Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion

TL;DR

This paper investigates boundary gradient blow-up phenomena in a Hamilton-Jacobi equation with nonlinear p-Laplacian diffusion, demonstrating that under certain conditions, the blow-up occurs at a single boundary point for p>2.

Contribution

It extends the understanding of boundary gradient blow-up to nonlinear diffusion cases with p>2, showing single-point blow-up for localized, monotone initial data.

Findings

Gradient blow-up occurs at a single boundary point for p>2.

The result generalizes previous linear diffusion cases to nonlinear p-Laplacian diffusion.

Analysis requires handling more delicate nonlinear diffusion effects.

Abstract

We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion $u_t=\Delta_p u+|\nabla u|^q$ in a two-dimensional domain for $q>p>2$. It is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded. We show that, for suitably localized and monotone initial data, the gradient blow-up occurs at a single point of the boundary. Such a result was known up to now only in the case of linear diffusion ($p=2$). The analysis in the case $p>2$ is considerably more delicate.