Díaz Díaz, Jesús Ildefonso and Kersner, R.
(1988)
*On the behavior and cases of nonexistence of the free-boundary in a semibounded porous-medium.*
Journal of Mathematical Analysis and Applications, 132
(1).
pp. 281-289.
ISSN 0022-247X

PDF
Restricted to Repository staff only until 31 December 2020. 254kB |

Official URL: http://www.sciencedirect.com/science/article/pii/0022247X88900613

## Abstract

The authors consider the Fokker-Planck equation ut=(um)xx+b(uλ)x, x>0, t>0, with initial and boundary data u(x,0)=u0(x), x>0, u(0,t)=u1(t), t>0, u0 having its support in a bounded interval. They concentrate on the case 0<λ<1, m≥1 with the aim of investigating the behavior of the free boundary, i.e. the moving boundary of suppu, in various different cases. When b>0 it is shown that if u1 tends to zero as t→∞, then the free boundary tends to zero. If u1 vanishes in a finite time, so does the free boundary. The possibility that the free boundary tends to infinity is also discussed. Moreover, conditions are found on m,λ and on u1 such that the free boundary can be estimated from above (localization) and from below by a positive constant. When b<0 it is shown that the free boundary never exists (for λ≥1, m>1 the free boundary is known to start from the right endpoint of suppu0).

Item Type: | Article |
---|---|

Uncontrolled Keywords: | behavior; free boundary; semibounded porous medium; Cauchy-Dirichlet problem; Fokker-Planck equation; qualitative properties; free boundaries; interfaces |

Subjects: | Sciences > Mathematics > Differential geometry |

ID Code: | 16261 |

References: | D. G. ARONSON, Regularity properties of flows through porous media: The interface,Arch. Rational Mech. Anal. 37 (1970), l-l0. J. I. DÍAZ AND R. KERSNER, "On a Nonlinear Degenerate Parabolic Equation in Infiltration of Evaporation through a Porous Medium,"J.of Differential Equayions 69(1987),368-403. J. I. DíAZ AND R. KERSNER,Non cxistence d'une des frontiers libres dans une equation degenereé en theorie de la filtration, C. R. Acad. Sci. Paris 296 (1983), 505-508. B. H. GILDING, A nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa 4(1977), 393-432. B. H. GILDING, Properties of Sollltions of an equation in the theory of infiltration, Arch.Rational Mech. Anal. 65 (1977), 203-225. A. S. KALASHNIKOV, On the character of the propagation of perturbation in processes described by quasilinear degenerate parabolic equations, in "Proceedings, Seminars Dedicated to I. G. Petrovskogo, 1975," pp.135-144. [Russian] A. S. KALASHNIKOV, On the influence of the boundary values on the behaviour of temperature of nonlinear non-stationary mcdium, Vestnik Moskov. Univ. Ser. I Mat.Mekh.(1986),40-45. [Russian] R. KERSNER, Localization conditions for thermal perturbations in a semibounded moving medium with absorption, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 31 (1976), 52-58;transl. in Moscow Univ. Math. Bull. B. F, KNERR, The porous mediurn cquation in one dimension, Trans. Amer. Math. Soc.234 (1977), 381-415. J. R. PHILLIP, Evaporation, and moisture and heat fields in the soils, J Meterol. 14(1957), 354--366. D. SWARTZENDRUBER. The flow of water in unsaturated soils, in "Flow Through Porous Media" (R. J. M. Dewiested, Ed.) pp. 215-292, Academic Press, New York, 1969. J. L. VAZQUEZ, Asymptotic behaviour and propagation of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983). 507-527. |

Deposited On: | 10 Sep 2012 07:58 |

Last Modified: | 07 Feb 2014 09:26 |

Repository Staff Only: item control page