In this paper we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature, i.e. the existence for the Cauchy-Dirichlet problem to parabolic systems of the type
{
∂
t
u
−
Δ
u
=
−
2
(
H
∘
u
)
D
1
u
×
D
2
u
in
B
×
(
0
,
∞
)
,
u
=
u
o
on
∂
par
(
B
×
(
0
,
∞
)
)
,
\begin{equation*} \left \{ \begin {array}{c} \partial _t u-\Delta u =-2 (H\circ u)D_1u\times D_2u\quad \mbox {in $B\times (0,\infty )$,}\\[3pt] u=u_o\quad \mbox {on $\partial _\textrm {par} \big (B\times (0,\infty )\big )$}, \end{array} \right . \end{equation*}
where
H
:
R
3
→
R
H\colon \mathbb {R}^3\to R
is a bounded continuous function satisfying an isoperimetric condition,
B
B
is the unit ball in
R
2
\mathbb {R}^2
and
u
:
B
×
(
0
,
∞
)
→
R
3
u\colon B\times (0,\infty )\to \mathbb {R}^3
. As one of the possible applications we show that the problem has a solution with values in
B
R
⊂
R
3
B_R\subset \mathbb {R}^3
, whenever
u
o
(
B
)
⊆
B
R
u_o(B)\subseteq B_R
and furthermore there holds
∫
{
ξ
∈
B
R
:
|
H
(
ξ
)
|
≥
3
2
R
}
|
H
|
3
d
ξ
>
9
π
2
,
|
H
(
a
)
|
≤
1
R
for
a
∈
∂
B
R
.
\begin{equation*} \int _{\{ \xi \in B_R: |H(\xi )|\ge \frac {3}{2R}\}}|H|^3\, d\xi >\frac {9\pi }{2}, \qquad |H(a)|\le \tfrac {1}{R}\quad \mbox {for $a\in \partial B_R$.} \end{equation*}