The impact of intrinsic scaling on the rate of extinction for anisotropic non-Newtonian fast diffusion

We study the decay towards the extinction that pertains to local weak solutions to fully anisotropic equations whose prototype is $${\mathrm{\partial }}_{t}u=\sum _{i=1}^N{\mathrm{\partial }}_i(\mathrm{\mid }{\mathrm{\partial }}_{i}u{\mathrm{\mid }}^{p_i-2}{\mathrm{\partial }}_{i}u),1<p_i<2.$$ 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.