We consider the heat flow associated to the system for surfaces of prescribed mean curvature, more precisely
\[
\frac{\partial u}{\partial t} -\Delta u= -2(H\circ u) \frac{\partial u}{\partial x}\times \frac{\partial u}{\partial y}
\quad \mbox{ in \(B\times (0,\infty)\),}
\]
for a given function \(H\colon \mathbb R^3\to\mathbb R\).
Imposing an isoperimetric condition on the prescribed mean curvature function \(H\), we employ the method of minimal movements to construct a weak solution of the Cauchy-Dirichlet problem. The solution exists for all times and sub-converges to a solution of the stationary problem as time tends to infinity. Moreover, we show that the solution can be constructed in such a way that it develops singularities at most
at finitely many times.
All results were established in joint works with Verena Bögelein (Erlangen)
and Christoph Scheven (Duisburg-Essen).
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