Harnack-type estimates and extinction in finite time for a class of anisotropic porous medium type equations

In this work we are interested in the study of a class of anisotropic porous medium-type equations whose prototype is $$u_t=\sum _{i=1}^N{\left(m_i{u}^{m_i-1}{u}_{x_i}\right)}_{x_i},0<m_1\le \cdots \le m_N<1,$$ for which we derive several estimates, namely two Harnack-type inequalities; and, when considering the associated Dirichlet problem, we determine the finite time of extinction and thereby present a decay rate of extinction.