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

Abstract

This paper concerns the existence of global solutions for the following class of heat equation involving the 1-Laplacian operator of the Dirichlet problem $$ \left\{ \begin{array}{llc} u_{t}-Δ_1 u=f(u) & \text{in}\ & Ω\times (0, +\infty) , u =0 & \text{in} & \partialΩ\times (0, +\infty), u(x,0)=u_{0}(x)& \text{in} &Ω, \end{array}\right. \leqno{(P)} $$ where $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying some technical conditions, and ${\Delta }_{1}u=\text{div}\left(\frac{Du}{|Du|}\right)$ 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.