- Hacettepe Journal of Mathematics and Statistics
- Vol: 50 Issue: 2
- Blow up for non-Newtonian equations with two nonlinear sources
Blow up for non-Newtonian equations with two nonlinear sources
Authors : Burhan Selçuk
Pages : 541-548
Doi:10.15672/hujms.653805
View : 12 | Download : 5
Publication Date : 2021-04-11
Article Type : Research
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.Keywords : Heat equation, Nonlinear parabolic equation, blow up, maximum principles