Decay estimates for “anisotropic” viscous Hamilton-Jacobi equations in ℝ^N

Abstract

The large time behaviour of the L ^q-norm of nonnegative solutions to the “anisotropic” viscous Hamilton-Jacobi equation

$$ {u_{t}} - \Delta u + {\sum\limits_{{i = 1}}^{m} {|{u_{{xi}}}|} ^{{Pi}}} = 0 in {\mathbb{R}_{ + }} x {\mathbb{R}^{N}}, $$

is studied for q = 1 and q = ∞, where m ∈ {1,...,N} and p _i for i ∈ {1,...,m}. The limit of theL ¹-norm is identified, and temporal decay estimates for the L ^∞-norm are obtained, according to the values of the p _i ’s. The main tool in our approach is the derivation of L^∞-decay estimates for $\nabla ({u^{\alpha }}),\alpha \in (0,1]$, by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.