^{\infty}_{0}([0,T]\times (\mathbb{R}^N))$,
satisfies the generalized formula
\begin{equation} \label{e2.13n}
\begin{gathered}
 {\int_{0}^{T}}(u_{tt}(\tau ) ,
 v(\tau))_{L^{2}_{g}}d\tau + {\int_{0}^{T}}\Big(\|\nabla
 u(t)\|^{2}{\int_{\mathbb{R}^N}}\nabla u(\tau)\nabla v(\tau)
 dx\,d\tau\Big)\\
+ \delta  {\int_{0}^{T}}(u_{t}(\tau),
 v(\tau))_{L^{2}_{g}}d\tau  + {\int_{0}^{T}}(
 |u(\tau)|^{a}u(\tau),  v(\tau))_{L^{2}_{g}}d\tau =0,
 \end{gathered}
\end{equation}
\item[(iii)] $u$ satisfies the initial  conditions
$u(x,0)=u_{0}(x)$, $ u_{0}\in D(A)$,  $u_{t}(x,0)=u_{1}(x)$,
$u_{1} \in {\mathcal{D}^{1,2}}(\mathbb{R}^N)$.
\end{itemize}
 \end{definition}

\section{ Existence of an Absorbing Set.}

In this section we prove existence of an absorbing set for our
problem \eqref{e2.4}-\eqref{e2.5} in the energy space $\mathcal{X}_{0}$.
First, we give existence and uniqueness results for the problem
\eqref{e2.4}-\eqref{e2.5} using the space setting established
previously.

\begin{theorem}[Local Existence] \label{Th.1}
  Consider that $(u_{0},u_{1}) \in D(A) \times {\mathcal{D}^{1,2}}$
and satisfy the  nondegeneracy condition
\begin{equation} \label{nondeg}
 \|\nabla u_{0}\|>0.
\end{equation}
Then there exists  $T=T(\|u_{0}\|_{D(A)}$, $\|\nabla u_{1}\|)>0$
 such that the problem \eqref{e2.4}-\eqref{e2.5} admits a unique
local weak solution $u$ satisfying
$$
u \in C(0,T ; D(A)) \; \; and \; \; u_{t} \in C(0,T; {\mathcal{D}^{1,2}}).
$$
Moreover, at least one of the following two statements holds:
\begin{itemize}
\item[(i)] $T= {+ \infty }$,

\item[(ii)] $e(u(t)):=\|u(t)\|^{2}_{D(A)}+\|u_{t}(t)\|^{2}_{\mathcal{D}^{1,2}}
\to {\infty}$, as $t \to T_{-}$.
\end{itemize}
\end{theorem}

For the proof of the above theorem, we refer to \cite[Theorem 3.2]{pap-stav}.

\begin{remark} \label{rem.1} \rm
The nondegeneracy condition (\ref{nondeg}) is imposed by the method which
is used even for the proof  of existence of  local solutions of the
problem \eqref{e2.4}-\eqref{e2.5}.
For more details we refer to the proof of Theorem 3.2 in \cite{pap-stav}.
Also we must notice that this condition is necessary even in the case
of bounded domains (e.g., see \cite{ono97} and \cite{ono97-2}).
\end{remark}

\begin{lemma} \label{lem4.1}
Assume that $a\geq 0$, $N\geq 3$. If the initial data
$(u_{0}, u_{1})\in D(A)\times \mathcal{D}^{1,2}$  and satisfy the condition
\begin{equation}
\label{e^{*}}
 \|\nabla u_{0}\| > 0,
 \end{equation}
then
\begin{equation}
\label{e^{**}}
 \|\nabla u(t)\|>0, \quad \mbox{for all }  t\geq 0.
 \end{equation}
 \end{lemma}

 \begin{proof}
Let  $u(t)$  be a unique solution of the problem
 \eqref{e2.4}-\eqref{e2.5} in the sense of Theorem \ref{Th.1} on  $[0,T)$.
 Multiplying \eqref{e2.4}  by  $-2\Delta u_{t}$  (in the sense of the inner
 product in the space  $L^{2}$) and integrating it over
  $\mathbb{R}^N$,  we have
 \begin{equation}  \label{e4.18}
\begin{aligned}
&\frac{d}{dt}\|\nabla u_{t}(t)\|^{2}+\|\nabla  u(t)\|^{2} \frac{d}{dt}
 \|u(t)\|^{2}_{D(A)} \\
+ 2\|\nabla u_{t}(t)\|^{2}+2(|u|^{a}u,  \Delta u_{t}(t))=0
\end{aligned}
\end{equation}
Since  $\|\nabla u_{0}\|>0$, we see that  $\|\nabla u(t)\|>0$
near  $t=0$. Let
\[
T:=\sup \{t \in [0,+\infty):\|\nabla u(s)\|>0 \quad  \mbox{for }
  0\leq s<t\},
\]
then  $T>0$  and  $\|\nabla u(t)\|>0$   for   $0\leq t<T$. By contradiction
we may prove that  $T=+\infty$.
\end{proof}

\begin{theorem}[Absorbing Set] \label{them4.2}
Assume that  $0 \leq a < 2/(N-2)$,
$N \geq 3$,   $M_{0}:=\frac{1}{2}\|\nabla u_{0}\|^{2}>0$,
$(u_{0}, \; u_{1}) \in D(A)\times \mathcal{D}^{1,2}$  and
 \begin{equation}  \label{e.e}
 \frac{\delta}{4} > 4\alpha^{-1/2}R^{2}c_{3}^{2},
 \end{equation}
 where  $c_{3}:=(\max \{1, M_{0}^{-1}\})^{1/2}$ and
 $R$  a given positive constant. Then the ball
$\mathcal{B}_{0}:=B_{\mathcal{X}_{0}}(0,\bar{R_{*}})$,
 for any  $\bar{R_{*}}
>R_{*}$,  is an absorbing set in the energy space  $\mathcal{X}_{0}$, where
\[
R_{*}^{2}:=\frac{2k_{2}R^{2(a+1)}}{\delta}\big(\frac{\delta}{4}-
\frac{4R^{2}c_{3}^{2}}{\sqrt{\alpha}}\big)^{-1}.
\]
\end{theorem}

\begin{proof}
Given the constants  $T >0, \; R >0$, we introduce the two
parameter space of solutions
\[
X_{T,R}:= \{ u \in C (0,T;D(A)) :u_{t}  \in C
(0,T; {\mathcal{D}^{1,2}} ),\, e(u) \leq R^{2},\, t \in [0,T]\},
\]
where  $e(u):=\|u_{t}\|^{2}_{\mathcal{D}^{1,2}}+\|u\|^{2}_{D(A)}$.
The set  $X_{T,R}$  is a complete metric space under the
distance  $d(u, v):=\sup_{0 \leq t \leq T}e(u(t)-v(t))$.
Following \cite{ono97} we introduce the notation
$$
T_{0}:=\sup\{ t\in [0, \infty):\, \|\nabla u(s)\|^{2}
>M_{0}, \;  0 \leq s \leq t\}.
$$
Condition $\frac{1}{2}\|\nabla u_{0}\|^{2}=M_{0}>0$ implies
  $T_{0} >0$  and  $\|\nabla u(t)\|^{2}> M_{0}>0$, for all
$t \in [0,T_{0}]$.  Next, we set  $v=u_{t}+\varepsilon u$  for
sufficiently small  $\varepsilon$.  Then, for calculation needs,
equation  \eqref{e2.4} is rewritten as
 \begin{equation}
\label{e4.1} v_{t}+(\delta -
\varepsilon)v +(-\phi(x)\|\nabla u\|^{2}\Delta -\varepsilon
(\delta -\varepsilon))u+f(u)=0.
 \end{equation}
Multiplying equation (\ref{e4.1}) by
$$
gAv=g(-\varphi \Delta)v=-\Delta v=-\Delta (u_{t}+\varepsilon u),
$$
and integrating over  $\mathbb{R}^N$,  we obtain (using H\"older
inequality with  $ p^{-1}=\frac{1}{N}$,
$q^{-1}=\frac{N-2}{2N}$,  $r^{-1}=\frac{1}{2}$)
\begin{equation} \label{e4.22}
\begin{aligned}
&{\frac{1}{2}}{\frac{d}{dt}}\big\{\|u\|^{2}_{\mathcal{D}^{1,2}}\|u\|^{2}_{D(A)}
+ \|v\|^{2}_{\mathcal{D}^{1,2}}
+\frac{\varepsilon (\delta -\varepsilon)}{2}\|u\|^{2}_{\mathcal{D}^{1,2}}
\big\}\\
&+(\delta -\varepsilon)\|v\|^{2}_{\mathcal{D}^{1,2}}+\varepsilon
\|u\|^{2}_{\mathcal{D}^{1,2}}\|u\|^{2}_{D(A)} +
\varepsilon^{2}(\delta -\varepsilon)\|u\|^{2}_{\mathcal{D}^{1,2}}\\
&\leq \big|\big({\frac{d}{dt}}\|u\|^{2}_{\mathcal{D}^{1,2}}\big)
 \|u\|^{2}_{D(A)}\big|
+\|u\|^{a}_{L^{Na}} \|\nabla u\|_{L^{\frac{2N}{N-2}}}\|\nabla v\|.
\end{aligned}
\end{equation}
We observe that
\begin{equation} \label{e4.23}
\begin{aligned}
\theta(t)&:= \|u\|^{2}_{\mathcal{D}^{1,2}}\|u\|^{2}_{D(A)}
 +\|v\|^{2}_{\mathcal{D}^{1,2}}
+\frac{\varepsilon (\delta -\varepsilon)}{2}\|u\|^{2}_{\mathcal{D}^{1,2} }\\
 &\geq M_{0}\|u\|^{2}_{D(A)}+\|u_{t}\|^{2}_{\mathcal{D}^{1,2}} \geq
c_{3}^{-2}e(u).
 \end{aligned}
\end{equation}
Also
\begin{equation} \label{e.aa}
\begin{aligned}
 \big|\big({\frac{d}{dt}}\|u\|^{2}_{\mathcal{D}^{1,2}}\big)\|u\|^{2}_{D(A)}
\big|&= \big|\big(2\int_{\mathbb{R}^N}\Delta u u_{t} \varphi g\,dx\big)
\|u\|^{2}_{D(A)}\big| \\
 &\leq 2\big(\|u\|^{2}_{D(A)}\big)^{1/2}
 \big(\|u_{t}\|^{2}_{L^{2}_{g}}\big)^{1/2}
 \|u\|^{2}_{D(A)}\\
 &\leq 2 \alpha^{-1/2}R^{2}e(u) \leq 2 \alpha^{-1/2}R^{2} c_{3}^{2} \theta(t).
 \end{aligned}
\end{equation}
Applying Young's inequality in the last term of (\ref{e4.22}) and
using  relations (\ref{e4.23}),  (\ref{e.aa}) and the estimates
 \begin{equation}  \label{e4.25}
 \|u\|^{a}_{L^{Na}}\leq R^{a} \quad  \mbox {and} \quad
 \|\nabla u\|_{L^{\frac{2N}{N-2}}} \leq \|u\|_{D(A)} \leq R,
 \end{equation}
inequality (\ref{e4.22}) becomes (for suitably small $\varepsilon$)
 \begin{equation}  \label{e4.a}
 \frac{d}{dt}\theta(t)+C_{*}\theta(t) \leq \frac{C(R)}{\delta},
 \end{equation}
 where  $C_{*}=\frac{1}{2}
\left(\delta/4-4\alpha^{-1/2}R^{2}c_{3}^{2}\right)
>0$ and   $C(R)=R^{2(a+1)}$. An application of Gronwall's inequality
 in the relation (\ref{e4.a}) gives
 \begin{equation} \label{e4.b}
 \theta(t) \leq \theta(0)
e^{-C_{*}t}+{\frac{1-e^{-C_{*}t}}{C_{*}}}\frac{C(R)}{\delta}.
 \end{equation}
Following the reasoning developed by K. Ono (see \cite{ono97}),
the nondegeneracy condition  $\|\nabla u_{0}\|>0$  and the
relation (\ref{e^{**}}), imply that  $\|\nabla u(s)\|> M_{0}> 0$,
$ 0\leq s\leq  t$,  $t \in [0, +\infty)$.  Now, letting
$t \to \infty$,  in the relation (\ref{e4.b}) conclude
that
 \begin{equation}  \label{e4.c}
 \lim_{t \to \infty}\sup \theta(t)
 \leq \frac{R^{2(a+1)}}{\delta C_{*}}:=R^{2}_{*}.
 \end{equation}
 So, the ball  $B_{0}:=B_{\mathcal{X}_{0}}(0,\bar{R_{*}})$,  for any
 $\bar{R_{*}}>R_{*}$, is an  absorbing set for the associated semigroup
$S(t)$  in the energy
 space of solutions  $\mathcal{X}_{0}$.
\end{proof}

\begin{corollary}[Global Existence] \label{coroll3.4}
  The unique local  solution the problem \eqref{e2.4}-\eqref{e2.5}
defined by Theorem \ref{Th.1}  exists globally in time.
 \end{corollary}

\begin{proof}
Combining  inequality (\ref{e4.c}) and  the
arguments developed in the proof of  \cite[Theorem 3.2]{pap-stav},
we conclude that the solution
 of the problem \eqref{e2.4}-\eqref{e2.5} exists globally in time.
\end{proof}

\section{Strong Global Attractor in the space ${\mathcal{X}}_{1}$}

In this section we  study the problem
\eqref{e2.4}-\eqref{e2.5} from a dynamical system point of view.
 We need the following results.

\begin{theorem}\label{Th.5.1}
Assume that  $0 \leq a \leq 4/(N-2)$, where $N\geq 3$.
If $(u_{0},  u_{1}) \in D(A) \times {\mathcal{D}^{1,2}}$
and satisfy the  nondegeneracy condition
\begin{equation} \label{nondeg-2}
 \|\nabla u_{0}\|>0,
\end{equation}
then there exists  $T>0$  such that the problem
\eqref{e2.4}-\eqref{e2.5} admits local weak solutions $u$
satisfying
\begin{equation}  \label{cp}
 u \in C(0,T ;{\mathcal{D}^{1,2}})  \quad\text{and}\quad
 u_{t} \in C(0,T; L^{2}_{g}).
 \end{equation}
\end{theorem}

\begin{proof}
The proof follows the lines of \cite[Theorem 3.2]{pap-stav},
so we just sketch the proof. The compactness of
the  embedding  ${\mathcal{X}}_{0}\subset {\mathcal{X}}_{1}$  implies
$ e_{1}(u(t)) \leq e(u(t))$,
where the associated norms are
\[
e_{1}(u(t)):=\|u\|^{2}_{\mathcal{D}^{1,2}}+\|u_{t}\|^{2}_{L^{2}_{g}}\quad
\text{and}\quad
e(u(t)):=\|u\|^{2}_{D(A)}+\|u_{t}\|^{2}_{\mathcal{D}^{1,2}}.
\]
 Then, for some positive constant   $R$  an a priori bound can be found
of the form
\[
 e_{1}(u(t)) \leq e(u(t))\leq R^{2}.
 \]
Hence the solutions $u$ of the problem \eqref{e2.4}-\eqref{e2.5}
satisfy
 \[
 u \in L^{\infty}(0,T; \; \mathcal{D}^{1,2} ), \quad
 u_{t} \in L^{\infty}(0,T;  L^{2}_{g}).
 \]
Finally, the continuity properties (\ref{cp}), are proved
following ideas from \cite[Sections II.3 and II.4]{tem88}.
\end{proof}

 Next, the strong continuity of the semigroup $S(t)$ is
proved in the space $\mathcal{X}_{1}$.

\begin{lemma} \label{lem.5.2}
 The mapping $S(t): \mathcal{X}_{1} \to \mathcal{X}_{1}$ is continuous,
for all  $t \in \mathbb{R}$.
 \end{lemma}

 \begin{proof}
Let  $u,v$  two solutions of the problem \eqref{e2.4}-\eqref{e2.5}
  such that
 \begin{gather*}
 u_{tt}-\phi (x) \|\nabla u\|^{2}\Delta u+\delta u_{t}=-|u|^{a}u, \\
 v_{tt}-\phi (x) \|\nabla v\|^{2}\Delta v+\delta v_{t}=-|v|^{a}v.
 \end{gather*}
Let $w=u-v$. So, we have
\begin{gather*}
 w_{tt}-\phi \|\nabla u\|^{2}\Delta
 w+ \delta w_{t}=\phi\{\|\nabla u\|^{2}-\|\nabla
 v\|^{2}\}\Delta v -(|u|^{a}u-|v|^{a}v)\\
 w(0)=0, \quad w_{t}(0)=0.
\end{gather*}
Multiplying the previous equation by $2gw_{t}$ and integrating
over $\mathbb{R}^N$, we get
\begin{equation} \label{e5.23}
\begin{gathered}
 {\int_{\mathbb{R}^N}}gw_{t}w_{tt} dx - 2{\int_{\mathbb{R}^N}}\|\nabla
 u\|^{2}\Delta ww_{t} dx  + 2\delta
 {\int_{\mathbb{R}^N}}gw_{t}^{2} dx \\
 =\{\|\nabla u\|^{2} - \|\nabla
 v\|^{2}\}{\int_{\mathbb{R}^N}}\Delta vw_{t} dx
 -2{\int_{\mathbb{R}^N}}g(|u|^{a}u-|v|^{a}v))w_{t}dx.
 \end{gathered}
\end{equation}
Hence
\begin{equation}  \label{e5.24}
 \begin{aligned}
&{\frac{d}{dt}}e^{*}(w)+2\delta \|w_{t}\|^{2}_{L^{2}_{g}}\\
&= ({\frac{d}{dt}}\|\nabla u\|^{2})\|\nabla w\|^{2}+2\{\|\nabla
u\|^{2}-\|\nabla v\|^{2}\}  (\Delta v\;,w_{t})-2(|u|^{a}u-|v|^{a}v,
w_{t})_{L^{2}_{g}}\\
&\equiv I_{1}(t)+I_{2}(t)+I_{3}(t).
 \end{aligned}
\end{equation}
So
\begin{equation} \label{e5.25}
 {\frac{d}{dt}}e^{*}(w) \leq I_{1}(t)+I_{2}(t)+I_{3}(t),
 \end{equation}
where $e^{*}(w)=\|w_{t}\|^{2}_{L^{2}_{g}}+C_{u}\|w\|^{2}_{\mathcal{D}^{1,2}}$
and $C_{u}=\|u\|^{2}_{\mathcal{D}^{1,2}}$.
To estimate the above integrals, more smoothness of the solutions
$u, v$ is needed.  Theorem \ref{Th.1} guarantees the uniqueness
of local solutions in the space $\mathcal{X}_{0}$, if the initial
conditions $(u_{0},u_{1}) \in \mathcal{X}_{0}$. To improve these
results, it is assumed that $(u_{0}, u_{1}) \in \mathcal{X}_{1}$.
Then, applying again Theorem \ref{Th.1}, it could be proved the
existence of a local solution $(u, u_{t})$ in $\mathcal{X}_{1}$.
Furthermore, we may obtain
\begin{equation}  \label{e5.26}
 \begin{aligned}
 I_{1}(t)&=(2\int_{\mathbb{R}^N}\Delta u u_{t}\phi(x) g(x)dx)\|\nabla
 w\|^{2} \\
 &\leq  2(\|u\|^{2}_{D(A)})^{1/2}(\|u_{t}\|^{2}_{L^{2}_{g}})^{1/2}\|\nabla
 w\|^{2} \\
 &\leq 2R_{*}k(\|u_{t}\|^{2}_{\mathcal{D}^{1,2}})^{1/2}\|\nabla
 w\|^{2} \\
&\leq  2R_{*}^{2}k\|\nabla w\|^{2} \leq C_{2}e^{*}(w),
 \end{aligned}
\end{equation}
where $C_{2}=2R^{2}_{*}k$. Also, the following estimation is valid
\begin{equation} \label{e5.27}
\begin{aligned}
I_{3}(t)\leq |I_{3}(t)|
&\leq \alpha^{-1}(\|\nabla u\|^{2}-\|\nabla
 v\|^{2})\|\nabla (u-v)\| \; \|w_{t}\|_{L^{2}_{g}} \\
 &\leq  \alpha^{-1}2R_{*}^{2}\|w\|_{\mathcal{D}^{1,2}} \|w_{t}\|_{L^{2}_{g}} \\
&\leq C_{A}(\frac{C_{u}}{2C_{u}}\|w\|^{2}_{\mathcal{D}^{1,2}}
 +\frac{1}{2}\|w_{t}\|^{2}_{L^{2}_{g}}) \\
&\leq C_{A}C_{B}e^{*}(w),
 \end{aligned}
\end{equation}
 where we have used Young's inequality and
$ C_{A}=2 \alpha^{-1}R_{*}^{2}$,
$ C_{B}=\max (\frac{1}{2}$, $\frac{1}{2C_{u}})$.
Hence,
\begin{equation} \label{e5.28}
\begin{aligned}
I_{2}(t)&\leq {(\|\nabla u\|+\|\nabla
 v\|)(\|\nabla (u-v)\|)}\Big(\int_{\mathbb{R}^N}\Delta vw_{t}dx\Big) \\
 &\leq 2R_{*}\|w\|_{\mathcal{D}^{1,2}}(\|v\|^{2}_{D(A)})^{1/2}
 (\|w_{t}\|^{2}_{L^{2}_{g}})^{1/2}\\
 &\leq 2R^{2}_{*}\|w\|_{\mathcal{D}^{1,2}}(\|w_{t}\|^{2}_{L^{2}_{g}})^{1/2}
 \\
 &\leq 2R^{2}_{*}(\frac{C_{u}}{2C_{u}}\|w\|^{2}_{\mathcal{D}^{1,2}}
 +\frac{1}{2}\|w_{t}\|^{2}_{L^{2}_{g}})
 \leq C_{\Gamma}C_{B}e^{*}(w),
 \end{aligned}
\end{equation}
 where  $C_{\Gamma}=2R^{2}_{*}$. Finally, using relations
 (\ref{e5.26})-(\ref{e5.28}), estimation
(\ref{e5.25}) becomes
\begin{equation} \label{e5.29}
 {\frac{d}{dt}}e^{*}(w) \leq  (C_{2}+C_{A}C_{B}+C_{\Gamma}C_{B})e^{*}(w)
 \leq C_{**}e^{*}(w),
 \end{equation}
 where $C_{**}=C_{2}+C_{A}C_{B}+C_{\Gamma}C_{B}$ and the proof
 is completed.
\end{proof}

\begin{remark}[Continuity in  $\mathcal{X}_{1}$] \label{rem.4.3} \rm
 It is important to state that the operator $S(t): \mathcal{X}_{0}
\to \mathcal{X}_{0}$ associated to the problem
\eqref{e2.4}-\eqref{e2.5} is weakly continuous in the space
$\mathcal{X}_{0}$, but it is strongly continuous in the space
$\mathcal{X}_{1}$. Therefore, we will study  problem
\eqref{e2.4}-\eqref{e2.5} as a dynamical system in the space
$\mathcal{X}_{1}:=\mathcal{D}^{1,2}(\mathbb{R}^N)
\times L^{2}_{g}(\mathbb{R}^N)$.
  \end{remark}

\begin{remark}[Uniqueness in  $\mathcal{X}_{1}$] \label{rem.4.4} \rm
Assuming that the initial data are from the space $\mathcal{X}_{1}$,
 relation (\ref{e5.29}) guarantees the  uniqueness of the  solutions for the
problem  \eqref{e2.4}-\eqref{e2.5}.  Indeed, if
$\widehat{u}_{a}=(u_{0},u_{1})$, $\widehat{u}_{b}=(u_{0}',u_{1}')$,
from  inequality (\ref{e5.29}) take
 \begin{equation}  \label{e5.30}
 \|S(t)\widehat{u}_{a}-S(t)\widehat{u}_{b}\|_{\mathcal{X}_{1}}
\leq  C(\|\widehat{u}_{a}\|_{\mathcal{X}_{1}},
 \|\widehat{u}_{b}\|_{\mathcal{X}_{1}}) \|\widehat{u}_{a}
 -\widehat{u}_{b}\|_{\mathcal{X}_{1}}.
 \end{equation}
 \end{remark}


 \begin{remark} \label{rem.5.5} \rm
According to Theorem \ref{them4.2} we have that
the ball $\mathcal{B}_{0}:=B_{\mathcal{X}_{0}}(0,\overline{R}_{*})$  is
an absorbing set in the space $\mathcal{X}_{0}$, so and in
$\mathcal{X}_{1}$  by the compact embedding.
\end{remark}

So, we obtain the following theorem.

 \begin{theorem}  \label{them.5.6}
The dynamical system given by the semigroup  $(S_{t})_{t \geq 0}$,
 possesses an invariant set, which attracts all
bounded sets of   $\mathcal{X}_{1}$, denoted by
 \[
 \mathcal{A}=\cap_{t\geq 0}
 \cup_{s\geq t}\overline{S_{s}{\mathcal{B}}_{0}} \subset \mathcal{X}_{1}.
 \]
The above set is also compact, so it is \emph{global attractor}
for the strong topology of  $\mathcal{X}_{1}$.
\end{theorem}

\begin{proof}
First, we have that operators  $(S_{t})_{t\geq 0}$  form a semigroup on
 $\mathcal{X}_{1}$  and  that  $S_{t}:\mathcal{X}_{1} \to
\mathcal{X}_{1}$  is continuous, for all  $t\in \mathbb{R}$ (Lemma \ref{lem.5.2}).
Also, we have that the ball  $\mathcal{B}_{0}$,  is an absorbing set in   $\mathcal{X}_{1}$  (Remark \ref{rem.5.5}). Our goal is to prove that the functional invariant
set $\mathcal{A}$ is compact for the strong topology of $\mathcal{X}_{1}$.
So, we  must show that for a point  $w_{1} \in \mathcal{A}$,  the sequence
$S(t_{j})u_{0}^{j}$  converges strongly to  $w_{1}$ in $\mathcal{X}_{1}$.
Here, we have that  $(u_{0}^{j})_{j \in N}$  and  $(t_{j})_{j \in N}$,  are two
sequences such that  $(u_{0}^{j})$  is bounded in  $\mathcal{X}_{1}$,
$t_{j}$  goes to  $+ \infty$, as  $j$  goes to  $+ \infty$
 and  $S(t_{j})u_{0}^{j}$  converges weakly to  $w_{1}$  in the space  $\mathcal{X}_{1}$,
as  $j$  goes to  $+ \infty$ (for more details we refer to \cite{Gil-2}
and \cite{hale88}). We fix  $T>0$  and note that the sequence
$S(t_{j}-T)u_{0}^{j}$  is bounded in  $\mathcal{X}_{1}$  thanks to
the existence of an absorbing set in  $\mathcal{X}_{1}$. Hence from this sequence
we may extract a
subsequence  $j_{1}$  such that, for some  $v_{1} \in \mathcal{X}_{1}$,
\begin{equation} \label{e4.41}
 S(t_{j_{1}}-T)u_{0}^{j_{1}} \rightharpoonup
 v_{1}, \quad \mbox{as } j_{1} \to  \infty.
\end{equation}
Introducing the  notation
\begin{equation} \label{e4.42}
u_{j_{1}}(t) := S(t_{j_{1}}+t-T)u_{0}^{j_{1}},
\end{equation}
 we deduce from (\ref{e4.41}) that
\begin{equation} \label{e4.43}
u_{j_{1}}(t)  \rightharpoonup
S(t)v_{1}, \quad  \mbox{as }  j_{1} \to \infty,
\end{equation}
since  $S(t)$  is weakly continuous on  $\mathcal{X}_{1}$.
Using the  energy type estimate (\ref{e4.b}) and the fact that the
sequence  $\theta (u_{j_{1}}(0))=\theta (S(t_{j_{1}}-T)u_{0}^{j_{1}})$
 is bounded by a constant, let say $C$, we  obtain
 \begin{equation}  \label{e4.45}
 \lim_{j_{1}\to \infty} \sup \theta(S(t_{j_{1}})u_{0}^{j_{1}}) \leq
 C e^{-C_{*}T}+{\frac{1-e^{-C_{*}T}}{C_{*}}}{\frac{C(R)}{\delta}}.
 \end{equation}
 Applying the invariance of the set  $\mathcal{A}$, for
$v_{1}(t)=S(t)v_{1}$, we get
 \begin{equation}  \label{e4.46}
 \theta (w_{1})=\theta (S(T)v_{1}) \leq e^{-C_{*}T} \theta
 (v_{1})+{\frac{1-e^{-C_{*}T}}{C_{*}}}{\frac{C(R)}{\delta}}.
 \end{equation}
 Subtracting by parts relations (\ref{e4.45}) and (\ref{e4.46}) we get
 \begin{equation}  \label{e4.47}
\lim_{j_{1}\to \infty} \sup
\theta(S(t_{j_{1}})u_{0}^{j_{1}}) \leq \theta
(w_{1})+e^{-C_{*}T}(C-\theta(v_{1})).
 \end{equation}
 Since  $T$  is chosen arbitrarily, for $T=0$ we have
\begin{equation}
 \label{e4.48}
\lim_{j_{1}\to \infty} \sup
\theta(S(t_{j_{1}})u_{0}^{j_{1}}) \leq \theta (w_{1}).
\end{equation}
On the other hand, since  $S(t_{j_{1}})u_{0}^{j_{1}}$  converges weakly
to  $w_{1}$ in $\mathcal{X}_{1}$,  we have that  $\lim
\inf_{j_{1}\to \infty} \theta (S(t_{0}^{j_{1}}) \geq
\theta (w_{1})$. So we get
\begin{equation} \label{e4.49}
 \lim_{j\to \infty} \theta (S(t_{j})u_{0}^{j})=\theta(w_{1}).
 \end{equation}
Using again the fact that  $S(t)\mathcal{A}=\mathcal{A}$  and  that
$\theta (t)$  is weakly continuous, we obtain
 \begin{equation} \label{e4.50}
\lim_{j \to \infty}\|S(t_{j})u_{0}^{j}\|^{2}_{\mathcal{X}_{1}}
=\|w_{1}\|^{2}_{\mathcal{X}_{1}}.
 \end{equation}
Therefore,  $S(t_{j})u_{0}^{j}$  converges strongly to  $w_{1}$
in the space $\mathcal{X}_{1}$  as  $j\to \infty$. Thus, we
obtain that  $\mathcal{A}$  is a global attractor in the strong
topology of  $\mathcal{X}_{1}$ (see also \cite{tem88}).
\end{proof}

\subsection*{Acknowledgments}
This work was  supported by the Pythagoras
project  68/831 from the EPEAEK program on Basic Research from
the Ministry of Education, Hellenic Republic (75\%  by European
Funds and 25\%  by National Funds).


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\end{document}