Advances in Differential Equations

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

Matania Ben-Artzi, Saïd Benachour, and Philippe Laurençot

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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

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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.

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