  We consider a prototypical nonlinear parabolic equation whose flux has three
distinguished features: it is nonlinear with respect to both the unknown and
its gradient, it is homogeneous, and it depends only on the direction of the
gradient. For such equation, we obtain existence and uniqueness of entropy
solutions to the Dirichlet problem, the homogeneous Neumann problem, and the
Cauchy problem. Qualitative properties of solutions, such as finite speed of
propagation and the occurrence of waiting-time phenomena, with sharp bounds,
are shown. We also discuss the formation of jump discontinuities both at the
boundary of the solutions' support and in the bulk.
