Equation tag on the LHS is done by the option leqno to the article class.
Arbitrary Tags can be set with the commands \tag{} and \tag*{} (the first has parentheses, the second not).
If you want to refer to it later, you have use $…$ in the \tag{} command instead of \(…\).
Lastly, if you want to treat a set of equations as single object, for example if you want to tag it, you should use the aligned environment from amsmath package.
The code:
\documentclass[leqno,fleqn]{article}
\usepackage[showframe]{geometry}
\usepackage{amsmath}
\begin{document}
\begin{equation}\tag{$\text{P}_\text{W}$}\label{one}
\left\{
\begin{aligned}
u_t&=F(x)\\
u(0)&=u_0
\end{aligned}
\right.
\end{equation}
\paragraph{Some Text} Hello World! \eqref{one} is a d*** good pair of equations.
\end{document}
The subscript in the tag can be created by several ways. For example, if you use xelatex, you can also use the xltxtra package with its \textsubscript command, in order to circumvent the use of math inside the \tag.
casesconstruction for the brace. – Ethan Bolker Apr 25 '13 at 12:06amsmathpackage and use the command\tag{...}. E.g., you could write<body of equation> \tag{$P_w$} \label{eq:pw}; note that the argument of\tagis in text mode by default, hence the use of$...$to create the subscripted material. Elsewhere in your document, you can then create a cross-reference to this equation with the command\eqref{eq:pw}. – Mico Apr 25 '13 at 13:00