Blow up for non-Newtonian equations with two nonlinear sources

Abstract

This paper studies the following two non-Newtonian equations with nonlinear boundary conditions. Firstly, we show that finite time blow up occurs on the boundary and we get a blow up rate and an estimate for the blow up time of the equation $k_{t}=(\left \vert k_{x}\right \vert ^{r-2}k_{x})_{x}$, $(x,t)\in (0,L)\times (0,T)\ $with $k_{x}(0,t)=k^{\alpha }(0,t)$, $k_{x}(L,t)=k^{\beta }(L,t)$,$\ t\in (0,T)\ $and initial function $k\left(x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $r\geq 2$, $\alpha ,\beta \ $and $L\ $are positive constants. Secondly, we show that finite time blow up occurs on the boundary, and we get blow up rates and estimates for the blow up time of the equation $k_{t}=(\left \vert k_{x}\right \vert ^{r-2}k_{x})_{x}+k^{\alpha }$, $(x,t)\in (0,L)\times (0,T)\ $with $k_{x}(0,t)=0$, $k_{x}(L,t)=k^{\beta }(L,t)$,$\ t\in (0,T)\ $ and initial function $k\left( x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $r\geq 2$, $\alpha ,\beta$ and $L$ are positive constants.