Gradient estimates for elliptic regularizations of semilinear parabolic and degenerate elliptic equations
Résumé
This article is chiefly concerned with elliptic regularizations of semilinear parabolic equations of the type $$\varepsilon u_{tt}-u_t+Lu+f(u)=0$$ where $L$ is an elliptic operator in the space variables $x$. We establish $L^{\infty}$ gradient estimates up to the boundary which 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 a previous work \cite{bh}.\par 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 $$L_xu+\varepsilon L_{xy}u+\beta(x,y)\cdot\nabla_{x,y}u+f(x,y,u)=0$$ of degenerate elliptic equations.