Quenching for Porous Medium Equations

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

How to Cite
Selcuk, B. (2022). Quenching for Porous Medium Equations. Tamkang Journal of Mathematics, 53(2), 175–185. https://doi.org/10.5556/j.tkjm.53.2022.3853
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Section
Papers
Author Biography

Burhan Selcuk, Department of Computer Engineering Karabuk University Karabuk / TURKEY

Assist. Prof. Dr. / Computer Engineering Department

References

C. Y. Chan, S. I. Yuen, Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput., 121 (2001), 203–209.

K. Deng and M. Xu, Quenching for a nonlinear diffusion equation with singular boundary condition, Z. Angew. Math. Phys., 50 (4) (1999), 574–584.

Z. Jiang, S. Zheng and X. Song, Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions, Appl. Math. Letters, 17 (2004), 193–199.

N. Ozalp and B. Selcuk, Blow up and quenching for a problem with nonlinear boundary conditions, Electron. J. Diff. Equ., 2015 (192) (2015), 1–11.

N. Ozalp and B. Selcuk, The quenching behavior of a nonlinear parabolic equation with a singular boundary condition, Hacettepe Journal of Mathematics and Statistics, 44 (3) (2015), 615–621.

C. V. Pao, Singular reaction diffusion equations of porous medium type, Nonlinear Analysis, 71 (2009), 2033–2052.

B. Selcuk and N. Ozalp, The quenching behavior of a semilinear heat equation with a singular boundary outflux, Quart. Appl. Math., 72 (4) (2014), 747–752.

B. Selcuk and N. Ozalp, Quenching behavior of semilinear heat equations with singular boundary conditions, Electron. J. Diff. Equ., 2015 (311) (2015), 1–13.

B. Selcuk and N. Ozalp, Quenching behavior of nonlinear diffusion equation with singular boundary outfluxes, Turkish Journal of Mathematics and Computer Science, 8 (2018), 65– 69.

J. L. Vazquez, The porous medium equation: Mathematical Theory, Oxford Science Publi- cations, (2007).

Z. Zhang and Y. Li, Quenching rate for the porous medium equation with a singular boundary condition, Applied Mathematics, 2 (2011), 1134–1139.

Y. Zhi and C. Mu, The quenching behavior of a nonlinear parabolic equation with a nonlinear boundary outflux, Appl. Math. Comput, 184 (2007), 624–630.

L. Zhu, The quenching behavior of a quasilinear parabolic equation with double singular sources, C. R. Acad. Sci. Paris, Ser. I, 356 (7) (2018), 725–731.