A Liouvilletype theorem in a halfspace and its applications to the gradient blowup behavior for superquadratic diffusive HamiltonJacobi equations
Abstract
We consider the elliptic and parabolic superquadratic diffusive HamiltonJacobi equations with homogeneous Dirichlet conditions. For the elliptic problem in a halfspace, we prove a Liouvilletype classification, or symmetry result, which asserts that any solution has to be onedimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blowup (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 blowup 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 spacedimension.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.05161
 Bibcode:
 2019arXiv190605161F
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 27 pages