\magnification = \magstephalf
\hsize=14truecm
\hoffset=1truecm
\parskip=5pt
\nopagenumbers
\overfullrule=0pt
\input amssym.def  % The R for Real nunbers. 
\font\eightrm=cmr8 \font\eighti=cmti8 \font\eightbf=cmbx8
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1998/25\hfil Global attractor and finite 
 dimensionality\hfil\folio}
\def\leftheadline{\folio\hfil Astaburuaga, Bisognin, Bisognin \& Fernandez
\hfil EJDE--1998/25}
\voffset=\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1998}(1998) No.~25, pp. 1--14.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 35B40.
\hfil\break
{\eighti Key words and phrases:} Periodic solution, global attractor,
\hfil\break dimension of attractor.
\hfil\break
\copyright 1998 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted April 20, 1998. Published October 13, 1998.
} }

\bigskip\bigskip

\centerline{GLOBAL ATTRACTOR AND FINITE DIMENSIONALITY FOR}
\centerline{A CLASS OF DISSIPATIVE EQUATIONS OF BBM'S TYPE}
\medskip
\centerline{M.A. Astaburuaga, E.~Bisognin,}
\centerline{V.~Bisognin, \& C. Fernandez }
\bigskip\bigskip

{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
In this work we study the Cauchy problem for a class of nonlinear dissipative 
equations of Benjamin-Bona-Mahony's type. We discuss the existence of 
a global attractor and estimate its  Hausdorff and fractal dimensions.
\bigskip}
\def\esssup{\mathop{\rm ess~sup}}
\def\limsup{\mathop{\rm lim~sup}}
\def\frac#1#2{{{#1}\over{#2}}}

\bigbreak
\centerline{\bf \S 1. Introduction} \medskip\nobreak

We consider a family of dispersive equations of Benjamin-Bona-Mahony's type 
under the effect of dissipation, and we study the existence of a global 
attractor and its dimension.  Our model can be written in the abstract form
$$
M u_t + u_x + u u_x + \alpha L u = f\eqno(1.1)
$$
where $-\infty < x < \infty$, $t\ge 0$ and $\alpha\ge 0$.  The operators
$M$ and $L$ can be differential operators or pseudo-differential
operators, and the function $f$ is an external excitation.

In the simplest case, when $M$ and $L$ are the differential
operators $M = I - \frac{\partial^2}{\partial x^2}$, 
$L = -\frac{\partial^2}{\partial x^2}$,
the equation (1.1) is the well-known
Benjamin-Bona-Mahony model, which describes the unidirectional propagation
of weakly nonlinear dispersive long waves where Burger's type
dissipation is considered.

If $f\equiv 0$, the existence of global solutions and asymptotic
behaviour in time have been studied by several authors.  
The asymptotic behavior of solutions to the generalized 
Korteweg-de-Vries-Burgers and Benjamin-Bona-Mahony-Burgers equations 
in one space dimension was studied by Amick, Bona
and Schonbek in [3], by B.~Wang and W.~Yang in [13], and by Bona and Luo
in [7]. These results were generalized by Zhang [14] to multiple spatial
dimensions.  In [4], [5], [6] the authors considered a family of equations
of KdV and BBM's type described by pseudo-differential operators, and
studied the asymptotic behaviour in one space dimension.

In [8], Ghidaglia showed that the behaviour of the periodic solution
of the KdV equation is described by a global attractor that has finite
Hausdorff and fractal dimensions.  The author obtained similar results
for the Schr\"odinger equation in [9].

The aim of this work is to investigate the existence of a global attractor
and estimate its dimension, using the techniques of [8], [9] and [10]. 
More specifically, we consider the solutions of (1.1) that are periodic in
the spatial variable, that is, solutions $u(x,t)$ such that
$$
u (x + \beta, t) =  u (x, t)\eqno(1.2)
$$
where $\beta$ is a real number.  In the case that the external excitation
$f$ is independent of time, and the orders of pseudo-differential
operators $M$ and $L$ are $\mu$ and $s$ with $s\ge \mu\ge 2$, we show
that the behaviour, for large $t$, of the infinite dimensional dynamical
system (1.1) is, in fact, described by an attractor of finite dimension.

In [11] B.~Wang proved the existence of a weak attractor which is also
strong, working directly with the BBM equation in $H^2$.  In [12], the
same author proved the existence of a global attractor for the generalized
Benjamin-Bona-Mahony equation in $H^k$ for every integer $k\ge 2$.  He
also proved that the attractor has finite Hausdorff and fractal dimensions,
and constructed approximate inertial manifolds. Here, we consider a family
of dispersive equations of BBM's type, and we present a proof that applies
in an abstract context which includes the BBM equation.  We also prove
that the Hausdorff dimension is finite.

We shall use standard notation.  By $L^q (\Omega)$ we shall denote the space
of functions in $\Omega$ whose $q^{th}$ power is integrable, with
the norm $\|g\|^q_{L^q} = \int_\Omega  |f (x)|^q dx$, $1\le q < +\infty$.  The
norm in $L^2 (\Omega)$ we will denote by $\|\cdot\|_{L^2} = \|\cdot\|$.  By
$L^\infty (\Omega)$ we denote the space of measurable essentially bounded
functions in $\Omega$ with the norm
$$
\|g\|_{L^\infty} = \esssup_{x\in \Omega} \; |g (x)|\,.
$$

For each $\sigma\in {\Bbb R}$ we shall denote by $H^\sigma (\Omega)$ the usual
Sobolev space of order $\sigma$.  By $H^\sigma_p (\Omega)$, $\sigma \ge 0$,
$\Omega = (0, 1)$ we shall indicate the space of functions periodic in the
sense of (1.2), with $\beta=1$.  If $g\in H^\sigma_p (\Omega)$ then $g$
has an expansion in Fourier series
$$
g (x) = \sum_{k\in\,{\Bbb Z}} g_k \, \exp (2k i \pi x)\,.\eqno(1.3)
$$

The norm of $g$ in $H^\sigma_p (\Omega)$ will be denoted by
$$
\|g\|^2_\sigma = \sum_{k\in\,{\Bbb Z}} (1 + |k|^2)^\sigma |g_k|^2,\eqno(1.4)
$$
which is equivalent to $H^\sigma (\Omega)$ norm, $\sigma \ge 0$, according
to Temam ([10]).
We shall denote by $\dot{L}^2 (\Omega)$ and $\dot{H}^\sigma (\Omega)$ the space of
functions $g\in L^2 (\Omega)$ or $H^\sigma (\Omega)$ such that
$$
\int_\Omega  g (x) \, dx = 0.\eqno(1.5)
$$
The space $\dot{H}^\sigma_p (\Omega)$, $\sigma \in {\Bbb R}_{+}$, is the
space of functions $g\in L^2 (\Omega)$ such that $g$ satisfies (1.5) and
$$
\sum_{k\in\, {\Bbb Z}} (1 + |k|^2)^\sigma |g_k|^2 < +\infty\,.\eqno(1.6)
$$

In $\dot{H}^1_p (\Omega)$ the Poincar\'e inequality holds, that is, if
$g\in \dot{H}^1_p (\Omega)$ then
$$
\|g\| \le C (\Omega)\, \|g'\|\,.\eqno(1.7)
$$
The inequality (1.7) shows that $\dot{H}^1_p (\Omega)$ is a Hilbert space
with scalar product of $H^1_0(\Omega)$, and $\|u\|_1 = \{ (u, u)_1\}^{1/2}$
is a norm on this space equivalent to that induced by $H^1 (\Omega)$.

The operators $M$ and $L$ of (1.1) are pseudo-differential operators of
orders $\mu$ and $s$, respectively, with
$$
\eqalign{
&M:\; \dot{H}^\mu_p (\Omega) \; \to \; \dot{L}^2_p (\Omega),\qquad \mu\ge 1,
\quad \mu\in {\Bbb R}\cr
&L:\; \dot{H}^s_p (\Omega) \; \to \; \dot{L}^2_p (\Omega),\qquad s\ge 0,
\quad s\in {\Bbb R}\cr}
$$
and 
$$
\eqalignno{
M g (x) &= \sum_{k\in\, {\Bbb Z}} m (k) \, g_k \, \exp (2 k \pi i
x)&(1.8)\cr
L g (x) &= \sum_{k\in\, {\Bbb Z}} \ell (k) \, g_k \, \exp (2 k \pi i
x)&(1.9)\cr}
$$
where $m$ and $\ell$ are the principal symbols of the operators $M$ and $L$
respectively.  We assume from now on that the symbols $m$ and $\ell$ are even
functions of $k$ that satisfy the growth conditions:

\item{(~i)} There exist constants $c_1$, $c_2 > 0$ such that
$$
c_1 (1 + |k|)^\mu \le m (k) \le c_2 (1 + |k|)^\mu.\eqno(1.10)
$$

\item{(ii)} There exist constants $c_3$, $c_4 > 0$ such that
$$
c_3 |k|^s \le \ell (k) \le c_4 |k|^s.\eqno(1.11)
$$
The domains of operators $M$ and $L$ are given by
$$
\eqalign{
D (M) &= \left \{ g \in \dot{H}^\mu_p (\Omega), \; \sum_{k\in\, {\Bbb Z}} |m
(k)|^2 |g_k|^2 < \infty \right \}\cr
D (L) &= \left \{ g \in \dot{H}^s_p (\Omega), \; \sum_{k\in\, {\Bbb Z}} |\ell
(k)|^2 |g_k|^2 < \infty \right \}.\cr}
$$

If $X$ is a Banach space then we denote by $C (0, T;\; X)$ the space of
continuous functions $u:\; [0, T] \to X$.  Various positive constants
will be denoted by $C$; they may vary from line to line.

This paper is organized as follows.  In Section 2 we study the existence
and uniqueness of global solutions of the Cauchy problem associated to
equation (1.1).  Then in Section 3 we provide a priori bounds for the
nonlinear semigroups given by the evolution equation, and we use them
to establish the existence of a global attractor.  Finally, we show in
Section 4 that this set has finite Hausdorff dimension.
\bigbreak

\centerline{\bf \S 2. The Cauchy Problem} 
 \medskip\nobreak
In this section we consider Problem (1.1) with initial data $u (x,
0) = u_0 (x)$. We prove that the Cauchy problem is globally
well-posed in the Sobolev space $\dot{H}^r_p (\Omega)$ where $\Omega = (0,
1)$ and $r = \max \{ \mu, s \}$.

The lemma below is useful in proving the existence of a solution.

\proclaim Lemma 1.2.
Let $M:\; \dot{H}^\mu_p (\Omega) \to \dot{L}^2_p (\Omega)$, $\mu \ge 1$
satisfy assumptions (1.10) above. Then
\item{a)} 
$$ M^{-1} \; \hbox{ exists.} \eqno(2.1)$$
\item{b)} $M^{-1} \left ( \frac{dg}{dx} \right ) \in \dot{H}^\mu_p (\Omega)$
whenever $g\in \dot{H}^\mu_p (\Omega)$ and there exists a constant $C>0$
such that
$$
\| M^{-1} \,\frac{dg}{dx} \|_\mu \le C \, \|g\|_\mu.\eqno(2.2)
$$

\noindent The proof of this lemma follows directly from the
definition and (1.10).

\proclaim{Theorem 2.1 (Local Existence)}.  
Let $u_0 \in \dot{H}^r_p (\Omega)$ with $r = \max
\{ \mu, s \}$, $\mu \ge 2$, $s\ge 0$.  Assume $f\in \dot{H}^r_p (\Omega)$, and
suppose
$M$ and $L$ satisfy the assumptions (1.10), (1.11).  Then, for each
$T > 0$, there exists a unique function $u\in C (0, T;\;
\dot{H}_p^{\mu/2} (\Omega))$, with $u$ and $u_t$ in the class
 $C (0, T;\;\dot{H}^r_p (\Omega))$, that solves
(1.1) in $\Omega \times [0, T]$ with $u (x, 0) = u_0(x)$.  The mapping
that associates to $u_0\in \dot{H}^r_p (\Omega)$ the solution of (1.1) is
continuous from $\dot{H}^r_p (\Omega)$ to $C (0, T;\; \dot{H}^r_p (\Omega))$.

\noindent{\bf Proof }
First, we consider the linear problem
$$\displaylines{ \hfill
 w_t + \alpha M^{-1} L w = 0 \hfill\llap{(2.3)} \cr
 w (x, 0) = u_0 (x) \in \dot{H}^r_p (\Omega)\cr}
$$  
that has a unique solution $w$ given by $w (x, t) = E (t) u_0 (x)$,
where $\{ E (t) \}_{t\ge 0}$ is the strongly continuous semigroup of
linear operators generated by $A = -\alpha M^{-1} L$, $\alpha> 0$.  The
solution $w$ lies in the class $C (0, \infty;\; \dot{H}^r_p (\Omega)$ and
$w_t$ lies in $C (0, \infty;\; \dot{H}^r_p (\Omega)$.

Next, we consider the nonlinear problem (1.1).  Using Lemma 1.2 and the
observations concerning the solution of (2.3), we can write the integral
equation associated with (1.1):
$$
u (x, t) = E (t) u_0(x) - \int^t_0 E (t - \sigma) M^{-1} \, \frac{\partial}{\partial x}
\, \left ( u + \frac{u^2}{2} \right ) \, d\sigma + \int^t_0 E (t - \sigma)
M^{-1} f\, d\sigma.\eqno(2.4)
$$

Let $R>0$, $T>0$, and define the space of functions
$$
y_R (T) = \left\{ \sup_{[0,T]} w\in C (0, T;\,\dot{H}^r_p (\Omega)  : 
\sup_{[0, T]} \|w (\cdot, t) - E (t) u_0(\cdot)\|_r
\le R, \, w (x, 0) = u_0(x)\right\}.
$$
We define the map $P:\; y_R (T) \to C (0, T;\; \dot{H}^r_p(\Omega)$ by
$$
Pw (x,t) = E (t) u_0(x) - \int^t_0 E (t - \sigma) M^{-1} \, \frac{\partial}{\partial x}
\, \left ( u + \frac{u^2}{2} \right ) \, d\sigma + \int^t_0 E (t-\sigma)
M^{-1} f\, d\sigma\eqno(2.5)
$$
for all $0\le t \le T$.  Using well-known techniques we can easily prove
that $P$ is a contraction as long as  $T = T_0$ is chosen sufficiently
small.  Thus, $P$ has a fixed point, which gives  us a local solution of
the integral equation (2.4).  Next, since $u$ satisfies equation (2.4),
we can calculate $u_t$ explicitly and obtain that $u$ satisfies (1.1)
with $u (x, 0) = u_0(x)$, and that $u_t \in C (0,T_0;\; \dot{H}^r_p(\Omega))$.

Multiplying equation (1.1) by $u$, integrating in space, and using
Poincar\'e and H\"older's inequalities and (1.10)--(1.11), we obtain
the estimate
$$
\|u (\cdot, t)\|^2_{\mu/2} + \frac{\alpha}{c_1} \, \int^t_0 \,
\|L^{1/2} u (\cdot, \sigma)\|^2 \, d\sigma \le C \,
\|f\|^2 \, T + \|u_0\|^2_{\mu/2}\eqno(2.6)
$$
for all $0\le t\le T$ and $\mu\ge 2$.
Therefore, $u\in C (0, T;\; \dot{H}^{\mu/2}_p (\Omega)$, $\mu\ge 2$.

Multiplying equation (1.1) by $Mu$, integrating in space, using Poincar\'e
and H\"older's inequalities and properties (1.10)--(1.11), we have
$$
\|u (\cdot, t)\|^2_\mu + C_0 \int^t_0 \|L^{1/2} M^{1/2} u\|^2 \, d\sigma \le
C_1 \,\|u_0\|^2_\mu + \|f\|^2 T + C_2 \int^t_0 \|u (\cdot, \sigma)\|^2_\mu
d\sigma.
$$
From Gronwall's inequality
$$
\|u (\cdot, t)\|^2_\mu \le C \, \left ( \|u_0\|_\mu, \|f\|, T \right )
\; e^{C_2 T}.\eqno(2.7)
$$
Therefore, if $s\le \mu$ then $r=\mu$ and consequently $u\in C (0, T;\;
\dot{H}^r_p (\Omega)$.

Now, suppose that $s > \mu \ge 2$.  Multiplying equation (1.1) by $u_t$,
integrating in space and using the results above, we obtain
$$
\frac12 \, \|M^{1/2} u_t\|^2 + \frac{\alpha}{2} \; \frac{d}{dt} \, \|L^{1/2} u\|^2
\le C \left ( \|f\|, T \right ) + \|L^{1/2} u\|^2.
$$
Gronwall's inequality and (2.6) imply that $u\in C (0, T;\;
\dot{H}_p^{\frac{s}{2}} (\Omega)$ for $s > \mu \ge 2$ and all $0 \le t \le T$.

Finally, multiplying equation (1.1) by $Lu_t$ and using the same
sequence of ideas, we obtain
$$
\frac12 \, \|L^{1/2} M^{1/2} u_t\|^2 + \frac{\alpha}{2} \; \frac{d}{dt} \, \|Lu\|^2
\le C \left ( \|f\|, + \|u_0\|_r, T \right ) + C_1 \|Lu\|^2.
$$
From Gronwall's inequality and (2.6) we get
$$
u\in C (0, T;\; \dot{H}^s_p (\Omega) \; \hbox{ for } \; s > \mu \ge 2
\; \hbox{ and all } \; 0 \le t \le T.
$$

Since we know that $u\in C (0, T;\; \dot{H}^r_p (\Omega)$, we can use the
integral equation (2.4) to find $u_t$, and it follows from these that
$u_t\in C (0, T;\; \dot{H}^r_p (\Omega)$.  Uniqueness is a direct consequence
of Gronwall's inequality.
\bigbreak

\centerline{\bf \S 3. Existence of a global attractor} 
  \medskip\nobreak
In this section we study the existence of a global attractor.  The
first step is to prove the existence of an absorbing set in $\dot{H}^s_p
(\Omega)$, $s\ge \mu \ge 2$.

We consider the Cauchy problem
$$ \displaylines{
 Mu_t + u_x + uu_x + \alpha Lu = f \cr
\hfill u (x, 0) = u_0(x) \hfill\llap{(3.1)}\cr
 u (x + 1,\, t) = u (x, t).\cr}
$$
If the function $f$ is time independent, the system (3.1) is autonomous,
and for each $t\in {\Bbb R}^+$ we define the mapping
$$
\eqalignno{
E (t):\; &\dot{H}^r_p (\Omega) \; \to \; \dot{H}^r_p (\Omega)\cr
&u_0\; \mapsto \; E (t) u_0= u (x,t).&(3.2)\cr}
$$
The family $\{ E (t) \}_{t\in{\Bbb R}^{+}}$ forms a semigroup.

\proclaim Proposition 3.1.
If $E (t)$ is the mapping defined in (3.2), then there exists a constant
$C = C (\|u_0\|_r, \|f\|_r, T)$ such that
$$
\sup_{0\le t\le T} \|E (t)\, u_0\|_r \le C \left ( \|u_0\|_r, \|f\|_r, T
\right )
$$
with $u_0\in \dot{H}^r_p (\Omega)$, $f\in \dot{H}^r_p(\Omega)$, $r = \max
\{ \mu, s \}$, $\mu \ge 2$, $s\ge 2$.


\noindent The proof of this proposition follows directly from Theorem 2.1.


The next result is related to the existence of bounded absorbing set
for semigroup $\{ E (t) \}_{t\ge 0}$ in $\dot{H}^s_p (\Omega)$, $s\ge \mu\ge 2$.

\proclaim Proposition 3.2.
Let $f\in \dot{H}^s_p (\Omega)$, $s\ge \mu\ge 2$.  There exists a constant
$\rho_0 = \rho_0 (\|f\|_0)$ such that for every $R>0$ there exists
$T>0$, $T = T (R)$ such that
$$
\|E (t) u_0\|_s \le \rho_0 \; \hbox{ for all } \; u_0\in \dot{H}^s_{per}
(\Omega) \; \hbox{ with } \; \|u_0\|_s \le R
$$
and $t\ge T (R)$, where $E (t) u_0(x) = u (x, t)$ is the solution
of Cauchy problem (3.1).

\noindent{\bf Proof }
Multiplying the equation in (3.1) by $u$ and integrating in $(\Omega)$,
we have
$$
\frac{d}{dt} \; \|M^{1/2} u\|^2 + \alpha\|L^{1/2} u\|^2 \le C\, \|f\|^2.\eqno(3.3)
$$
Poincar\'e's inequality and the fact that $s\ge \mu$ imply that
$\|M^{1/2} u\| \le C \|L^{1/2} u\|$.
Therefore, there exists a constant $\beta>0$ such that
$$
\frac{d}{dt} \; \|M^{1/2} u\|^2 + \beta \, \|M^{1/2} u\|^2 \le 
C \, \|f\|^2.\eqno(3.4)
$$
From (3.4) and properties (1.10)--(1.11) we obtain
$$
\|u (t)\|^2_{\mu/2} \le C_0 \, \|u_0\|^2_{\mu/2} e^{-\beta t} + C\,
\|f\|^2 \left ( 1 - e^{-\beta t} \right ) \le C_0 R^2 e^{-\beta t} +
C \|f\|^2 (1 - e^{-\beta t}).\eqno(3.5)
$$
This shows that $E(t) u_0$ is uniformly bounded in ${\dot H}^{\mu/2}_p 
(\Omega)$ and
$$
\| u (t)\|^2_{\mu/2} \le C \|f\|^2 = \rho^2_1\eqno(3.6)
$$
for all $t \ge T_0 (R) = \frac{1}{\beta} \, \ln \, \frac{C_0 R^2}{C\|f\|^2}$.

Multiplying the equation (3.1) by $Mu$ and integrating in $\Omega$, we have,
for all $t \ge T_0 (R)$,
$$
\frac{d}{dt} \, \|Mu\|^2 + 2\alpha\|L^{1/2} Mu^{1/2} u\|^2 = 2\int_\Omega  f \, Mu\, dx - 2 \int_\Omega 
uu_x Mu \,dx.\eqno(3.7)
$$
Using H\"older's inequality and the embedding ${\dot H}_p^{\mu/2} (\Omega)
\hookrightarrow L^\infty (\Omega)$, $\mu\ge 2$, we deduce from (3.7) that
$$
\frac{d}{dt} \, \|Mu\|^2 + 2\alpha\|L^{1/2} Mu^{1/2} u\|^2 \le 2 \|f\|\, \|Mu\| + 2
\|u\|_{L^\infty} \|u_x\|\, \|Mu\|.\eqno(3.8)
$$
Therefore, for $\mu\ge 2$ we have
$$
\eqalignno{
\frac{d}{dt} \, \|Mu\|^2 + 2\alpha\|L^{1/2} Mu^{1/2} u\|^2 &\le 2 \|f\|\, \|Mu\| + C \|u\|^2_{\mu/2}  \|Mu\|\cr
&\le \left ( 2 \|f\| + C \|u\|^2_{\mu/2} \right ) \, \|Mu\|.&(3.9)\cr}
$$
Poincar\'e's inequality implies that
$$
\|Mu\| \le C \|L^{1/2} M^{1/2} u\| \; \hbox{ for } \; u\in {\dot H}^s_p (\Omega),\qquad
s\ge \mu \ge 2.\eqno(3.10)
$$
From (3.10) we have for (3.9)
$$
\frac{d}{dt} \|Mu\|^2 + 2\alpha\|L^{1/2} M^{1/2} u\|^2 \le C \left ( \|f\| + C \|u\|^2_{\mu/2}
\right ) \|L^{1/2} M^{1/2} u\|.
$$
Using the inequality $ab\le \frac{a^2}{2} + \frac{b^2}{2}$, we have
$$
\frac{d}{dt} \|Mu\|^2 + \alpha\|L^{1/2} M^{1/2} u\|^2 \le C \left ( \|f\| + C \|u\|^2_{\mu/2}
\right )^2.
$$
From (3.10) and (3.6) it follows that
$$
\frac{d}{dt} \|Mu\|^2 + \beta_1 \|Mu\|^2 \le C \left ( \|f\| + \rho^2_1 \right )^2,\qquad
\beta_1 > 0,\eqno(3.11)
$$
and for all $t\ge T_0$ we obtain
$$
\|Mu\|^2 \le \|Mu (T_0)\|^2 e^{-\beta_1 (t - T_0)} + C \left ( \|f\| + \rho_1^2
\right )^2 \left ( 1 - e^{-\beta_1 (t - T_0)} \right ).\eqno(3.12)
$$

We choose $T_1 = T_1 (R) \ge T_0 (R)$ such that
$$
\|Mu (T_0)\|^2 e^{-\beta_1 (t - T_0)} \le C (\|f\| + \rho^2_1)^2\eqno(3.13)
$$
holds for every $u_0$ satisfying $\|u_0\|_s \le R$, $s \ge \mu \ge 2$.  This is possible
since we know from (2.7) that $\|u (T_0)\|_\mu$ is bounded by a quantity that only depends
on $R$ and the data of the problem.
Then, according to (3.12) and (3.13),
$$
\|Mu (t)\|^2 \le C \left ( \|f\| + \rho^2_1 \right )^2 = \rho^2_2 \; \hbox{ for all } \;
t\ge T_1 (R).\eqno(3.14)
$$
If $s = \mu \ge 2$ the proof is concluded.

We now consider the case $s > \mu \ge 2$.
Taking into account Lemma~1.2, we can write the equation (3.1) as
$$
u_t + M^{-1} u_x + M^{-1} (uu_x) + \alpha M^{-1} Lu = M^{-1} f.\eqno(3.15)
$$
Multiplying (3.15) by $Lu$ and integrating in space, we have
$$
\frac{d}{dt} \|L^{1/2} u\|^2 + 2\alpha\|M^{1/2} Lu\|^2 
= 2 \left [ \left ( M^{-1} f, Lu \right ) -
\left ( M^{-1} \frac{\partial}{\partial x} \left ( \frac{u^2}{2} \right ), 
Lu \right )
\right ].\eqno(3.16)
$$
Integrating by parts and using the Cauchy-Schwarz inequality yields
$$
\frac{d}{dt} \|L^{1/2} u\|^2 + 2\alpha\|M^{-\frac12} Lu\|^2 \le 2 \left [ \|M^{-\frac12} f\|\, \|M^{-\frac12} Lu\|
+ \| M^{-\frac12} \frac{\partial}{\partial x} \left ( \frac{u^2}{2} \right )\|\, \|M^{-\frac12} Lu\|
\right ].
$$
Using the inequality $ab\le \frac{a^2}{2} + \frac{b^2}{2}$, we have
$$
\frac{d}{dt} \|L^{1/2} u\|^2 + \alpha\|M^{-\frac12} Lu\|^2 \le C \left ( \|M^{-\frac12} f\|^2 + 
\| M^{-\frac12} \frac{\partial}{\partial x} \left ( \frac{u^2}{2} \right )\|^2 \right ).
\eqno(3.17) 
$$
From (1.10) we obtain
$$
\|M^{-\frac12} f\| \le C \|f\|.\eqno(3.18)
$$
For $\mu\ge 2$ and the embedding ${\dot H}^\mu_p (\Omega) \hookrightarrow L^\infty (\Omega)$, 
it follows that
$$
\| M^{-\frac12} \frac{\partial}{\partial x} \left ( \frac{u^2}{2} \right )\|^2 \le \|u^2\|^2 \le C
\|u\|^2_{L^\infty} \|u\|^2 \le C \|u\|^4_\mu.\eqno(3.19)
$$
From (3.19), (3.18), and (3.14), we have for (3.17)
$$
\frac{d}{dt} \|L^{1/2} u\|^2 + \alpha\|M^{-\frac12} Lu\|^2 \le C \left ( \|f\|^2 + \rho^4_2  \right )
\hbox{ for all } \; t\ge T_1.\eqno(3.20)
$$
On the other hand, Poincar\'e's inequality implies
$$
\|M^{-\frac12} Lu\|^2 \ge C \|L^{\frac12} u\|\; \hbox{ for } \; s > \mu \ge 2.
\eqno(3.21) 
$$
Therefore, from (3.21) we have for (3.20)
$$
\frac{d}{dt} \|L^{1/2} u\|^2 + \beta_2 \|L^{1/2} u\|^2 \le C \left ( \|f\|^2 + \rho^4_2  \right )
\eqno(3.22)
$$
for all $t \ge T_1$.
Integrating (3.22) in time for $t\ge T_1$ we obtain
$$
|L^{1/2} u (t)\|^2 \le \|L^{\frac12} u (T_1)\|^2  e^{-\beta_2 (t - T_1)} + C 
\left ( \|f\|^2 + \rho^4_2  \right )\, \left ( 1 - e^{-\beta_2 (t - T_1)}
\right ).\eqno(3.23)
$$
As  in (3.13), we choose $T_2 = T_2 (R) \ge T_1 (R)$
such that $$
\|L^{1/2} u (T_1)\|^2 e^{-\beta_2 (t - T_1)} \le C \left ( \|f\|^2 + \rho^4_2 
\right ).
\eqno(3.24)
$$
Therefore, from (3.24) we have
$$
\|L^{1/2} u (t)\|^2 \le C \left ( \|f\|^2 + \rho^4_2  \right ) = \rho^2_3 \; \hbox{ for all }
\; t\ge T_2.\eqno(3.25)
$$

Next, we consider the equation
$$
Lu_t + L M^{-1} u_x + LM^{-1} (uu_x) + \alpha LM^{-1} Lu = LM^{-1} f.\eqno(3.26)
$$
Multiplying (3.26) by $Lu$ and integrating in space we obtain, after
integration by parts and the use of H\"older's inequality,
$$
\frac{d}{dt} \|Lu\|^2 + \alpha\|M^{-\frac12} L^{\frac{3}{2}} u\|^2 \le C \left ( \|M^{-\frac12} L^{1/2} f\|^2 
+ \| L^{1/2} M^{-\frac12} \frac{\partial}{\partial x} \left ( \frac{u^2}{2} \right )\|^2
\right ).\eqno(3.27)
$$
From (1.10) and (1.11) we have
$$
\|M^{-1/2} L^{1/2} f\| \le C \|f\|_{\frac{s}{2}}\eqno(3.28)
$$
and
$$
\| M^{-1/2} L^{1/2} \frac{\partial}{\partial x} \left ( \frac{u^2}{2} \right )\| \le C
\|u^2\|_{s/2}\le C \|u\|^2_{s/2} \eqno(3.29)
$$
because ${\dot H}^s_p (\Omega)$ is an algebra for $s\ge 1$.

Using (3.28), (3.29), and Poincar\'e's inequality, we obtain for (3.27)
$$
\frac{d}{dt} \|Lu\|^2 + \beta_3 \|Lu\|^2 \le C \left ( \|f\|^2_{s/2} 
+ \|u\|^2_{s/2} 
\right ),\qquad \forall t\ge T_2.\eqno(3.30)
$$
From (3.25) we have
$$
\frac{d}{dt} \|Lu\|^2 + \beta_3 \|Lu\|^2 \le C \left ( \|f\|^2_{\frac{s}{2}} + \rho^2_3 
\right ),\qquad \forall t\ge T_2
$$
and
$$
\|Lu\|^2 \le \|Lu (T_2)\|^2 e^{-\beta_3 (t - T_2)} + C \left ( \|f\|^2_{s/2} + 
\rho^2_3 \right )\, \left ( 1 - e^{-\beta_3 (t - T_2)} \right )\eqno(3.31)
$$
for all $t\ge T_2$.
Choosing $T = T (R) \ge T_2 (R)$, we have
$$
\|Lu\|^2 \le C \left ( \|f\|^2_{s/2} + \rho^2_3 \right ) = \rho^2_4 \; \hbox{ for all }
\; t\ge T.
$$

On the other hand, we know that $\|u\| \le \rho_1$; therefore,
$$
\|u\|^2 + \|Lu\|^2 \le \rho_1^2 + \rho^2_4.
$$
Using (1.11) we deduce that
$$
\|u\|^2_s \le C \, (\rho^2_1 + \rho^2_4) = \rho^2_0 \; \hbox{ for all }
\; t \ge T (R).
$$
This completes the proof of Proposition 3.2. \medskip


Proposition 3.2 shows that $E (t)\, u_0$ is uniformly bounded in $\dot{H}^s_p
(\Omega)$, for $s\ge \mu \ge 2$ and every $t \ge T (R)$, when $\|u_0\|_s \le R$.
In other words, every solution with initial data $u_0$ in the ball
$\{ \|u_0\|_s \le R \}$ is absorbed at time $t \ge T (R)$, by the ball
$$
B_0 = \left \{ v \in \dot{H}^s_p (\Omega), \qquad  \|v\|_s \le\rho_0 \right \}.
$$
It is natural to consider then the $w$-limit set of $B_0$, which is
defined as
$$
w (B_0) = \bigcap_{\ell \ge 0} \; \overline{\cup_{t\ge \ell} E (t) B_0},
$$
where the closure is taken in $\dot{H}^s_p (\Omega)$.

In order to obtain the existence of a global attractor for the equation
(1.1), we next prove that the flow $E (t)$ is
uniformly compact, for $t$ large.

\proclaim Proposition 3.3.
Let $f\in \dot{H}^s_p (\Omega)$, $s\ge \mu\ge 2$.  For every bounded set $B$ of
$\dot{H}^s_p (\Omega)$ there exists $T>0$, $T = T (B)$, such that $\cup_{t\ge T}
E (t) B$ is relatively compact in $\dot{H}^s_p (\Omega)$.

\noindent{\bf Proof }
The idea is to prove that $E (t) = E_1 (t) + E_2 (t)$, where
the operator $E_1 (\cdot)$ is uniformly compact for $t$ large, and the
norm of $E_2 (\cdot)$ as a bounded operator goes to zero as $t\to\infty$.

Decompose the solution $u$ of (1.1) as $u = v + w$, where $v (x, t) =
E_2 (t) v (x, 0)$ is the solution of the linear problem
$$\displaylines{ \hfill
 M v_t + v_x + \alpha L v = 0 \hfill\llap{(3.32)}\cr
v (x, 0) = u_0(x),\cr}
$$
and $w = w (x, t)$ is the solution to
$$\displaylines{ \hfill
 M w_t + w_x + \alpha L w = f - u u_x \hfill\llap{(3.33)}\cr
w (x, 0) = 0.\cr}
$$
In order to prove that $E_2 (t)$ has decaying norm we consider the equation
$$
L v_t + L M^{-1} v_x + \alpha L M^{-1} Lv = 0.\eqno(3.34)
$$
Multiplying by $Lv$, integrating in $\Omega$, and using the properties
of operators $M$ and $L$, we obtain
$$
\|E_2 (t)\|_{{\cal L} (\dot{H}^s_p (\Omega), \dot{H}^s_p (\Omega))} \le C_0\,
e^{-Ct},\qquad \forall t \ge 0.\eqno(3.35)
$$

Now consider the equation
$$
LM w_t + L w_x + \alpha LLw = Lf - L (uu_x).
$$
Multiplying by $Lw$ and using the properties of operators $M$ and $L$
and the fact that $\|u (t)\|_s \le \rho_0$, we have
$$
\frac{d}{dt} \, \|L M^{1/2} w\|^2 + \beta_5 \|L M^{1/2} w\|^2 \le C \left (
\|f\|_s, \rho_0 \right ),\qquad \beta_5 > 0.\eqno(3.36)
$$
From (3.36) we can conclude that $w$ is uniformly bounded in
$\dot{H}_p^{\frac{\mu}{2} + s} (\Omega)$.
Using the compact embedding from $\dot{H}_p^{\frac{\mu}{2} + s} (\Omega)$ into
$\dot{H}^s_p (\Omega)$ it follows that $\cup_{t\ge T} E_1 (t) \, B$ is
relatively compact in $\dot{H}^s_p (\Omega)$.  The proposition follows
as in the proof of Theorem~1.1, Chapter~1 of Temam [10]. 

\proclaim Theorem 3.2.
Let $f$, $u_0\in \dot{H}^s_p(\Omega)$, $s\ge \mu \ge 2$.  Then the
semigroup $\{ E (t) \}_{t\ge 0}$ has a global attractor ${\cal A} = w (B)$
in $\dot{H}^s_p (\Omega)$.  The set ${\cal A}$ is compact in $\dot{H}^s_p (\Omega)$
and has the properties:

\item{~i)} The set ${\cal A}$ is invariant under $E (t)$, that is, $E (t) \,
{\cal A} = {\cal A}$, $\forall t\ge 0$.

\item{ii)} For every bounded set $B$ in $\dot{H}^s_p (\Omega)$, $d (E (t) B,
{\cal A}) \to 0$ as $t \to +\infty$.


The proof this theorem is a consequence of Temam [10], Theorem~1.1, 
Chapter~I. 
\bigbreak
\centerline{\bf \S 4. Dimension of the global attractor} 
\medskip\nobreak
Our aim in this section is to study the finite dimensionality of the
global attractor.  In the first part we shall prove the differentiability
property of $E (t)$ and in the second part we will establish the finite
dimension of the attractor.

We consider the following non-autonomous evolution equation, which
corresponds to a linearized version of the equation (1.1):
$$\displaylines{
Mv_t + v_x + (uv)_x + \alpha Lv = 0\cr
\hfill v (x, 0) = v_0 (x) \hfill\llap{(4.1)}\cr
v (x + 1, t) = v (x, t)\cr}
$$
where $u (t) = E (t) u_0$, $u_0\in \dot{H}^s_p (\Omega)$ is a trajectory
solution of (1.1), and $v_0 \in \dot{H}^s_p (\Omega)$.  It is not difficult
to prove that, since $u\in C^1 ([0, \infty); \; \dot{H}^s_p (\Omega))$, the
problem (4.1) has a unique solution $v\in C^1 ([0, \infty);\;
\dot{H}^s_p (\Omega))$.

Next we show, with the aid of the linearized problem (4.1),
that the linear mapping $(DE (t) u_0) \, v_0  \equiv 
v (t)$ is the uniform differential of $E (t)$.

\proclaim Theorem 4.1.
For every $0 < R$, $T < \infty$, there exists a positive constant 
$C = C(R, T)$ such that for all $u_0$, $v_0 \in \dot{H}^s_p (\Omega)$, 
$s \ge \mu \ge 2$, that satisfy $\|u_0\|_s \le R$, $\|u_0+ v_0\|_s \le R$
and $0\le t \le T$, we have
$$
\|E (t) (u_0+ v_0) - E (t) u_0- (DE (t) u_0)\, v_0\|_s \le C
\|v_0\|^2_s.
$$

\noindent{\bf Proof }
Let $u_0$, $v_0\in \dot{H}^s_p(\Omega)$, $s\ge \mu\ge 2$, with 
$\|u_0\|_s \le R$, $\|u_0+ v_0\|_s \le R$.  We consider the
solutions $u_1 (t) = E (t) u_0$, $u_2 (t) = E (t) (u_0+ v_0)$ and $v (t) =
(DE (t) u_0) \,v_0$.
Then $w = u_2 - u_1 - v$ satisfies the problem
$$\displaylines{\hfill
Mw_t + w_x + u_2 u_{1x} - (u_1 v)_x + \alpha Lw = 0 \hfill\llap{(4.2)}\cr
w (0) = 0.\cr}
$$
Since $u_1$, $u_2$, $v\in C^1 ([0, \infty); \; \dot{H}^s_p (\Omega))$,
we may use the sequence of ideas in the proof of Theorem~2.1 to obtain
$$
\|w\|^2_s \le C (R, T) \|v_0\|^2
$$
for the solution $w$ of the problem (4.2).
Therefore, $E (t)$ is uniformly differentiable in the bounded sets
of $\dot{H}^s_p (\Omega)$. \hfill$\diamondsuit$
\medskip

Now we study how the operators $D (E (t)) \, u_0$ transform
the $m$-dimensional volumes in $\dot{H}^s_p (\Omega)$, $s\ge \mu \ge 2$
where $u_0\in {\cal A}$.
Let $v^1_0$, $v^2_0, \cdots, v^m_0$ in $\dot{H}^s_p (\Omega)$. We study
the evolution of the quantities
$$
\|v^1 (t) \wedge \cdots \wedge v^m (t)\|^2_s = \det_{1\le i, j\le m} \;
(v^i (t), v^j (t))_s\eqno(4.3)
$$
where $v^i (t) = (D E (t) u_0)\, v^i_0$.

The expression (4.3) is the Gram determinant, and it represents the
square of $m!$-times the volume of the $m$-dimensional polyhedron
defined by the vectors $v^1 (t), \cdots, v^m (t)$.  The aim is to show
that for sufficiently large $m$ this determinant decays exponentially as
$t \to +\infty$.  More precisely, we consider an invariant set $X$ which
is bounded in $\dot{H}^s_p (\Omega)$, $s\ge \mu \ge 2$.
We have:

\proclaim Theorem 4.2.
Let $X \subset \dot{H}^s_p (\Omega)$ be an invariant bounded set.  Assume
$s\ge \mu \ge 2$.  Then there exist positive constants $b_0$, $b_1$, $\gamma$
such that for every $u_0\in X$, $t\ge 0$ and integer $m\ge 1$, the
functions $v^i (t) = (DE (t) u_0) \, v^i_0$ satisfy
$$
\|v^1 (t) \wedge \cdots \wedge v^m (t)\|_s \le \|v^1_0 \wedge
\cdots v^m_0\|_s \, b^{-m}_1 \exp \, (b_0 m^{1-2\mu} - \gamma m) \, t
$$
for all $v^i_0 \in \dot{H}^s_p (\Omega)$.

\noindent{\bf Proof } 
We consider $w^i (t) = v^i (t) \, e^{\gamma t}$, where $\gamma > 0$ is to be
chosen.
For simplicity, we omit the index $i$ in this part of the proof.

Clearly, $w (t)$ is the unique solution of
$$\displaylines{\hfill
Mw_t + w_x + (uw)_x + \alpha Lw - \gamma Mw = 0\hfill\llap{(4.4)}\cr
w (0) = v_0.\cr}
$$
Since $M$ is invertible, we have that
$$
Lw_t + L M^{-1} w_x + L M^{-1} (uw)_x + \alpha L M^{-1} Lw - \gamma Lw =
0.\eqno(4.5)
$$
Multiplying (4.5) by $Lw$ and integrating in space gives
$$
\frac{d}{dt} \; \|Lw\|^2_0 + 2 \left ( LM^{-1} (uw)_x, Lw \right ) + 2 \alpha
\|M^{-1/2} L^{3/2} w\|^2 - 2\gamma \|Lw\|^2 = 0\,.\eqno(4.6)
$$
Since $\|Lw\| \le C \|M^{-{1/2}} L^{3/2} w\|$, we can choose
$\gamma > 0$ such that
$$
\gamma \, \|Lw\|^2 \le 2\alpha\, \|M^{-1/2} L^{3/2} w\|^2.
$$
Hence,
$$
\frac{d}{dt} \, \|Lw\|^2 \le -2 \, \left ( L M^{-1} (uw)_x, Lw \right ).
$$

We now consider the following quadratic forms on $\dot{H}^s_p (\Omega)$:
$$
g (\xi) = \|L\xi\|^2,
$$
and
$$
z (t, \xi) = -2 \, \langle LM^{-1} \, (u (t) \, \xi)_x, \; L\xi \rangle
\eqno(4.7)
$$
for any $t$.
Clearly, by Poincar\'e's inequality there exist nonnegative numbers
$b_3$ and $b_4$ such that
$$
b_3 \, \|\xi\|^2_s \le g (\xi) \le b_4 \, \|\xi\|^2_s,
$$
for all $\xi\in \dot{H}^s_p (\Omega)$.
Moreover, the function $t \to g \, (e^{\gamma t} (DE (t) \, u_0) \, v_0)
= g (w (t))$ is differentiable and its derivative satisfies
$$
\frac{d}{dt} \, g (w (t)) \le z (t, w (t)).
$$

On the other hand, since the order of $M$ is $\mu \ge 2$, we have that
$M^{-\frac12} \, \frac{d}{dx}$ is a bounded operator and therefore, by the
Schwarz inequality, (4.7) implies that
$$
|z (t, \xi)| \le C \|\xi\|_s \|M^{-1/2} \xi\|_s = C \|\xi\|_s \,
\left ( M^{-1} \xi, \xi \right )_s^{1/2}
$$
where we have used that $u(t)$ is bounded in $\dot{H}^s_p (\Omega)$ and also
in $L^\infty (\Omega)$.  We note that $M^{-1}$ is a continuous linear operator
from $\dot{H}^s_p (\Omega)$ to $\dot{H}^{s+\mu}_p (\Omega)$ and, therefore,
it is a compact operator on $\dot{H}^s_p (\Omega)$.

The hypotheses of Theorem A in the appendix of the paper [8] by Ghidaglia
are then fulfilled, where we have taken $\alpha= b_3$, $\beta = b_4$, $\sigma = \frac12$,
$q = g$, $r = z$ and $K = M^{-1}$.  Here we are using an extension of this
theorem to the case where $\frac{dg}{dt} \le r$ (instead of $\frac{dg}{dt}
= r$), which follows immediately from the arguments given in [8]
(pg.~387).

Thus,
$$
\det \, (w^i (t), w^j (t))_s \le \left ( \frac{b_4}{b_3} \right )^m \,
\exp\left\{\frac{Ct}{b_3} \, \sum^m_{\ell=1} \, K_\ell\right\} \,
\det \, (v^i_0, v^j_0)_s,
$$
where $\{ K_\ell \}^\infty_{\ell=1}$ are the eigenvalues of the operator
$M^{-1}$, namely $K_\ell = (1 + C \ell^{2\mu})^{-1}$.
Since $\sum^m_{\ell=1} \, K_\ell \le C m^{1-2\mu}$, we conclude that
$$
\det_{1\le ij\le m} \left ( w^i (t), w^j (t) \right )_s \le C
e^{Ct m^{1-2\mu}} \det_{1\le i, j \le m} (v^i_0, v^j_0)_s.
$$
The theorem follows from the fact that $w^i (t) = e^{\gamma t} v^i (t)$.


\proclaim Theorem 4.3.
The global attractor ${\cal A}$ has finite fractal and Hausdorff
dimensions in $\dot{H}^s_p (\Omega)$, $s\ge \mu \ge 2$.

\noindent{\bf Proof }
This result is a consequence of an abstract result according to [8].
The main idea is to apply Theorem 4.2 with $X = {\cal A}$ and to choose $m$
such that $b_0 m^{1-2\mu} - \gamma m < 0$,  that is, $m > \left (
\frac{b_0}{\gamma} \right )^{\frac{1}{2\mu}}$.  For such $m$, according to
Theorem 4.2, the mapping $DE (t) u_0$ contracts
$m$-dimensional volumes in $\dot{H}^s_p (\Omega)$ for sufficiently large
$t$,   uniformly for $u_0\in {\cal A}$.  With this result, according to
Temam [10],  Chapter V, and Ghidaglia [8], Theorem~3.2, it follows that
${\cal A}$ has finite fractal dimension.

\bigskip

\noindent{\bf Acknowledgments}
The authors' research is partially financed by grants Fondecyt 1940700,
CNPq 910089/94-9 and FAPERGS 93/3051-7.  The authors would like to express
their gratitude to the referee for the suggestions
and comments on the original manuscript.

\bigbreak
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\medskip

M.A. Astaburuaga \& C. Fernandez\hfill\break
Facultad de Matem\'atica, 
Pontificia Universidad Catolica de Chile\hfill\break
Av. Vicu\~na Mackenna 4860, Casilla 306, 
Santiago, Chile. \smallskip

E.~Bisognin (evbisog@zaz.com.br) \& V.~Bisognin\hfill\break
Departamento de Ci\^encias Exatas, Faculdades Franciscanas.\hfill\break
Rua dos Andradas 1614, 97010-032, Santa Maria - RS, Brasil.
\bye