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

TL;DR

This paper investigates anisotropic porous medium equations, deriving Harnack inequalities and establishing finite-time extinction results along with decay rates, enhancing understanding of solution behavior in these nonlinear PDEs.

Contribution

It introduces new Harnack-type estimates and characterizes finite-time extinction for anisotropic porous medium equations with multiple exponents.

Findings

Derived two Harnack-type inequalities for the equations.

Established finite time of extinction for solutions.

Provided decay rates for the extinction time.

Abstract

In this work we are interested in the study of a class of anisotropic porous medium-type equations whose prototype is \[ u_t =\sum_{i=1}^N \left( m_i u^{m_i-1} u_{x_i} \right)_{x_i} \ , \qquad 0<m_1 \leq \cdots \leq m_N <1 \ , \] 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.