Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in \( \mathbb{R}^{N} \)

Abstract.

The large time behaviour of the \( L^q \)-norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶\(u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,\)¶¶is studied for \( q=1 \) and \( q=\infty \), where \( m\in\{1,\ldots,N\} \) and \( p_i\in [1,+\infty) \) for \( i\in\{1,\ldots,m\} \). The limit of the \( L^1 \)-norm is identified, and temporal decay estimates for the \( L^\infty \)-norm are obtained, according to the values of the $ p_i $'s. The main tool in our approach is the derivation of \( L^\infty \)-decay estimates for \( \nabla\left(u^\alpha \right), \alpha\in (0,1] \), by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.