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

TL;DR

This work establishes a Liouville-type symmetry for the elliptic equation $-\Delta u=|\nabla u|^p$ in the half-space with $p>2$, showing all solutions are one-dimensional in the normal direction and enabling explicit final profiles. It then translates this elliptic rigidity into sharp, universal descriptions of gradient blow-up for the parabolic problem in general domains, proving that near the boundary the solution obeys an ODE-type normal behavior with $u_{\nu\nu}\sim -|u_\nu|^p$ and that the normal blow-up profile is universal while tangential directions are more singular. The analysis yields optimal Bernstein-type bounds, a refined elliptic bound in the inhomogeneous case, and a rigorous description of the post-GBU continuation via global viscosity solutions, including generic loss of boundary conditions after GBU. Additionally, the results generalize one-dimensional threshold phenomena to multi-dimensional domains and provide a detailed space-time characterization of the GBU set and boundary behavior. Overall, the paper connects Liouville theory, boundary layer analysis, and viscosity solution techniques to illuminate the gradient blow-up mechanism for superquadratic diffusive Hamilton-Jacobi equations in bounded domains.

Abstract

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations with 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) 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 blow-up 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 space-dimension.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.2
• Remark 1.3
• proof