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 ∂tvε + H(|∇xvε|) = ε ∆vε in R × (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 (vε) as ε → 0 to a viscosity solution, the initial condition being only continuous and either bounded or non-negative. The main requirement on H is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r for r ∈ [0,∞) and p ∈ (0,∞), p 6= 1.