  This paper concerns the existence of global solutions for the following class
of heat equation involving the 1-Laplacian operator of the Dirichlet problem
  $$
  \left\{
  \begin{array}{llc}
  u_{t}-\Delta_1 u=f(u) & \text{in}\ & \Omega\times (0, +\infty) ,
  u =0 & \text{in} & \partial\Omega\times (0, +\infty),
  u(x,0)=u_{0}(x)& \text{in} &\Omega ,
  \end{array}\right.
  \leqno{(P)}
  $$
  where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N \geq 1$
and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying some
technical conditions, and $\Delta_1 u=\text{div}\left(\frac{Du}{|Du|}\right)$
denotes the 1-Laplacian operator. The existence of global solution is done by
using an approximation technique that consists in working with a class of
$p$-Laplacian problem associated with $(P)$ and then taking the limit when $ p
\to 1^+$ to get our results.
