Sharp decay estimates and vanishing viscosity for diffusive hamilton-jacobi equations

Sharp temporal decay estimates are established for the gradient and time derivative of solutions to the Hamilton-Jacobi equation ∂_tυ_ε + H({pipe}▶_xυ_ε{pipe}) = εΔυ_ε in R^N × (0,∞), the parameter ε being either positive or zero. Special care is given to the dependence of the estimates on ε. As a by-product, we obtain convergence of the sequence (υ_ε) as ε→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 ∈ [0,∞) and p ∈ (0,∞), p ≠ 1.