# 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 paper proves a Liouville-type theorem in a half-space and applies it to analyze boundary gradient blow-up in superquadratic diffusive Hamilton-Jacobi equations, revealing universal blow-up profiles and boundary condition loss.

## Contribution

It establishes a one-dimensional symmetry result for elliptic solutions in a half-space and applies this to understand boundary blow-up behavior in higher dimensions.

## Key findings

- Solutions near the boundary exhibit a universal blow-up profile in the normal direction.
- The tangential derivatives dominate the normal derivatives at blow-up points.
- Viscosity solutions generally lose boundary conditions after blow-up.

## 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.