Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{X},d,\mu )$ satisfying a doubling condition and supporting a Poincar'e inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $\mathrm{\Omega }\times (0,T)$ with $\Omega \subset $\mathcal{X}$$anopensetand$T > 0$, we prove existence in the weak parabolic function space $L_w^1(0,T;\mathrm{B}\mathrm{V}(\mathrm{\Omega }))$. In this paper, we generalize results from a previous work by B"ogelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{B}\mathrm{V}$-valued parabolic function spaces. We argue completely on a variational level.