  We study whether the solutions of a parabolic equation with diffusion given
by the fractional Laplacian and a dominating gradient term satisfy Dirichlet
boundary data in the classical sense or in the generalized sense of viscosity
solutions. The Dirichlet problem is well posed globally in time when boundary
data is assumed to be satisfied in the latter sense. Thus, our main results are
\emph{a)} the existence of solutions which satisfy the boundary data in the
classical sense for a small time, for all H\"older-continuous initial data,
with H\"older exponent above a critical a value, and \emph{b)} the nonexistence
of solutions satisfying the boundary data in the classical sense for all time.
In this case, the phenomenon of loss of boundary conditions occurs in finite
time, depending on a largeness condition on the initial data.
