Boundary Estimates for Doubly Nonlinear Parabolic Equations

TL;DR

This paper establishes boundary regularity estimates, decay rates, and Harnack inequalities for non-negative solutions to a class of doubly nonlinear parabolic equations in the super-critical fast diffusion regime.

Contribution

It provides new boundary estimates and decay properties for solutions to doubly nonlinear parabolic equations, extending understanding in the super-critical regime.

Findings

Solutions satisfy Carleson estimates at the boundary.

Under boundary regularity, solutions exhibit power-like decay.

A boundary Harnack inequality is established.

Abstract

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions vanish continuously at the Lipschitz boundary of a parabolic cylinder $\Omega_T$, they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain $\Omega_T$, we obtain a power-like decay at the boundary and a boundary Harnack inequality.