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Blow up for Porous Medium Equations

Burhan SELÇUK [1]

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.
Heat equation, Nonlinear parabolic equation, nonlinear boundary condition, blow up, maximum principles
• [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).
Primary Language en Mathematics Articles Orcid: 0000-0002-5141-5148Author: Burhan SELÇUK (Primary Author)Institution: Karabük ÜniversitesiCountry: Turkey Application Date : February 7, 2020 Acceptance Date : December 18, 2020 Publication Date : March 1, 2021
 Bibtex @research article { mathenot686065, journal = {Mathematical Sciences and Applications E-Notes}, issn = {}, eissn = {2147-6268}, address = {}, publisher = {Murat TOSUN}, year = {2021}, volume = {9}, pages = {22 - 27}, doi = {10.36753/mathenot.686065}, title = {Blow up for Porous Medium Equations}, key = {cite}, author = {Selçuk, Burhan} } APA Selçuk, B . (2021). Blow up for Porous Medium Equations . Mathematical Sciences and Applications E-Notes , 9 (1) , 22-27 . DOI: 10.36753/mathenot.686065 MLA Selçuk, B . "Blow up for Porous Medium Equations" . Mathematical Sciences and Applications E-Notes 9 (2021 ): 22-27 Chicago Selçuk, B . "Blow up for Porous Medium Equations". Mathematical Sciences and Applications E-Notes 9 (2021 ): 22-27 RIS TY - JOUR T1 - Blow up for Porous Medium Equations AU - Burhan Selçuk Y1 - 2021 PY - 2021 N1 - doi: 10.36753/mathenot.686065 DO - 10.36753/mathenot.686065 T2 - Mathematical Sciences and Applications E-Notes JF - Journal JO - JOR SP - 22 EP - 27 VL - 9 IS - 1 SN - -2147-6268 M3 - doi: 10.36753/mathenot.686065 UR - https://doi.org/10.36753/mathenot.686065 Y2 - 2020 ER - EndNote %0 Mathematical Sciences and Applications E-Notes Blow up for Porous Medium Equations %A Burhan Selçuk %T Blow up for Porous Medium Equations %D 2021 %J Mathematical Sciences and Applications E-Notes %P -2147-6268 %V 9 %N 1 %R doi: 10.36753/mathenot.686065 %U 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 AMA Selçuk B . Blow up for Porous Medium Equations. Math. Sci. Appl. E-Notes. 2021; 9(1): 22-27. Vancouver Selçuk B . Blow up for Porous Medium Equations. Mathematical Sciences and Applications E-Notes. 2021; 9(1): 22-27. IEEE B. Selçuk , "Blow up for Porous Medium Equations", Mathematical Sciences and Applications E-Notes, vol. 9, no. 1, pp. 22-27, Mar. 2021, doi:10.36753/mathenot.686065

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