  The aim of this thesis is to derive new gradient estimates for parabolic
equations. The gradient estimates found are independent of the regularity of
the initial data. This allows us to prove the existence of solutions to
problems that have non-smooth, continuous initial data. We include existence
proofs for problems with both Neumann and Dirichlet boundary data. The class of
equations studied is modelled on mean curvature flow for graphs. It includes
anisotropic mean curvature flow, and other operators that have no uniform
non-degeneracy bound. We arrive at similar estimates by three different paths:
a 'double coordinate' approach, an approach examining the intersections of a
solution and a given barrier, and a classical geometric approach.
