  We study the qualitative properties of the unique global viscosity solution
of the superquadratic diffusive Hamilton-Jacobi equation with (generalized)
homogeneous Dirichlet conditions. We are interested in the phenomena of
gradient blow-up (GBU), loss of boundary conditions (LBC), recovery of boundary
conditions and eventual regularization, and in their mutual connections.
  In any space dimension, we establish the sharp minimal rate of GBU. Only
partial results were previously known except in one space dimension. We also
obtain the corresponding minimal regularization rate.
  In one space dimension, under suitable conditions on the initial data, we
give a quite detailed description of the behavior of solutions for all $t>0$.
In particular, we show that nonminimal GBU solutions immediately lose the
boundary conditions after the blow-up time and are immediately regularized
after recovering the boundary data. Moreover, both GBU and regularization occur
with the minimal rates, while loss and recovery of boundary data occur with
linear rate. We describe further the intermediate singular life of those
solutions in the time interval between GBU and regularization.
  We also study minimal GBU solutions, for which GBU occurs {\it without} LBC:
those solutions are immediately regularized, but their GBU and regularization
rates are more singular.
  Most of our one-dimensional results crucially depend on zero-number
arguments, which do not seem to have been used so far in the context of
viscosity solutions of Hamilton-Jacobi equations.
