  We study whether the solutions of a fully nonlinear, uniformly parabolic
equation with superquadratic growth in the gradient satisfy initial and
homogeneous boundary conditions in the classical sense, a problem we refer to
as the classical Dirichlet problem. Our main results are: the nonexistence of
global-in-time solutions of this problem, depending on a specific largeness
condition on the initial data, and the existence of local-in-time solutions for
initial data $C^1$ up to the boundary. Global existence is know when boundary
conditions are understood in the viscosity sense, what is known as the
generalized Dirichlet problem. Therefore, our result implies loss of boundary
conditions in finite time. Specifically, that a solution satisfying homogeneous
boundary conditions in the viscosity sense eventually becomes strictly positive
at some point of the boundary.
