Boundary Estimates for Certain Degenerate and Singular Parabolic Equations

TL;DR

This paper investigates the boundary behavior of solutions to degenerate and singular parabolic equations, establishing Carleson-type estimates and their implications for equations like the parabolic p-Laplacian and porous medium type.

Contribution

It provides new boundary estimates for degenerate and singular parabolic equations, extending classical results to more general and complex equations.

Findings

Established Carleson-type boundary estimates for solutions.

Derived consequences under additional assumptions on equations and domains.

Extended boundary behavior analysis to porous medium type equations.

Abstract

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.