Formatting piecewise function

Question

I need to write this

So far I started my code this way:

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

 \begin{document}
\begin{equation*}
\begin{cases}
 u_t+2uu_x=0,              & x \in \mathbb{R}, t>0\\
 u(x,0)=g(x)=\begin{cases}
-1/2, & x<0,\\
1/2, & x>0.
\end{cases}
\end{cases}
\end{equation*}
\end{document}

I get a similar result, which is not really in the same format. The equation are not aligned according to the equal sign, and I have an extra open parenthesis which I don't know how to remove.

Answer 1

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}
\begin{align*}
  u_t+2uu_x   & = 0 ,\quad          x \in \mathbb{R}, t>0\\
u(x,0) = g(x) & = \begin{cases}
                -1/2, & x<0,\\
                 1/2, & x>0.
                    \end{cases}
\end{align*}
\end{document}

or maybe (stole from Werner's answer):

\documentclass{article}
\usepackage{mathtools, amssymb}

\makeatletter
\newcases{mycases}{\quad}{%
  \hfil$\m@th\displaystyle{##}$}{$\m@th\displaystyle{##}$\hfil}{\lbrace}{.}
\makeatother

\begin{document}
\begin{align*}
  u_t+2uu_x   & = 0 ,\quad          x \in \mathbb{R}, t>0\\
u(x,0) = g(x) & = \begin{mycases}
                -1/2, & x<0,\\
                 1/2, & x>0.
                    \end{mycases}
\end{align*}
\end{document}