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.

Keywords

• Heat equation
• Nonlinear parabolic equation
• blow up
• maximum principles

References

1. [1] Z. Cui, Critical curves of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions, Nonlinear Anal. 68 (10), 3201–3208, 2008.
2. [2] S.C. Fu, J.S. Guo and J-C. Tsai, Blow-up behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoku Math. J. (2), 55 (4), 565–581, 2003.
3. [3] Y. Giga and R.V. Kohn, Asymptotic self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38, 297–319, 1985.
4. [4] Z. Liang, On the blow-up set for non-Newtonian equation with a nonlinear boundary condition, Abstr. Appl. Anal. 2010, 1–12, 2010.
5. [5] Z. Lin and M. Wang, The blow up properties of solutions to semilinear heat equation with nonlinear boundary conditions, Z. Angew. Math. Phys. 50, 361–374, 1999.
6. [6] E. Novruzov, On blow up of solution of nonhomogeneous polytropic equation with source, Nonlinear Anal. 71 (9), 3992–3998, 2009.
7. [7] N. Ozalp and B. Selcuk, Blow up and quenching for a problem with nonlinear boundary conditions, Electron. J. Differential Equations, 2015 (192), 1–11, 2015.
8. [8] Z. Wang, J. Yin and W.C. Wang, Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition, Appl. Math. Lett. 20, 142–147, 2007.

9. [9] Z.Yang and Q. Lu, Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a non-Newtonian filtration system, Appl. Math. Lett. 16 (4), 581–587, 2003.
10. [10] H. Zhang, Z. Liu and W. Zhan, Growth estimates and blow-up in quasilinear parabolic problems, Appl. Anal. 86 (2), 261–268, 2007.
11. [11] J. Zhou, Global existence and blow-up of solutions for a Non-Newton polytropic filtra- tion system with special volumetric moisture content, Comput. Math. Appl. 71 (5), 1163–1172, 2016.