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 diffusive Hamilton–Jacobi equation∂tu−Δpu=|∇u|q,∂tu−Δpu=|∇u|q, with Dirichlet boundary conditions in a bounded domain Ω⊂RNΩ⊂RN, where p>2p>2 and q>p−1q>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 W1,∞W1,∞ 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 ∂tu∂tu is also obtained.