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

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.