\newenvironment{problemdesc}
{\renewcommand{\abstractname}{Problem Description}%
\begin{abstract}}
{\end{abstract}}
In this way you can also use the normal abstract
environment.
If many kinds of this generalized abstract environment are needed, one can do the following
\newenvironment{genabstract}[1][]
{\ifx\relax\detokenize{#1}\relax
% no optional argument
\else
% provide the desired title
\renewcommand{\abstractname}{#1}%
\fi
\begin{abstract}}
{\end{abstract}}
The calls can then be
\begin{genabstract}
<regular abstract>
\end{genabstract}
\begin{genabstract}[Problem Description]
<problem description>
\end{genabstract}
Don't feed \abstractname
as the optional argument, as this would cause infinite recursion (it would be possible to avoid this, but it seems overkill).
pdftex
option; the packages needing it are usually able to provide it implicitly.