Gradient Estimates for Elliptic Regularizations of Semilinear Parabolic and Degenerate Elliptic Equations

Abstract

This article is chiefly concerned with elliptic regularizations of semilinear parabolic equations of the type where L is an elliptic operator in the space variables x. We establish L ∞ gradient estimates up to the boundary that are uniform with respect to the small elliptic regularization parameter ϵ. Such estimates were used for instance in proving the existence of pulsating travelling front solutions for reaction–diffusion equations in Berestycki and Hamel (2002). Similar x-gradient estimates are also obtained, both in the interior of the domain and up to the boundary, for elliptic (in (x, y) variables) regularizations of degenerate elliptic equations.