# Quenching for Porous Medium Equations

## Main Article Content

## Abstract

This paper studies the following two porous medium equations with singular boundary conditions. First, we obtain that finite time quenching on the boundary, as well as $k_{t}\ $blows up at the same finite time\ and lower bound estimates of the quenching time of the equation $k_{t}=(k^{n})_{xx}+(1-k)^{-\alpha }$,\ $(x,t)\in (0,L)\times (0,T)\ $with $(k^{n})_{x}\left( 0,t\right) =0$, $\ (k^{n})_{x}\left( L,t\right)=(1-k(L,t))^{-\beta }$,$\ t\in (0,T)\ $and initial function $k\left(x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants. Second, we obtain that finite time quenching on the boundary, as well as $k_{t}\ $blows up at the same finite time\ and a local existence result by the help of steady state of the equation $k_{t}=(k^{n})_{xx}$,\ $(x,t)\in (0,L)\times (0,T)\ $with $(k^{n})_{x}\left( 0,t\right) =(1-k(0,t))^{-\alpha }$, $\ (k^{n})_{x}\left(L, t\right) =(1-k(L,t))^{-\beta }$,$\ t\in (0,T)\ $and initial function $k\left( x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants.

## Article Details

*Tamkang Journal of Mathematics*,

*53*(2), 175–185. https://doi.org/10.5556/j.tkjm.53.2022.3853

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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