The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation

TL;DR

This paper characterizes the precise boundary gradient blow-up profile for the diffusive Hamilton-Jacobi equation in two dimensions, revealing a novel anisotropic behavior and scale invariance violation in the tangential direction.

Contribution

It provides the first detailed asymptotic description of the boundary profile in the tangential direction for solutions with gradient blow-up, highlighting new anisotropic phenomena.

Findings

Derived the exact asymptotic boundary profile near blow-up point.

Discovered strong anisotropy with different exponents in normal and tangential directions.

Showed the tangential profile violates scale invariance, unlike the normal profile.

Abstract

We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the precise final asymptotic profile, under the form $$u_y(x,y,T) \sim d_p\Bigl[y+C|x|^{2(p-1)/(p-2)}\Bigr]^{-1/(p-1)},\quad\hbox{as $(x,y)\to (0,0)$.}$$ Interestingly, this result displays a new phenomenon of strong anisotropy of the profile, quite different to what is observed in other blowup problems for nonlinear parabolic equations, with the exponents $1/(p-1)$ in the normal direction $y$ and $2/(p-2)$ in the tangential direction $x$. Furthermore, the tangential profile violates the (self-similar) scale invariance of the equation, whereas the normal profile remains self-similar.