We consider the superquadratic diffusive Hamilton-Jacobi equation ut-Δu=|∇u|p with p > 2, in a smooth bounded domains of ℝn (n ≥ 2) under homogeneous Dirichlet conditions, which is known to undergo boundary gradient blow-up (BGBU) phenomena. First, 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 1-dimensional. This turns out to be an efficient tool to study the BGBU behaviour for the parabolic problem. Namely, we show that in a neighbourhood of the boundary, at leading order, solutions display a global ODE type behaviour of the form uνν ~ -|uν|p, 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 BGBU point, and moreover implies that the behaviour in the tangential direction is more singular. In single-point BGBU cases, the tangential profile can be precisely determined under additional assumptions. Related results on the time rates will be also presented. The ODE type behaviour and its connection with the Liouville-type theorem can be considered as an analogue of the well-known results of Merle and Zaag (1998) for the subcritical semilinear heat equation, with the significant difference that for the latter, u itself blows up and the ODE behaviour is in the time direction (instead of the normal spatial direction).
This is based on a series of joint works with R. Filippucci, P. Pucci, A. Porretta, A. Attouchi.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
