Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion

TL;DR

This paper investigates the well-posedness and boundary gradient blow-up behavior of solutions to a degenerate viscous Hamilton-Jacobi equation with Dirichlet boundary conditions, providing precise boundary estimates and regularizing effects.

Contribution

It establishes local well-posedness with blow-up criteria, derives boundary-specific gradient estimates, and demonstrates that gradient blow-up occurs only at the boundary for the first time.

Findings

Gradient blow-up occurs only on the boundary.

Established local well-posedness with blow-up alternative.

Derived a boundary distance-based gradient estimate.

Abstract

This paper is concerned with weak solutions of the degenerate viscous Hamilton-Jacobi equation $$\partial_t u-\Delta_p u=|\nabla u|^q,$$ with Dirichlet boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$, where $p>2$ and $q>p-1$. With the goal of studying the gradient blow-up phenomenon for this problem, we first establish local well-posedness with blow-up alternative in $W^{1, \infty}$ norm. We then obtain a precise gradient estimate involving the distance to the boundary. It shows in particular that the gradient blow-up can take place only on the boundary. A regularizing effect for $u_t$ is also obtained.