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

Abstract

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 $p\in (0,\infty)$, $p\ne 1$.