\documentclass[11pt]{article}
\hyphenation{con-firm} \textheight24cm
\setlength{\textwidth}{16.5cm} \addtolength{\topmargin}{-1.8cm}
\addtolength{\oddsidemargin}{-1.9cm}
\addtolength{\evensidemargin}{-1cm} \setcounter{enumi}{1}
\usepackage{amsfonts,amssymb,amsmath}

\begin{document}
\baselineskip13.5pt

\parindent=25pt

\title{\bf
Influence of boundary conditions on the qualitative property of a reaction diffusion equation
}
\author{ {Bendong Lou}\\
\small Department of Mathematics, Tongji University\\ \small Shanghai 200092, China.\\
\small E-mail: blou@tongji.edu.cn}


\date{}
\maketitle

\begin{abstract}
We study a reaction diffusion equation $u_t=u_{xx}+f(u)\ (x\in [0,h(t)])$ with
Robin boundary condition $u(0,t) = b u_x(0,t)$ and with a Stefan free boundary condition
at $x=h(t)$. When $f$ is an unbalanced bistable nonlinearity we prove a 
trichotomy result on the long time behavior of the solutions, that is,
any solution converges either to $0$ (i.e. vanishing), or an active solution 
(i.e. spreading), or a ground state $V(x-y(t))$ with finite or infinite shift $y(t)$
(i.e. transition). In the last case, we show that $y(t)\to z$ for some real 
$z$ when $b$ is large, and $y(t) =A \ln t +B + o(1)$ for some $A, B$
depending on $b$ and $f$ when $b$ is small.
\end{abstract}
\end{document}





















