Convergence of solutions to the $p$-Laplace evolution equation as $p$ goes to 1

We prove that the set of solutions to the parabolic singular $p$-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain $\mathrm{\Omega }$ for all space dimensions is continuous in the parameter $p\in [1,+\mathrm{\infty })$ and the initial data. The highly singular limit case $p=1$ is included. In particular, we show that the solutions $u_p$ converge strongly in $L^2(\mathrm{\Omega })$, uniformly in time, to the solution $u_1$ of the parabolic $1$-Laplace equation as $p\to 1$.