\documentclass[twoside]{article}
\usepackage{amsfonts, amsmath, amssymb} 
\pagestyle{myheadings}

\markboth{\hfil Uniqueness for the second Painlev\'e equation
\hfil EJDE--2001/49}
{EJDE--2001/49\hfil Mohammed Guedda \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 49, pp. 1--4. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
  \vspace{\bigskipamount} \\
%
  Note on the uniqueness of a global positive solution to the second
  Painlev\'e equation
%
\thanks{ {\em Mathematics Subject Classifications:} 34B15, 35B05, 82D55.
\hfil\break\indent
{\em Key words:} Second  Painlev\'e equation, Neumann condition, global existence.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted February 08, 2001. Published July 9, 2001.} }
\date{}
%
\author{Mohammed Guedda}
\maketitle

\begin{abstract}
  The purpose of this note is to study  the  uniqueness of solutions to
  $ u'' -u^3 + (t-c)u = 0$, for $ t \in (0,+\infty)$ with
  Neumann condition at $0$. Assuming a certain conditon at infinity,
  Helfer and Weissler \cite{HW} have found  a unique solution.
  We show that, without any assumptions at infinity,
  this problem has exactly one global positive solution.
  Moreover, the solution behaves like $\sqrt{t}$ as $t$ approaches infinity.
\end{abstract}


\newcommand{\grad}{\mathop{\rm grad}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{Prop}{Proposition}[section]
\newtheorem{Rm}{Remark}[section]
\newtheorem{lemma}{Lemma}[section]

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11
\@addtoreset{equation}{section}
\catcode`@=12


\section{Introduction }

The existence and uniqueness of solution to the Painlev\'e equation
\begin{equation}\label{eq:1.1}
u'' = u^3-(t-c)u,
\end{equation}
posed in the semi-infinite interval $ (0,+\infty)$, with a Neumann
condition at 0
\begin{equation}\label{eq:1.2}
u'(0) = 0,
\end{equation}
and having a prescribed behavior at $ +\infty $
\begin{equation}\label{eq:1.3}
u(t)\approx \sqrt{t},\end{equation}
has recently been considered by Helffer and Weissler \cite{HW}. Equation
(\ref{eq:1.1}) appears  in the study of the superheating
field attached to a semi-infinite superconductor \cite{Ch,HW}. When
$ c = 0 $, Equation (\ref{eq:1.1}) has a connection with
the Korteweg-de Vries equation; see \cite{Wh,Mi}.
The following theorem presents the family of solutions to
(\ref{eq:1.1})--(\ref{eq:1.3}), in terms of $c$, obatined by
Helfer and Weissler \cite{HW}.

\begin{theorem}\label{Th:1.1}
For each $ c \in \mathbb{R}$, there exists a unique solution, $ u_c $,  to
{\rm (\ref{eq:1.1})--(\ref{eq:1.3}).} This solution
is positive, strictly increasing  and at infinity it satisfies
\begin{equation}\label{estimation1}
   u_c(t) = \sqrt{t} + O(t^{-5/2}),\end{equation}
and
\begin{equation}\label{estimation2}
   u'_c(t) = 2^{-1/2}(t)^{-1/2} + O(t^{-2}).\end{equation}
\end{theorem}

The  proof is similar to the one used by Hastings and McLeod
\cite{HMc} for
constructing the unique strictly positive solution, defined on $\mathbb{R}$,
to
\begin{equation}\label{eq:1.6}
u'' + t u - u^3 = 0,\end{equation}
such that $ \lim_{t\to -\infty}u(t)= 0 $ and $ u(t) \approx \sqrt{t} $ at
$+\infty$. \medskip


The main objective of the present note is to prove the uniqueness of a
global positive solution to (\ref{eq:1.1})--(\ref{eq:1.2})
without any conditions at $ +\infty$.

\section{Main Result}

As in \cite{HW}, $ u(.,\alpha) $ denotes the unique maximal solution $ u \in
C^2((0,T(\alpha)), \mathbb{R})$, to
(\ref{eq:1.1})--(\ref{eq:1.2}) satisfying  $ u(0,\alpha) = \alpha$.
    To prove Theorem 1.1 Helffer and Weissler showed the existence of a
unique $ \alpha = \alpha(c) $
such that $ u(.,\alpha(c)) $ is global, positive and the quantity $
u(t,\alpha(c)) - \sqrt{t} $ tends to 0 as $ t $ approaches $
+\infty$.  The parameter $ \alpha(c) $ satisfies
\[ 0 < \alpha(c)(\alpha(c)^2+c) \leq 1.\]
The idea of the proof is to demonstrate that
\[ {{\cal N}}= (0,\alpha(c)),\quad\mbox{and}\quad
  {{\cal P}}= (\alpha(c),+\infty),\]
where
\begin{eqnarray*}
{{\cal P}} &=& \big\{ \alpha > 0 ;u(.,\alpha) > 0 \mbox{ on }
[0,T(\alpha)), \mbox{ and }\\
&& u(.,\alpha) >h(t)\mbox{ on } (t_0,T(\alpha))\mbox{ for some }
t_0 \in (0,T(\alpha)) \big\},
\\
{{\cal N}} &=& \big\{ \alpha > 0 ; \mbox{ there exists } 0 < t_0<
T(\alpha) ,u(.,\alpha) > 0 \mbox{ on } [0,t_0) \\
&&\mbox{and }   u(t_0,\alpha)= 0\big\},
\end{eqnarray*}
with
\[ h(t) = \sqrt{(t-c)_+}.\]
Our main result is the following.

\begin{theorem}\label{Th:guedda}
For every $ c \in \mathbb{R} $ there exists a unique global positive
solution, $ u_g$, to
\begin{equation} \label{eq:2.2}
\begin{gathered}
u'' = u^3 -(t-c)u,\quad\ t \in (0,+\infty),\\
u'(0) = 0\,.\end{gathered}
\end{equation}
Moreover
\[ \lim_{t\to\infty} (u_g(t) -\sqrt{t}) = 0,\]
and then
\[ u_g \equiv u(.,\alpha(c)).\]
\end{theorem}

This theorem is an immediate consequence of Theorem 1.1 and of the
following propositon.

\begin{Prop}
For every  $\alpha \in {\cal P}$, the maximal interval of definition
$[0,T(\alpha))$
satisfies $ T(\alpha) < +\infty$.
\end{Prop}

\paragraph{Proof.}
Let $ \alpha \in {{\cal P}}. $ Assume on the contrary that $ u(.,\alpha) =:
u $ is global. Since $ u >  h $ for $ t $
large,
$ u $ goes to infinity with $ t,  u' > 0 , u'' > 0 $ for
$ t $ large and the limit
$ \lim_{t\to+\infty} u'(t) $ exists in $ (0,+\infty]$.
Next fix $\varepsilon \in (0,1). $ Because $ \lim_{t\to\infty}
\frac{\sqrt{\varepsilon} u'(t)}{h'(t)} = +\infty $ we
deduce
that
\[ \lim_{t\to\infty} \frac{\sqrt{\varepsilon} u(t)}{h(t)} = +\infty,\]
thanks to the l'H\^opital rule. Therefore,
\[ u'' = u(u^{2} - h^2) \geq (1-\varepsilon) u^{3},\]
for $ t $ large, and then $ u $ is not global. This is a contradiction
that completes the Proof.
\quad$\Box$


\begin{Rm} \rm
Now it is clear that the unique global positive solution to
(1.1)--(1.2) is the one required by Chapman; this
confirms the previous condition at infinity.  By  similar argument, we can
prove that any global positive solution to
(\ref{eq:1.1}) satisfies (\ref{eq:1.3}) at infinity.
\end{Rm}

\begin{Rm} \rm
In the same spirit, we can show that the  problem
\begin{gather*}
(\vert u'\vert^{p-2}u')' = u\vert u\vert^{p-2}\left(\vert
u\vert^q-\vert h\vert^{q-1}h\right),\\
u'(0) = 0,\quad p > 1,\quad q > 0,
\end{gather*}
possesses a unique positive global solution, under some restrictions on
$h$  \cite{GG}. Moreover, this solution behaves like $h $ at infinity.
\end{Rm}

\begin{Rm}\rm
A similar classification is obtained in \cite{GGV} for the
problem
\[ \vert y'\vert^{p-2}y' = y^q - \beta y.\]
This equation is satisfied by similarity solutions to
\[u_t = (\vert u_x\vert^{p-2}u_x)_x -(u^q)_x,\quad q=2(p-1).\]
\end{Rm}


\paragraph{Acknowledgments.}
The author is grateful for a partial support from
the  Department of Mathematics of the Faculty of Sciences and Technique,
FST Marrakech Morroco, during his visit there.  This work was
also partially supported  by DRI ( UPJV) Amiens, France.

\begin{thebibliography}{99} \frenchspacing

\bibitem{Wh}
M. J. Ablowitz and H. Segur, { \it Exact linearization of a Painlev\'e
transcendent,} Phys. Rev. Lett., 38, (1977), pp.
1103--1106.

\bibitem{Ch}
S. J. Chapman, { \it Superheating field of type II superconductors,} SIAM J.
Appl. Math..
55, (1995), pp. 1233--1258.

\bibitem{GG}
   A. Gmira and M. Guedda, { \it Classification of solutions to a
class of nonlinear
differential equations,}  International
J. Diff. Equat. Appl., Vol 1., No 2, (2000), pp. 223--238,

\bibitem{GGV}
A. Gmira, M. Guedda et L. Veron,{ \it Source-type solution for the
one-dimensional diffusion-convection equation, } NoDEA, 7, No.
2,  (2000), pp. 127--142.


\bibitem{HMc}
S. P. Hastings and  J. B. McLeod,  {\it A boundary value problem
associated with the
second Painlev\'e transcendent and the Korteweg-de  Vries equation,} Arch.
Rational Mech. Anal., 73, No. 1, (1980),  pp. 31--51.

\bibitem{HW}
B. Helffer and F. B. Weissler, {\it  On a family of solutions of the second
Painlev\'e equation related to superconductivity,}
European J. Appl. Math.,  Vol. 9,  No. 3, (1998), pp. 223--243.

\bibitem{Mi}
R. M. Miura, { \it  The Korteweg-de  Vries equation; a survey of results,}
SIAM Rev., 18, (1976), pp. 412--459.


\end{thebibliography}

\noindent\textsc{Mohammed Guedda }\\
Lamfa,  CNRS FRE 2270,  Universit\'e de  Picardie Jules Verne, \\
Facult\'e  de  Math\'ematiques et d'Informatique,\\
33, rue Saint-Leu 80039 Amiens, France \\
e-mail: Guedda@u-picardie.fr


\end{document}