Quenching Rate for the Porous Medium Equation with a Singular Boundary Condition

ABSTRACT

We study the porous medium equation*u*_{t}=(*u*^{m}). 0＜*x*＜∞, *t*＞0 with a singular boundary condition (*u*^{m}) (0,*t*)=*u*^{-β}(0,*t*). We prove finite time quenching for the solution at the boundary *χ*=0. We also establish the quenching rate and asymptotic behavior on the quenching point.

We study the porous medium equation

Cite this paper

nullZ. Zhang and Y. Li, "Quenching Rate for the Porous Medium Equation with a Singular Boundary Condition,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1134-1139. doi: 10.4236/am.2011.29157.

nullZ. Zhang and Y. Li, "Quenching Rate for the Porous Medium Equation with a Singular Boundary Condition,"

References

[1] L. S. Leibenzon, “The Motion of a Gas in a Porous Medium,” Russian Academy of Sciences, Moscow, 1930.

[2] M. Muskat, “The Flow of Homogeneous Fluids through Porous Media,” McGraw-Hill, New York, 1937.

[3] J. L. Vázquez, “Asymptotic Behaviour for the Porous Medium Equation Posed in the Whole Space,” Journal of Evolution Equations, Vol. 3, No. 1, 2003, pp. 67-118. doi:10.1007/s000280300004

[4] H. Kawarada, “On Solutions of Initial Boundary Value Problem for ut = uxx = 1/(1–u),” Publications of the Research Institute for Mathematical Sciences, Vol. 10, 1975, pp. 729-736. doi:10.2977/prims/1195191889

[5] C. Y. Chan and M. K. Kwong, “Quenching Phenomena for Singular Nonlinear Parabolic Equations,” Nonlinear Analysis, Vol. 12, No. 2, 1998, pp. 1377-1383. doi:10.1016/0362-546X(88)90085-5

[6] C. Y. Chan, “New Results in Quenching,” Proceeding of 1st World Congress of Nonlinear Analysts, Tampa, Vol. 1, 19-26 August 1992, pp. 427-434.

[7] K. Deng and M. X. Xu, “Quenching for a Nonlinear Diffusion Equation with a Singular Boundary Condition,” Zeitschrift fur Angewandte Mathematik und Physik, Vol. 50, No. 4, 1999, pp. 574-584. doi:10.1007/s000330050167

[8] K. Deng and M. X. Xu, “On Solutions of a Singular Diffusion Equation,” Nonlinear Analysis, Vol. 41, No. 3-4, 2000, pp. 489-500. doi:10.1016/S0362-546X(98)00292-2

[9] H. A. Levine, “Advances in Quenching,” Proceeding of International Conference on Reaction-Diffusion Equations and Their Equilibrium States, Vol. 7, 1992, pp. 319-346.

[10] H. A. Levine and G. M. Lieberman, “Quenching of Solutions of Parabolic Equations with Nonlinear Boundary Conditions in Several Dimensions,” Journal für Die Reine und Angewandte Mathematik, Vol. 1983, No. 345, 1983, pp. 23-38.

[11] Z. C. Zhang and B. Wang, “Blow-up Rate Estimate Parabolic Equation with Nonlinear Gradient Term,” Applied Mathematics and Mechanics, Vol. 31, No. 6, 2010, pp. 787-796. doi:10.1007/s10483-010-1313-6

[12] V. A. Galaktionov and H. A. Levine, “On Critical Fujita Exponents for Heat Equations with Nonlinear Flux Conditions on the Boundary,” Israel Journal of Mathematics, Vol. 94, No. 1, 1996, pp. 125-146.

[13] A. De Pablo, F. Quiros and J. D. Rossi, “Nonsimultaneous Quenching,” Applied Mathematics Letters, Vol. 15, No. 3, 2002, pp. 265-269. doi:10.1016/S0893-9659(01)00128-8

[14] S. N. Zheng and X. F. Song, “Quenching Rates for the Heat Equatons with Coupled Singular Nonlinear Boundary Flux,” Science in China Series A-Mathematics, Vol. 51, No. 9, 2008, pp. 1631-1643.

[15] M. Fila and H. A. Levine, “Quenching on the Boundary,” Nonlinear Analysis, Vol. 21, No. 10, 1993, pp. 795-802. doi:10.1016/0362-546X(93)90124-B

[16] J. Filo, “Difusivity versus Absorption through the Boundary,” Journal of Differential Equations, Vol. 99, No. 2, 1992, pp. 281-305.

[1] L. S. Leibenzon, “The Motion of a Gas in a Porous Medium,” Russian Academy of Sciences, Moscow, 1930.

[2] M. Muskat, “The Flow of Homogeneous Fluids through Porous Media,” McGraw-Hill, New York, 1937.

[3] J. L. Vázquez, “Asymptotic Behaviour for the Porous Medium Equation Posed in the Whole Space,” Journal of Evolution Equations, Vol. 3, No. 1, 2003, pp. 67-118. doi:10.1007/s000280300004

[4] H. Kawarada, “On Solutions of Initial Boundary Value Problem for ut = uxx = 1/(1–u),” Publications of the Research Institute for Mathematical Sciences, Vol. 10, 1975, pp. 729-736. doi:10.2977/prims/1195191889

[5] C. Y. Chan and M. K. Kwong, “Quenching Phenomena for Singular Nonlinear Parabolic Equations,” Nonlinear Analysis, Vol. 12, No. 2, 1998, pp. 1377-1383. doi:10.1016/0362-546X(88)90085-5

[6] C. Y. Chan, “New Results in Quenching,” Proceeding of 1st World Congress of Nonlinear Analysts, Tampa, Vol. 1, 19-26 August 1992, pp. 427-434.

[7] K. Deng and M. X. Xu, “Quenching for a Nonlinear Diffusion Equation with a Singular Boundary Condition,” Zeitschrift fur Angewandte Mathematik und Physik, Vol. 50, No. 4, 1999, pp. 574-584. doi:10.1007/s000330050167

[8] K. Deng and M. X. Xu, “On Solutions of a Singular Diffusion Equation,” Nonlinear Analysis, Vol. 41, No. 3-4, 2000, pp. 489-500. doi:10.1016/S0362-546X(98)00292-2

[9] H. A. Levine, “Advances in Quenching,” Proceeding of International Conference on Reaction-Diffusion Equations and Their Equilibrium States, Vol. 7, 1992, pp. 319-346.

[10] H. A. Levine and G. M. Lieberman, “Quenching of Solutions of Parabolic Equations with Nonlinear Boundary Conditions in Several Dimensions,” Journal für Die Reine und Angewandte Mathematik, Vol. 1983, No. 345, 1983, pp. 23-38.

[11] Z. C. Zhang and B. Wang, “Blow-up Rate Estimate Parabolic Equation with Nonlinear Gradient Term,” Applied Mathematics and Mechanics, Vol. 31, No. 6, 2010, pp. 787-796. doi:10.1007/s10483-010-1313-6

[12] V. A. Galaktionov and H. A. Levine, “On Critical Fujita Exponents for Heat Equations with Nonlinear Flux Conditions on the Boundary,” Israel Journal of Mathematics, Vol. 94, No. 1, 1996, pp. 125-146.

[13] A. De Pablo, F. Quiros and J. D. Rossi, “Nonsimultaneous Quenching,” Applied Mathematics Letters, Vol. 15, No. 3, 2002, pp. 265-269. doi:10.1016/S0893-9659(01)00128-8

[14] S. N. Zheng and X. F. Song, “Quenching Rates for the Heat Equatons with Coupled Singular Nonlinear Boundary Flux,” Science in China Series A-Mathematics, Vol. 51, No. 9, 2008, pp. 1631-1643.

[15] M. Fila and H. A. Levine, “Quenching on the Boundary,” Nonlinear Analysis, Vol. 21, No. 10, 1993, pp. 795-802. doi:10.1016/0362-546X(93)90124-B

[16] J. Filo, “Difusivity versus Absorption through the Boundary,” Journal of Differential Equations, Vol. 99, No. 2, 1992, pp. 281-305.