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

This paper is concerned with weak solutions of the degenerate viscous Hamilton-Jacobi equation $\partial_t u-Δ_p u=|\nabla u|^q,$ with Dirichlet boundary conditions in a bounded domain $Ω\subset\mathbb{R}^N$, 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.

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