Advances in Differential Equations

Sharp decay estimates and vanishing viscosity for diffusive Hamilton-Jacobi equations

Matania Ben-Artzi, Saïd Benachour, and Philippe Laurençot

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Sharp temporal decay estimates are established for the gradient and time derivative of solutions to the Hamilton-Jacobi equation $\partial_t {v_\varepsilon} + H(|\nabla_x {v_\varepsilon} |)= \varepsilon \Delta {v_\varepsilon}$ in ${\mathbb{R}^N\times(0,\infty)}$, the parameter $\varepsilon$ being either positive or zero. Special care is given to the dependence of the estimates on $\varepsilon$. As a by-product, we obtain convergence of the sequence $({v_\varepsilon})$ as $\varepsilon\to 0$ to a viscosity solution, the initial condition being only continuous and either bounded or nonnegative. The main requirement on $H$ is that it grows superlinearly or sublinearly at infinity, including in particular $H(r)=r^p$ for $r\in [0,\infty)$ and $p\in (0,\infty)$, $p\ne 1$.

Article information

Adv. Differential Equations, Volume 14, Number 1/2 (2009), 1-25.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35F25: Initial value problems for nonlinear first-order equations 35K15: Initial value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 49L25: Viscosity solutions


Benachour, Saïd; Ben-Artzi, Matania; Laurençot, Philippe. Sharp decay estimates and vanishing viscosity for diffusive Hamilton-Jacobi equations. Adv. Differential Equations 14 (2009), no. 1/2, 1--25.

Export citation