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We consider the diffusive Hamilton-Jacobi equation, with homogeneousDirichlet 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 studythe 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 onthe initial data and reveal a rather rich variety of phenomena. For any smoothbounded domain, we construct initial data such that the loss of boundaryconditions occurs everywhere on the boundary, as well as initial data for whichno loss of boundary conditions occurs in spite of gradient blowup. Actually, weshow that the latter possibility is rather exceptional. More generally, we showthat the set of the points where boundary conditions are lost, can beprescribed to be arbitrarily close to any given open subset of the boundary.

Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation