\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 $$\label{eq:1.1} u'' = u^3-(t-c)u,$$ posed in the semi-infinite interval $(0,+\infty)$, with a Neumann condition at 0 $$\label{eq:1.2} u'(0) = 0,$$ and having a prescribed behavior at $+\infty$ $$\label{eq:1.3} u(t)\approx \sqrt{t},$$ 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 $$\label{estimation1} u_c(t) = \sqrt{t} + O(t^{-5/2}),$$ and $$\label{estimation2} u'_c(t) = 2^{-1/2}(t)^{-1/2} + O(t^{-2}).$$ \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 $$\label{eq:1.6} u'' + t u - u^3 = 0,$$ 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 $$\label{eq:2.2} \begin{gathered} u'' = u^3 -(t-c)u,\quad\ t \in (0,+\infty),\\ u'(0) = 0\,.\end{gathered}$$ 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}