Blow up for Porous Medium Equations

Year 2021, Volume: 9 Issue: 1, 22 - 27, 01.03.2021

Burhan Selçuk

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.

Keywords

Heat equation, Nonlinear parabolic equation, nonlinear boundary condition, blow up, maximum principles

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).