  We consider the elliptic and parabolic superquadratic diffusive
Hamilton-Jacobi equations with homogeneous Dirichlet conditions. For the
elliptic problem in a half-space, we prove a Liouville-type classification, or
symmetry result, which asserts that any solution has to be one-dimensional.
This turns out to be an efficient tool to study the behavior of boundary
gradient blow-up (GBU) for the parabolic problem in general bounded domains.
Namely, we show that in a neighborhood of the boundary, at leading order,
solutions display a global ODE type behavior, with domination of the normal
derivatives upon the tangential derivatives. This leads to the existence of a
universal, sharp blow-up profile in the normal direction at any GBU point, and
moreover implies that the behavior in the tangential direction is more
singular. On the other hand, it is known that any GBU solution admits a weak
continuation, under the form of a global viscosity solution. As another
consequence, we show that these viscosity solutions {\it generically} lose
boundary conditions after GBU. This result, as well as the above GBU profile,
were up to now essentially known only in one space-dimension.
