On some properties of entropy solutions of degenerate non-linear anisotropic parabolic equations

We prove existence of the largest and smallest entropy solutions to Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish comparison principle in case when at least one initial functions is periodic. In function vanishes infinity (in sense strong average) long time decay an solution under exact nonlinearity-diffusivity condition.