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 $${\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{p}u=\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q,$$ with Dirichlet boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, 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 $W^{1,\mathrm{\infty }}$ 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.