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\markboth{\hfil Relaxation approximations \hfil EJDE--2002/19}
{EJDE--2002/19\hfil Francisco Caicedo, Yunguang Lu, \& Mauricio Sep\'ulveda
\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 19, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Relaxation approximations and bounded variation estimates
for some partial differential equations
%
\thanks{ {\em Mathematics Subject Classifications:} 35B40, 35D10,
35K15, 35K65, 35L65.
\hfil\break\indent
{\em Key words:} Degenerate parabolic equation, Hyperbolic conservation laws,
\hfil\break\indent Relaxation approximation.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted 15 November, 2001. Published February 19, 2002.} }
\date{}
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\author{Francisco Caicedo, Yunguang Lu, \& Mauricio Sep\'ulveda}
\maketitle
\begin{abstract}
In this paper, we introduce a new technique for studying solutions of
bounded variation for some conservation laws of first order partial
differential equations and for some degenerate parabolic equations
in multi-dimensional space. The connection between these
two types of equations is the vanishing relaxation method.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
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\section{Introduction}
We are concerned with solutions of bounded variation and the limiting
behavior of relaxation approximated solutions to the Cauchy problem
for the conservation system
\begin{equation} \label{e1}
G(u)_{t}+ \sum_{j=1}^{D} F_j(u)_{x_j}=\epsilon \Delta u
\end{equation}
with initial data
\begin{equation} \label{e2}
u(x,0)=u_0(x),
\end{equation}
where $ \epsilon=(\epsilon_1,\epsilon_2,\dots ,\epsilon_N)^T$ is a
nonnegative constant vector, $x \in \mathbb{R}^D$, $u \in \mathbb{R}^N$, $D$
denotes the space dimension,
$F_j(u)=(F_j^1(u), F_j^2(u),\ldots,F_j^N(u))^T$, and
$G(u)=(G^1(u),G^2(u),\ldots,G^N(u))^T$ are smooth nonlinear maps
from
$\mathbb{R}^N$ to $\mathbb{R}^N$.
For the hyperbolic case, $ \epsilon=0$, and for the scalar equation, $N=1$,
the behavior of the unique solution of (\ref{e1})--(\ref{e2}) have been
studied in many papers; see for example \cite{k4,s1} and their references.
For the interesting case $N \geq 2$, a partial list of results is as follows:
\noindent 1.) For $D=1$ and the system (\ref{e1}) is strictly hyperbolic and
genuinely nonlinear in the sense of Lax \cite{l1}. When
the total variation of initial data (\ref{e2}) is small, the
bounded variation of solutions to (\ref{e1})--(\ref{e2}) was obtained by
using the Glimm method
\cite{g1}. This method was developed by Glimm and extended by
Liu \cite{l4} to strictly hyperbolic systems whose
characteristic field is either genuinely nonlinear or linearly degenerate.
The Glimm method was use by many authors in the study of arbitrary data with
bounded variation for some special systems (See \cite{n1}), and for general
systems of Temple type in which shock waves and rarefaction waves coincide
(See \cite{t2}). The uniqueness of limits of Glimm scheme solutions was
proved by Bressan \cite{b2}.
One of the ideas in the Glimm's method is to use the explicit structure
of the nonlinear progressing wave solutions in a single space variable, i.e.
the solution of the Riemann problem constructed by Lax in \cite{l1}. However
for multi-dimensional case, $ D \geq 2$, even for the Riemann data, the global
structure of weak solutions is not clear. See \cite{z1} for the details about
the Riemann solution.
\noindent 2.) For the case $N=2$ and $D=1$, if the system
(\ref{e1}) is strictly hyperbolic and genuinely nonlinear, and
the approximated solutions of (\ref{e1}) are uniformly bounded,
then the global existence of $L^{ \infty}$ solutions for
(\ref{e1})--(\ref{e2}) was proved by DiPerna \cite{d2} using
the compensated compactness ideas developed by Tartar and Murat \cite{m3,t1}.
The most successful application of the theory of compensated compactness
is in the study of gas dynamics system
which was non-strictly hyperbolic at the vacuum line. The existence of
a global solution was proved for the case of a polytropic
gas \cite{b3,l2,l3}. See also \cite{k3,l6,l7} for a related system.
One of the strong restrictions on the applications of the compensated
compactness is that one must construct infinite pairs of entropy-entropy
flux for a given system, which makes this method work well only for
systems of two equations in a single variable.
\noindent 3.) The local existence in time of smooth solutions of
(\ref{e1})--(\ref{e2}) in multi-dimensional space was obtained by Kato for
symmetric hyperbolic systems and sufficiently smooth
data \cite{k1}. The short-time existence and stability
of multi-dimensional ``shock front" solutions of (\ref{e1}) with discontinuous
initial data were proved by Majda under some structural hypotheses
\cite{m1,m2}.
For the parabolic case, where $\epsilon$ is a fixed positive constant, (\ref{e1}) can
be considered as typical degenerate parabolic equation. For instance, let
$F(u)=0$, $\epsilon=1$ and $ G(u)=u^{1/m}$, then (\ref{e1}) is equivalent to
\begin{equation} \label{e3}
w_{t}= \Delta w^{m},
\end{equation}
which is so-called porous medium equation. This equation models the
non-stationary flow of a compressible Newtonian fluid in a porous medium
under polytropic conditions. The value of $w \geq 0$ is proportional
to the density of the fluid. It is the most typical case of degenerate
parabolic equations. The study of its regularity has a long history.
The optimal H\"older estimate in a single space variable was resolved by
Aronson \cite{a1} many years ago, but in the multi-dimensional space, it is
still an open problem. The
regularity of solutions for the Cauchy problem (\ref{e3}) in one dimension
and in multi-dimensions are quite different. We refer the readers to the paper
\cite{k2} and the papers cited therein for the details. Some recent regularity
results about the equation (\ref{e3}) can be found in \cite{l8,l9}.
In this paper, we study the bounded variation solutions of the Cauchy problem
(\ref{e1})--(\ref{e2}) for the case of $\epsilon=0$ or for the case of
$ \epsilon $ being a fixed positive constant. As the first one of a series,
in this paper we restrict our attention to the most simple case $N=1$.
Our method is to select solutions of the Cauchy problem (\ref{e1})--(\ref{e2})
as the singular perturbation limit of approximated solutions for the system
\begin{equation} \label{e4}
\begin{gathered}
u_{t}+ \sum_{j=1}^{D} F_j(u)_{x_j}+ \frac{H(u)-v}{ \tau}= \nu \Delta u \\
v_{t}+ \frac{v-H(u)}{ \tau}=\mu \Delta v,
\end{gathered}
\end{equation}
with initial data
\begin{equation} \label{e5}
(u(x,0),v(x,0))=(u_0(x),v_0(x)),
\end{equation}
where $\nu > 0$ and $ \mu \geq 0$ are constants.
The system (\ref{e4}) itself has great interests since it can be considered
as the relaxation problem which arises in many physical situations such as
kinetic theory, multiphase and phase transition, viscoelasticity,
river flows, traffic flows, the theory of
combustion and chromatography. In the physical background, $u$ and $ v$ are
vectors. In chromatography, $u_i$ represent the concentration of the solute
in the fluid phase and $v_i$ its concentration
in the solid phase, both being expressed in moles
per unit volume of their own phase.
When $ \tau=0$, the equilibrium relation $v_i$ which is usually called the
adsorption isotherm is, in general, a complicated nonlinear function of
$u_i$ in which the mutual influences among
different solutes are taken into account. The details of the physical
backgrounds can be found in \cite{r1,w1}. It is worth while
pointing that the unique solution for the system (\ref{e4}) without the
parameters $ \nu$ and $ \mu $ was recently studied in \cite{b3} by using
the Glimm method for a small initial date.
First of all, about the solutions of the Cauchy
problem (\ref{e4})--(\ref{e5}) we have the following result.
\begin{theorem} \label{thm1}
I.) Assume that $H(u)$ is a nondecreasing function and the initial data
$(u_0(x),v_0(x))$ have compact support or vanish sufficiently fast
as $ |x| \rightarrow \infty$.
If \begin{equation} \label{e6}
\iint _{\mathbb{R}^D} |(u_0(x))_{x_j}| dx \leq M, \quad
\iint _{\mathbb{R}^D} |(v_0(x))_{x_j}| dx \leq M
\end{equation}
for $j=1,2,\dots ,D$, then
for any fixed $ \nu > 0, \mu \geq 0$
and $ \tau > 0$, the solutions $(u^{ \tau}, v^{ \tau})$ of the Cauchy problem
(\ref{e4})--(\ref{e5}) satisfy
the a-priori estimates
\begin{equation} \label{e7}
(u^{ \tau}(x,t),v^{ \tau}(x,t)) \rightarrow (0,0), \quad |x| \rightarrow
\infty
\end{equation}
for fixed $ t > 0$ and
\begin{equation}\label{e8}
\iint _{\mathbb{R}^D} |(u^{ \tau})_{x_j}(x,t)| dx \leq M, \quad
\iint _{\mathbb{R}^D} |(v^{ \tau})_{x_j}(x,t)| dx \leq M;
\end{equation}
if
\begin{equation} \label{e9}
\iint _{\mathbb{R}^D} |(u^{ \tau}(x,0))_{t}| dx \leq M, \quad
\iint _{\mathbb{R}^D} |(v^{ \tau}(x,0))_{t}| dx \leq M,
\end{equation}
then
\begin{equation} \label{e10}
\iint _{\mathbb{R}^D} |(u^{ \tau})_{t}(x,t)| dx \leq M, \quad
\iint _{\mathbb{R}^D} |(v^{ \tau})_{t}(x,t)| dx \leq M,
\end{equation}
where $M$ denotes a positive constant which is independent of $ \nu, \mu$ and
$\tau$.
\noindent II.) If the conditions in I.) are satisfied and the space
dimension is $D=1$,
then the Cauchy problem (\ref{e4})--(\ref{e5}) has a unique classical
solution $(u^{\tau}, v^{ \tau})$ on $\mathbb{R} \times [0,T]$ for any
even time $T$, which satisfies (\ref{e8}), (\ref{e10}) and
the boundedness estimates
\begin{equation} \label{e11}
|u^{ \tau}(x,t)| \leq M, \quad |v^{ \tau}(x,t)| \leq M;
\end{equation}
If the space dimension $D \geq 2$ and there exist two large constants $M_1,
M_2$ such that
$H(M_1)=M_2, |u_0(x)|
\leq M_1, |v_0(x)| \leq M_2$, then the above boundedness estimate
(\ref{e11}) is true.
\end{theorem}
From the estimates given in (\ref{e8}), (\ref{e10}), we immediately have
the following theorem.
\begin{theorem} \label{thm2}
Under the conditions of Theorem \ref{thm1}, if
$ v_0(x)=H(u_0(x))$, then there
exists a subsequence (still denoted by $(u^{ \tau}(x,t), v^{ \tau}(x,t))$)
such that
\begin{equation} \label{e12}
(u^{ \tau}(x,t), v^{ \tau}(x,t))
\rightarrow (u(x,t),v(x,t)) \quad
\mbox{a.e.}
\end{equation}
as $\nu, \mu, \tau $ go to zero and the limit function $(u,v)$ satisfies
$v=H(u)$, a.e. and $u$ is a generalized solution of the scalar equation
\begin{gather} \label{e13}
(u+H(u))_{t}+ \sum_{j=1}^{D} F_j(u)_{x_j}=0, \\
u(x,0)=u_0(x) \label{e14}
\end{gather}
which satisfies the $BV$ estimates
\begin{equation} \label{e15}
|u(x,t)| \leq M, \quad
\iint _{\mathbb{R}^D} |u_{x}(x,t)| dx \leq M, \quad
\iint _{\mathbb{R}^D} |u_{t}(x,t)| dx \leq M.
\end{equation}
\end{theorem}
\begin{theorem} \label{thm3}
Assume that $0 \leq w_0(x) \leq M$, $ m > 1$,
$\iint _{\mathbb{R}^D} |w_0(x)_{x_j}|dx $,
and $ \iint _{\mathbb{R}^D} | \Delta (w^{m}_0(x))|dx$ are
bounded. Then the unique weak solution for the Cauchy problem
\begin{gather} \label{e16}
w_{t}= \Delta w^{m} \\
w(x,0)=w_0(x) \label{e17}
\end{gather}
satisfies the bounded variation estimates
\begin{equation} \label{e18}
|w(x,t)| \leq M, \quad \iint _{\mathbb{R}^D} |w(x,t)_{x_j}|dx \leq M, \quad
\iint _{\mathbb{R}^D} |w(x,t)_{t}|dx \leq M.
\end{equation}
\end{theorem}
The proofs of the above theorems are given in the next section.
\section{Proofs of Theorems}
\paragraph{Proof of Theorem \ref{thm1}}
The behavior (\ref{e7}) of the solutions can be seen from
the proof of the existence of a local solution; see \cite{s1} for the details.
To prove (\ref{e8}), we differentiate (\ref{e4}) with respect to $y$,
where $y=x_{l}$ for $ l=1,2,\dots ,D$, and multiply by
$ \tau \mathop{\rm sgn}(u_{y})$ in the first equation, and by
$ \tau \mathop{\rm sgn}(v_{y})$ in the second equation.
We obtain
\begin{equation} \label{e19}
\begin{gathered}
\tau |u_{y}|_{t}+ \tau \sum_{j=1}^{D} (F_j'(u)|u_{y}|)_{x_j}
+ (H'(u)|u_{y}|- \mathop{\rm sgn}(u_{y})v_{y})
= \nu \tau sgn u_{y}) \Delta (u_{y}), \\
\tau |v_{y}|_{t}+ (|v_{y}|- H'(u)u_{y} \mathop{\rm sgn}(v_{x}))= \tau \mu
\mathop{\rm sgn}(v_{y}) \Delta (v_{y}).
\end{gathered}
\end{equation}
Adding the above two equations, we obtain
\begin{multline} \label{e20}
\tau (|u_{y}|+|v_{y}|)_{t}+ \tau \sum_{j=1}^{D} (F_j'(u)|u_{y}|)_{x_j}
+ (H'(u)|u_{y}|+|v_{y}|)(1-\mathop{\rm sgn}(v_{y})\mathop{\rm sgn}(u_{y}))\\
= \tau ( \nu \mathop{\rm sgn} u_{y}) \Delta (u_{y})
+ \mu \mathop{\rm sgn}(v_{y}) \Delta (v_{y})).
\end{multline}
Integrating (\ref{e20}) on $\mathbb{R}^D \times [0,T_1]$ for a fixed time $T_1$,
and noticing (\ref{e7}), we have
\begin{equation} \label{e21}
\iint _{\mathbb{R}^D} (|u_{y}(x,t)|+|v_{y}(x,t)|) dx
\iint _{\mathbb{R}^D} (|(u_0(x))_{y}|+|(v_0(x))_{y}|)dx,
\end{equation}
which is the estimate (\ref{e8}).
Similarly we can get
\begin{equation} \label{e22}
\iint _{\mathbb{R}^D} (|u_{t}(x,t)|+|v_{t}(x,t)|) dx
\leq \iint _{\mathbb{R}^D} (|(u(x,0))_{t}|+|(v(x,0))_{t}|)dx,
\end{equation}
which is the estimate (\ref{e10}). So part I) of Theorem \ref{thm1} is proved.
For the one dimension case $D=1$ in II) of Theorem \ref{thm1}, we have
\begin{equation} \label{e23}
|u|= \big| \int_{ - \infty}^{x} u_{x}dx\big| \leq \int_{R}|u_{x}|dx \leq M,
\end{equation}
which imply the existence of global solutions in time for the Cauchy problem
(\ref{e4}),(\ref{e5}).
Using the condition $H(M_1)=M_2$ given in II), a maximum principle
applying to (\ref{e4}) gives the boundedness estimate (\ref{e11}), which
again implies the global existence of solutions for the Cauchy problem
(\ref{e4}),(\ref{e5}). So Theorem \ref{thm1} is proved.
\paragraph{Proof of Theorem \ref{thm2}}
Let $ \mu=0$ in (\ref{e4}).
Then if the total variation of $u_0(x)$ is bounded, we can smooth
$u_0(x)$ by a molifier such that $ \nu \Delta u_0(x)$ is $L^1$ bounded.
Then from the first equation in (\ref{e4}) and $ v_0(x)=H(u_0(x))$, $u^{\tau}(x,0)_{t}$ is also $L^1$
bounded; and from the second
equation in (\ref{e4}), $v^{\tau}(x,0)_{t}=0$.
From the second equation in (\ref{e4}) and the the estimate in (\ref{e10}),
we have
\begin{equation} \label{e24}
\iint _{\mathbb{R}^D} |H(u^{\tau})-v^{\tau}| dx \leq \tau M.
\end{equation}
So there exists a subsequence $(u^{ \tau_{k}}, v^{\tau_{k}}) $
such that $u^{ \tau_{k}}$ converges to a function $u$ as
$ \tau, \nu $ tend to zero and so $ v^{ \tau_{k}}$ converges to $H(u)$
from (\ref{e24}), where $u$ satisfies the estimates (\ref{e8}),(\ref{e10}).
Adding two equations in (\ref{e4}) together, we have
\begin{equation} \label{e25}
(u^{\tau}+v^{\tau})_{t}+ \sum_{j=1}^{D} F_j(u^{\tau})_{x_j}
= \nu \Delta u^{\tau}.
\end{equation}
which gives (\ref{e13}) in the sense of distributions as $ \tau $ and
$ \nu $ tend to zero. So Theorem \ref{thm2} is proved.
\paragraph{Proof of Theorem \ref{thm3}}
Let $F_j=0, \mu=0$ and $ \nu=c$ be a fixed positive
constant in (\ref{e4}). Then (\ref{e4}) is equivalent to
\begin{equation} \label{e26}
\begin{gathered}
u_{t}+ \frac{H(u)-v}{ \tau}=c \Delta u \\
v_{t}+ \frac{v-H(u)}{ \tau}=0,
\end{gathered}
\end{equation}
with the initial data
\begin{equation} \label{e27}
(u_0(x),v_0(x))=(w_0^{m}(x),H(w_0^{m}(x))),
\end{equation}
where $H(u)=cu^{1/m}-u$. Let
$0 \leq u_0(x) \leq ( \frac{c}{m})^{ \frac{m}{m-1}}=U_{+}$
and $V_{+}=H(U_{+})$, then $H'(u) \geq 0$
for $u \in [0,U_{+}]$. From the conditions in Theorem \ref{thm3},
$ \iint _{\mathbb{R}^D}|w_{0y}| dx$ is bounded, where $y$ denotes an
$x_j$, $j=1,2,\dots D$, then the integral
$ \iint _{\mathbb{R}^D} |(w^{m}_0)_{y}| dx$
is also bounded since $m > 1$ and the
boundedness of $w_0$ follows. Thus
$ \iint _{\mathbb{R}^D} |u_{0y}|+|v_{0y}| dx \leq $ is bounded.
Furthermore if $ \iint _{\mathbb{R}^D} | \Delta (w^{m}_0)| dx $
is bounded, then $ \iint _{\mathbb{R}^D} |u_{t}(x,0)| dx $ is bounded
from the first equation in (\ref{e26}) and $v_{t}(x,0)=0 $ from the second
in (\ref{e26}).
Therefore from the conclusions given in II) in Theorem \ref{thm1},
we have the following estimates for the solutions $(u^{ \tau},v^{ \tau})$
of the Cauchy problem (\ref{e26}),(ref{e27})
\begin{equation} \label{e28}
\begin{gathered}
0 \leq u^{ \tau} \leq U_{+}, \quad 0 \leq v^{ \tau} \leq V_{+} \\
\iint _{\mathbb{R}^D} |u^{ \tau}_{y}(x,t)|dx \leq M, \quad
\iint _{\mathbb{R}^D} |v^{\tau}_{y}(x,t)|dx \leq M \\
\iint _{\mathbb{R}^D} |u^{ \tau}_{t}(x,t)|dx \leq M, \quad
\iint _{\mathbb{R}^D} |v^{ \tau}_{t}(x,t)|dx \leq M.
\end{gathered}
\end{equation} \label{e29}
Using the second equation in (\ref{e26}), we have
\begin{equation}
\iint _{\mathbb{R}^D} |v^{ \tau}-H(u^{ \tau})| dx \leq \tau M.
\end{equation}
The estimates in (\ref{e28}),(\ref{e29}) imply the convergence of the
relaxation solutions $(u^{ \tau}, v^{ \tau})$ as the relaxation parameter
$ \tau $ goes to zero. Let the limit be $(u,v)$. Then $ v=H(u)$, a.e.
from (\ref{e29}).
Adding two equations in (\ref{e26}) together, we obtain
\begin{equation} \label{e30}
(c(u^{ \tau})^{1/m})_{t}+(v^{ \tau}-A(u^{ \tau}))_{t}
=c \Delta u^{ \tau}.
\end{equation}
This implies that (let $u^{1/m}=w$)
\begin{equation} \label{e31}
w_{t}= \Delta w^{m}.
\end{equation}
Since $ v=H(u)$, a.e. and $ H(u)=cu^{1/m}-u$, then from (\ref{e28}),
$ w=u^{1/m}$ satisfies the estimates in (\ref{e18}). Theorem \ref{thm3} is proved.
\paragraph{Remark 1}
The first strong and local regularity $ L^{p}$ estimate of
$ w_{t}$ for the solution $w$ of the porous media equation $ (31)$ was
obtained by B\'enilan in \cite{b1}. Here we extended the estimates to the
whole $\mathbb{R}^D$ space both for $ w_{t}$ and for $w_{x_i}$.
\paragraph {Remark 2} In Theorem \ref{thm3}, we only consider the
bounded variation solution for
the most typical degenerate parabolic model, i.e. the porous medium equation.
In fact, from its proof, we can see that the results are still true for more
general degenerate parabolic equation of the form
\begin{equation} \label{e32}
G(w)_{t}+ F(x,t,w,w_{x_i})= \Delta w,
\end{equation}
where the nonlinear function $G(w)$ is smooth and $G'(w) > c >0$ for a constant $c$. For instance,
the singular nonlinear partial differential equation
\begin{equation} \label{e33}
\begin{gathered}
\beta(u(x,t))_{t} \ni \Delta u(x,t), \quad (x,t) \in Q= \Omega \times (0,T), \\
u(x,t)=0, \quad (x,t) \in \partial Q = \partial \Omega \times (0,T), \\
u(x,0)=u_0(x), \quad x \in \Omega,
\end{gathered}
\end{equation}
where $ \Omega $ is a bounded smooth domain in $\mathbb{R}^D (D \geq 1), u_0$
is a given smooth function, and $ \beta $ is the multivalued mapping
\begin{equation} \label{e34}
\beta(x)= \left\{
\begin{array}{ll}
ax-1, & x \leq 0 \quad (a>0) \\
(-1,1), & x=0, \\
bx+1, & x \geq 0 \quad (b>0).
\end{array}\right.
\end{equation}
Equations (\ref{e33}), (\ref{e34}) are a formulation of the classical
two-phase Stefan problem, describing the flow of heat within a substance
(say water) which changes phase (melts or freezes) at the temperature zero.
The constants
$a$ and $b$ denote the respective thermal conductivities in the ice and water
regions, and the jump in $ \beta $ at zero corresponds to the latent heat of
fusion. The temperature is
\begin{equation} \label{e35}
T= \left\{
\begin{array}{ll}
u/b, & u>0, \\
0, & u=0, \\
u/a, & u<0.
\end{array}\right.
\end{equation}
The continuity of a unique weak solution of (\ref{e33}),(\ref{e34})
for all $D \geq 1 $
was obtained by Caffarelli and Evan \cite{c1}. Here if we omit the region $ \Omega$
and consider the solution in whole space, then the solution for the Cauchy
problem of two-phase Stefan problem has BV bounded from Theorem \ref{thm3}.
\paragraph{Acknowledgments:}
The authors are very grateful to the referee for his carefully reading the
manuscript and proposing many valuable suggestions.
This paper was partially supported NNSF 10071080-China, by the Program A
on Numerical Analysis of FONDAP-Chile in Applied Mathematics, and
Fondecyt \# 1000332.
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\end{thebibliography}
\noindent\textsc{Francisco Caicedo} (e-mail: jcaicedo@matematicas.unal.edu.co)\\
\textsc{Yunguang Lu} (e-amil: yglu@matematicas.unal.edu.co)\\
Departamento de Matem\'aticas y Estad\'\i stica \\
Universidad Nacional de Colombia, Bogota, Colombia
\smallskip
\noindent\textsc{Mauricio Sep\'ulveda}\\
Departamento de Ingenier\'\i a Matem\'atica\\
Facultad de Ciencias F\'\i sicas y Matem\'aticas\\
Universidad de Concepci\'on, \\
Casilla 160-C, Concepci\'on, Chile\\
e-mail: mauricio@ing-mat.udec.cl
\end{document}