Gradient estimates for elliptic regularizations of semilinear parabolic and degenerate elliptic equations

This article is chiefly concerned with elliptic regularizations of semilinear parabolic equations of the type epsilon u(n) - u(t) + Lu + f(u) = 0, where L is an elliptic operator in the space variables x. We establish L-infinity gradient estimates lip to the boundary that are uniform with respect to the small elliptic regularization parameter epsilon. 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 L(x)u + epsilon L(xy)u + beta(x, y) (.) del(x,y)u + f(x, y, u) = 0 of degenerate elliptic equations.