Nested cases, alignment and numbering

I don't think that much is gained by aligning the three equations on their respective = symbols. I'd left-align the expressions, using a numcases environment.

\documentclass{article}
\usepackage{newtxtext,newtxmath,mathrsfs} % optional
\usepackage{cases} % for 'numcases' env.
\begin{document}
\begin{numcases}{}
\partial_t u(\vec{x},t) - \alpha\Delta u(\vec{x},t) = f(\vec{x}), 
      &$(\vec{x},t)\in\Omega\times J$, \\
u(\vec{x},t) = 0, 
      &$(\vec{x},t)\in\partial\Omega\times J$, \\
u(\vec{x},0) =
          \left\{\begin{array}{@{}ll@{}}
          \rho,   &\vec{x} \in\mathscr{T} \\
          0,      &\vec{x} \in\mathscr{B}\setminus \mathscr{T}
          \end{array}\right.
      &$\vec{x} \in\mathscr{B}$
\end{numcases}
\end{document}

With the use of the empheq package:

\documentclass{article}

\usepackage{empheq}
\begin{document}
    \begin{empheq}[left=\empheqlbrace]{align}
\partial_t u(\vec{x},t) - \alpha \Delta u(\vec{x},t)
    &= f(\vec{x}), &&(\vec{x},t) \in \Omega \times J, \\
%
u(\vec{x},t) &= 0, &&(\vec{x},t) \in \partial \Omega \times J, \\
%
u(\vec{x},0)
    & = \begin{cases}
    \rho,   &\vec{x} \in T \\
    0,      &\vec{x} \in B \setminus T
        \end{cases}
    &&\vec{x} \in B
    \end{empheq}
\end{document}