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

摘要

This paper is concerned with weak solutions of the degenerate diffusive Hamilton-Jacobi equation ? _t u-δ_pu=|~?;u| q, with Dirichlet boundary conditions in a bounded domain Ω?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,∞) 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.