  We study a class of quadratic, infinite-dimensional dynamical systems,
inspired by models for viscoelastic fluids. We prove that these equations
define a semi-flow on the cone of positive, essentially bounded functions. As
time tends to infinity, the solutions tend to an equilibrium manifold in the
$L^2$-norm. Convergence to a particular function on the equilibrium manifold is
only proved under additional assumptions. We discuss several possible
generalizations.
