  We study the decay towards the extinction that pertains to local weak
solutions to fully anisotropic equations whose prototype is \[
  \partial_t u= \sum_{i=1}^N \partial_i (|\partial_i u|^{p_i-2} \partial_i u),
\qquad 1<p_i<2. \]
  Their rates of extinction are evaluated by means of several integral
Harnack-type inequalities which constitute the core of our analysis and that
are obtained for anisotropic operators having full quasilinear structure.
Different decays are obtained when considering different space geometries. The
approach is motivated by the research of new methods for strongly nonlinear
operators, hence dispensing with comparison principles, while exploiting an
intrinsic geometry that affects all the variables of the solution.
