A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations: and with p > 2 and 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) solutions of the parabolic problem in general bounded domains of with smooth boundaries. Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior of the form 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. A description of the space-time profile is also obtained. The ODE type behavior and its connection with the Liouville-type theorem can be considered as an analog of the well-known results of Merle and Zaag (1998) for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction). 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 generically lose boundary conditions after GBU. Namely, solutions without loss of boundary conditions after GBU are exceptional and can be characterized as thresholds between global classical solutions and GBU solutions which lose boundary conditions. This result, as well as the above GBU profile, was up to now essentially known only in one space-dimension. Finally, in the case of elliptic Dirichlet problems, we deduce from our Liouville theorem an optimal Bernstein-type estimate, which gives a partial improvement of a local estimate of P.-L. Lions (1985).