Qualitative properties of solutions to parabolic anisotropic equations: Part II -- The anisotropic Trudinger's equation

TL;DR

This paper establishes a parabolic Harnack inequality and Hölder continuity for solutions to a class of anisotropic doubly nonlinear parabolic equations, advancing understanding of their regularity properties without exponent restrictions.

Contribution

It proves a Harnack inequality and Hölder continuity for solutions to anisotropic Trudinger's equations, extending regularity results without restrictions on exponents.

Findings

Proved a parabolic Harnack inequality for nonnegative solutions.

Established Hölder continuity for solutions within certain exponent ranges.

Extended regularity theory to anisotropic equations with arbitrary exponents.

Abstract

In this paper we study the local regularity properties of weak solutions to a special class of anisotropic doubly nonlinear parabolic operators, whose prototype is the anisotropic Trudinger's equation $$ u_t- \sum\limits_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2} D_i u\Big)=0,\quad u\geqslant 0. $$ We prove a parabolic Harnack inequality for nonnegative local weak solutions, without any restrictions on the sparseness of the exponents $p_i$s. Moreover, for a restricted range of diffusion exponents, we prove that solutions are H\"{o}lder continuous.