AbstractWe 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 $$({\mathcal {X}}, d, \mu )$$
(
X
,
d
,
μ
)
satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$
u
0
on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$
Ω
×
(
0
,
T
)
with $$\Omega \subset {\mathcal {X}}$$
Ω
⊂
X
an open set and $$T > 0$$
T
>
0
, we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$
L
w
1
(
0
,
T
;
BV
(
Ω
)
)
. In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$
BV
-valued parabolic function spaces. We argue completely on a variational level.