Existence of solution for a class of heat equation involving the 1-Laplacian operator

This paper concerns the existence of global solutions for the following class of heat equation involving the 1-Laplacian operator of the Dirichlet problem $$\{\begin{array}{llccccc}{u}_t-{\mathrm{\Delta }}_{1}u=f(u)& \text{in }& \mathrm{\Omega }\times (0,+\mathrm{\infty }),u=0& \text{in}& \mathrm{\partial }\mathrm{\Omega }\times (0,+\mathrm{\infty }),u(x,0)=u_0(x)& \text{in}& \mathrm{\Omega },\end{array}\text{leqno}(P)$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying some technical conditions, and ${\mathrm{\Delta }}_{1}u=\text{div}\left(\frac{Du}{\mathrm{\mid }Du\mathrm{\mid }}\right)$ 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.