Research Article
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Year 2021, , 22 - 27, 01.03.2021
https://doi.org/10.36753/mathenot.686065

Abstract

References

  • [1] Chan, C.Y., Yuen, S.I.: Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput., 121, 203-209 (2001).
  • [2] Deng, K., Xu, M.: Remarks on blow-up behavior for a nonlinear diffusion equation with neumann boundary conditions, Proceedings of the American Mathematical Society, 127 (1), 167-172 (1999).
  • [3] Deng, K., Xu, M.: Quenching for a nonlinear diffusion equation with singular boundary condition, Z. Angew. Math. Phys., vol. 50, no. 4, (1999) 574-584.
  • [4] Ferreira, R., Pablo, A.D., Quiros, F., Rossi, J.D.: The blow-up profile for a fast diffusion equation with a nonlinear boundary condition, Rocky Mountain Journal of Mathematics, 33 (1), Spring 2003.
  • [5] Friedman, A., Mcleod, B.: Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-477 (1985).
  • [6] Fu, S.C., Guo, J.-S., Tsai, J.C.: Blow up behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoku Math. J., 55, 565-581 (2003).
  • [7] Jiang, Z., Zheng, S., Song, X.: Blow-up analysis for a nonlinear diffusion equation with a nonlinear boundary conditions, Applied Mathematics Letters, 17, 193-199 (2004).
  • [8] Ozalp, N., Selcuk, B.: Blow-up and quenching for a problem with nonlinear boundary conditions, Electron. J. Diff. Equ., 2015 (192), 1-11 (2015).
  • [9] Pao, C.V.: Singular reaction diffusion equations of porous medium type, Nonlinear Analysis, 71, 2033-2052 (2009).
  • [10] Vazquez, J.L.: The porous medium equation: Mathematical Theory, Oxford Science Publications, (2007).
  • [11] Zhang, Z., Li, Y.: Quenching rate for the porous medium equation with a singular boundary condition, Applied Mathematics., 2, 1134-1139 (2011).

Blow up for Porous Medium Equations

Year 2021, , 22 - 27, 01.03.2021
https://doi.org/10.36753/mathenot.686065

Abstract

In various branches of applied sciences, porous medium equations exist where this basic model occurs in a natural fashion. It has been used to model fluid flow, chemical reactions, diffusion or heat transfer, population dynamics, etc.. Nonlinear diffusion equations involving the porous medium equations have also been extensively studied. However, there has not been much research effort in the parabolic problem for porous medium equations with two nonlinear boundary sources in the literature. This paper adresses the following porous medium equations with nonlinear boundary conditions. Firstly, we obtain finite time blow up on the boundary by using the maximum principle and blow up criteria and existence criteria by using steady state of the equation $k_{t}=k_{xx}^{n},(x,t)\in (0,L)\times (0,T)\ $with $ k_{x}^{n}(0,t)=k^{\alpha }(0,t)$, $k_{x}^{n}(L,t)=k^{\beta }(L,t)$,$\ t\in (0,T)\ $and initial function $k\left( x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants.

References

  • [1] Chan, C.Y., Yuen, S.I.: Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput., 121, 203-209 (2001).
  • [2] Deng, K., Xu, M.: Remarks on blow-up behavior for a nonlinear diffusion equation with neumann boundary conditions, Proceedings of the American Mathematical Society, 127 (1), 167-172 (1999).
  • [3] Deng, K., Xu, M.: Quenching for a nonlinear diffusion equation with singular boundary condition, Z. Angew. Math. Phys., vol. 50, no. 4, (1999) 574-584.
  • [4] Ferreira, R., Pablo, A.D., Quiros, F., Rossi, J.D.: The blow-up profile for a fast diffusion equation with a nonlinear boundary condition, Rocky Mountain Journal of Mathematics, 33 (1), Spring 2003.
  • [5] Friedman, A., Mcleod, B.: Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-477 (1985).
  • [6] Fu, S.C., Guo, J.-S., Tsai, J.C.: Blow up behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoku Math. J., 55, 565-581 (2003).
  • [7] Jiang, Z., Zheng, S., Song, X.: Blow-up analysis for a nonlinear diffusion equation with a nonlinear boundary conditions, Applied Mathematics Letters, 17, 193-199 (2004).
  • [8] Ozalp, N., Selcuk, B.: Blow-up and quenching for a problem with nonlinear boundary conditions, Electron. J. Diff. Equ., 2015 (192), 1-11 (2015).
  • [9] Pao, C.V.: Singular reaction diffusion equations of porous medium type, Nonlinear Analysis, 71, 2033-2052 (2009).
  • [10] Vazquez, J.L.: The porous medium equation: Mathematical Theory, Oxford Science Publications, (2007).
  • [11] Zhang, Z., Li, Y.: Quenching rate for the porous medium equation with a singular boundary condition, Applied Mathematics., 2, 1134-1139 (2011).
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Burhan Selçuk 0000-0002-5141-5148

Publication Date March 1, 2021
Submission Date February 7, 2020
Acceptance Date December 18, 2020
Published in Issue Year 2021

Cite

APA Selçuk, B. (2021). Blow up for Porous Medium Equations. Mathematical Sciences and Applications E-Notes, 9(1), 22-27. https://doi.org/10.36753/mathenot.686065
AMA Selçuk B. Blow up for Porous Medium Equations. Math. Sci. Appl. E-Notes. March 2021;9(1):22-27. doi:10.36753/mathenot.686065
Chicago Selçuk, Burhan. “Blow up for Porous Medium Equations”. Mathematical Sciences and Applications E-Notes 9, no. 1 (March 2021): 22-27. https://doi.org/10.36753/mathenot.686065.
EndNote Selçuk B (March 1, 2021) Blow up for Porous Medium Equations. Mathematical Sciences and Applications E-Notes 9 1 22–27.
IEEE B. Selçuk, “Blow up for Porous Medium Equations”, Math. Sci. Appl. E-Notes, vol. 9, no. 1, pp. 22–27, 2021, doi: 10.36753/mathenot.686065.
ISNAD Selçuk, Burhan. “Blow up for Porous Medium Equations”. Mathematical Sciences and Applications E-Notes 9/1 (March 2021), 22-27. https://doi.org/10.36753/mathenot.686065.
JAMA Selçuk B. Blow up for Porous Medium Equations. Math. Sci. Appl. E-Notes. 2021;9:22–27.
MLA Selçuk, Burhan. “Blow up for Porous Medium Equations”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 1, 2021, pp. 22-27, doi:10.36753/mathenot.686065.
Vancouver Selçuk B. Blow up for Porous Medium Equations. Math. Sci. Appl. E-Notes. 2021;9(1):22-7.

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