# Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions

## TL;DR

This paper investigates the quenching behavior of a non-Newtonian filtration equation with singular boundary conditions, providing theoretical estimates and numerical validation for quenching rates and times.

## Contribution

It offers new theoretical insights and numerical validation for quenching phenomena in a non-Newtonian filtration model with singular boundary conditions.

## Key findings

- Conditions for quenching at boundaries are established.
- Theoretical quenching rates and bounds are derived for specific cases.
- Numerical experiments confirm the theoretical estimates.

## Abstract

In this paper, the quenching behavior of the non-Newtonian filtration equation $(\phi (u))_{t}=(\left \vert u_{x}\right \vert ^{r-2}u_{x})_{x}$ with singular boundary conditions, $u_{x}\left( 0,t\right) =u^{-p}(0,t)$, $u_{x}\left( a,t\right) =(1-u(a,t))^{-q}$ is investigated. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined when $\phi(u)=u$ and $r=2$. Numerical experiments are provided to illustrate and provide additional validation of the theoretical estimates to the quenching rates and times.