Localization of solutions of a nonlinear Fokker-Planck equation with Dirichlet boundary conditions. (English) Zbl 0698.35077

The paper is concerned with the Cauchy-Dirichlet problem for a nonlinear degenerate second-order parabolic equation in one space dimension. The equation possesses the property that if the initial data function for the Cauchy-Dirichlet problem is nonnegative and has compact support then at all later times the (generalized) solution of the problem also is nonnegative and has compact support as a function of the spatial variable. Such equations arise in the theory of soil-moisture infiltration. A solution of the Cauchy-Dirichlet problem is said to be localized if its support with respect to the spatial variable is uniformly bounded in time. The influence of the coefficients in the equation and of the lateral boundary data on the localization of solutions is investigated. With regard to the physical background to the problem, the results relate to the role of soil characteristics, gravity, and boundary conditions in determining the finite penetration of a wetting front during soil-moisture infiltration.
Reviewer: B.H.Gilding


35K57 Reaction-diffusion equations
35K65 Degenerate parabolic equations
35B99 Qualitative properties of solutions to partial differential equations
35Q99 Partial differential equations of mathematical physics and other areas of application
76S05 Flows in porous media; filtration; seepage
Full Text: DOI


[1] Aronson, D.G., The porous medium equation, (), 1-46, Lecture Notes in Mathematics · Zbl 0202.37901
[2] Barenblatt, G.I.; Zel’dovich, Ya.B., On the dipole-type solution in problems of unsteady gas filtration in the polytropic regime, Prikl. mat. mekh., 21, 718-720, (1957), (In Russian)
[3] Bear, J.; Zaslavsky, D.; Irmay, S., Physical principles of water percolation and seepage, (1968), United Nations Educational Scientific and Cultural Organization Paris
[4] Diaz, J.I.; Kersner, R., Non existence d’une des frontiéres libres dans une équation dégénérée en théorie de la filtration, C.R. acad. sci. Paris Sér t math., 296, 505-508, (1983) · Zbl 0543.35102
[5] Diaz, J.I.; Kersner, R., On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. diff. eqns, 69, 368-403, (1987) · Zbl 0634.35042
[6] Diaz, J.I.; Kersner, R., On the behaviour and cases of nonexistence of the free boundary in a semibounded porous medium, J. math. analysis applic., 132, 281-289, (1988) · Zbl 0673.35058
[7] Gilding, B.H., Properties of solutions of an equation in the theory of infiltration, Archs ration. mech. analysis, 65, 203-225, (1977) · Zbl 0366.76074
[8] Gilding, B.H., The occurance of interfaces in nonlinear diffusion-advection processes, Archs ration. mech. analysis, 100, 243-263, (1988) · Zbl 0672.76094
[9] Gilding, B.H., Improved theory for a nonlinear degenerate parabolic equation, Annali scu. norm. sup. Pisa cl. sci., 587, 4, (1986), Previously appearing in provisional form as: Twente University of Technology Department of Applied Mathematics Memorandum · Zbl 0702.35140
[10] Kalashnikov, A.S., The nature of the propagation of perturbations in processes that can be described by quasilinear degenerate parabolic equations, Trudy sem. petrovsk., 1, 135-144, (1975), (In Russian.) · Zbl 0318.35047
[11] Kalashnikov, A.S.; Kalashnikov, A.S., Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russ. math. survs, Usp. mat. nauk, 42, 135-176, (1987) · Zbl 0642.35047
[12] Kersner, R.; Kersner, R., Localization conditions for thermal perturbations in a semibounded moving medium with absorption, Moscow univ. math. bull., Vest. moskov. univ. ser. I mat. mekh., 31, 52-58, (1976) · Zbl 0342.35030
[13] Kersner, R., Private communication, (1986)
[14] Matano, H., Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. fac. sci. univ. Tokyo sect. IA math., 29, 401-441, (1982) · Zbl 0496.35011
[15] Peletier, L.A., The porous media equation, (), 229-241 · Zbl 0497.76083
[16] Swartzendruber, D., The flow of water in unsaturated soils, (), 215-292
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.