  We consider the diffusive Hamilton-Jacobi equation, with homogeneous
Dirichlet conditions and regular initial data. It is known from [Barles-DaLio,
2004] that the problem admits a unique, continuous, global viscosity solution,
which extends the classical solution in case gradient blowup occurs. We study
the question of the possible loss of boundary conditions after gradient blowup,
which seems to have remained an open problem till now.
  Somewhat surprisingly, our results show that the issue strongly depends on
the initial data and reveal a rather rich variety of phenomena. For any smooth
bounded domain, we construct initial data such that the loss of boundary
conditions occurs everywhere on the boundary, as well as initial data for which
no loss of boundary conditions occurs in spite of gradient blowup. Actually, we
show that the latter possibility is rather exceptional. More generally, we show
that the set of the points where boundary conditions are lost, can be
prescribed to be arbitrarily close to any given open subset of the boundary.
