Uniqueness of sign-changing solutions to Trudinger's equation

TL;DR

The paper addresses the problem of uniqueness for sign-changing weak solutions to Trudinger's equation $\partial_t(|u|^{p-2}u) = \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ in a bounded Lipschitz domain with time-dependent $C^2$ Dirichlet boundary data $\psi$. It extends Otto's time-independent boundary results by employing a Kružkov doubling-of-variables approach adapted to moving boundaries and introduces auxiliary barrier solutions to squeeze any candidate solution to a unique limit as $\varepsilon \to 0$. The main contribution is a rigorous proof of uniqueness for sign-changing solutions in $L^p(0,T;W^{1,p}(\Omega))$ under $\psi\in C^2(\overline{\Omega_T})$, including the Sobolev time-derivative regularity $\partial_t(|u|^{(p-2)/2}u)\in L^2(\Omega_T)$ for $p\ge 2$ and $\partial_t(|u|^{p-2}u)\in L^p(\Omega_T)$. This work advances the understanding of Trudinger's equation in the sign-changing regime and provides a robust framework for uniqueness under general time-dependent boundary data, with implications for the regularity of nonlinear parabolic flows.

Abstract

We establish uniqueness for sign-changing solutions to Trudinger's parabolic equation with time dependent $C^2$ Dirichlet boundary data.

Theorems & Definitions (14)

• Definition 2.1
• Theorem 3.1: Comparison Principle
• Lemma 3.2
• proof
• Lemma 3.3: Key lemma
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof