Blow up for non-Newtonian equations with two nonlinear sources

Authors : Burhan SELÇUK

Pages : 541-548

Doi:10.15672/hujms.653805

View : 53 | Download : 10

Publication Date : 2021-04-11

Article Type : Research Paper

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 $k_{t}=insert ignore into journalissuearticles values(\left \vert k_{x}\right \vert ^{r-2}k_{x});_{x}$, $insert ignore into journalissuearticles values(x,t);\in insert ignore into journalissuearticles values(0,L);\times insert ignore into journalissuearticles values(0,T);\ $with $k_{x}insert ignore into journalissuearticles values(0,t);=k^{\alpha }insert ignore into journalissuearticles values(0,t);$, $k_{x}insert ignore into journalissuearticles values(L,t);=k^{\beta }insert ignore into journalissuearticles values(L,t);$,$\ t\in insert ignore into journalissuearticles values(0,T);\ $and initial function $k\leftinsert ignore into journalissuearticles values(x,0\right); =k_{0}\leftinsert ignore into journalissuearticles values( 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 $k_{t}=insert ignore into journalissuearticles values(\left \vert k_{x}\right \vert ^{r-2}k_{x});_{x}+k^{\alpha }$, $insert ignore into journalissuearticles values(x,t);\in insert ignore into journalissuearticles values(0,L);\times insert ignore into journalissuearticles values(0,T);\ $with $k_{x}insert ignore into journalissuearticles values(0,t);=0$, $k_{x}insert ignore into journalissuearticles values(L,t);=k^{\beta }insert ignore into journalissuearticles values(L,t);$,$\ t\in insert ignore into journalissuearticles values(0,T);\ $ and initial function $k\leftinsert ignore into journalissuearticles values( x,0\right); =k_{0}\leftinsert ignore into journalissuearticles values( 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

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