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

Journal of Differential Equations 253 (2012) 2474–2492This paper is concerned with weak solutions of the degenerate diffusive Hamilton–Jacobi equation with Dirichlet boundary conditions in a bounded domain Ω ⊂ RN , 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 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 is also obtained.