  We study solutions of the system of PDE $D\psi({\bf v}_t)=\text{div}DF(D{\bf
v})$, where $\psi$ and $F$ are convex functions. This type of system arises in
various physical models for phase transitions. We establish compactness
properties of solutions that allow us to verify partial regularity when $F$ is
quadratic and characterize the large time limits of weak solutions. Special
consideration is also given to systems that are homogeneous and their
connections with nonlinear eigenvalue problems. While the uniqueness of weak
solutions of such systems of PDE remains an open problem, we show scalar
equations always have a preferred solution that is also unique as a viscosity
solution.
