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

2012/01/17 by Attouchi, Amal · 1 citation
#35A01 #35B44 #35K92 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1201.3566

Abstract

This paper is concerned with weak solutions of the degenerate viscous Hamilton-Jacobi equation ∂_t u-Δ_p u=|∇ 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 u_t is also obtained.

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