  We analyze existence and qualitative behavior of non-negative weak solutions
for fourth order degenerate parabolic equations on graph domains with
Kirchhoff's boundary conditions at the inner nodes and Neumann boundary
conditions at the boundary nodes. The problem is originated from industrial
constructions of spray coated meshes which are used in water collection and in
oil-water separation processes. For a certain range of parameter values we
prove convergence toward a constant steady state that corresponds to the
uniform distribution of coating on a fiber net.
