Díaz Díaz, Jesús Ildefonso and Kersner, R. (1987) On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. Journal of Differential Equations, 69 (3). pp. 368-403. ISSN 0022-0396
Restricted to Repository staff only until 31 December 2020.
The main result of the paper is the uniqueness of nonnegative solutions of the Cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut=(um)xx+(un)x, where m≥1 and n>0. In particular, n is allowed to be smaller than one. The proof is based on a refined test function argument. The condition that u be nonnegative is crucial, but the restriction to one space variable is not.
|Uncontrolled Keywords:||Dirichlet boundary conditions; initial data; infiltration; evaporation; porous medium; existence of limit solutions; weak solutions; modulus of continuity; uniqueness|
|Subjects:||Sciences > Mathematics > Differential geometry|
D. G. ARONSON, Regularily properties of flows through porous media. SIAM J. Appl.Math. 17 (1969), 461-467.
D. G. ARONSON. M, G, CRANDALL, AND L A. PELETIER, Stabilization of solutions of a degenerate nonlinear diffusion prob1em, Nonlinear Anal. 6 (1982), 1001--1022.
J. BEAR. "Dynamics of Fluids in Porous Media" Elsevier, New York, 1971.
PH. BENILAN. On the equation u = (u)xx + (u)x, unpublished.
PH, BENILAN AND H. TOURÉ, Sur l’equation généra1e u, =(u)xx - (u)x. + v, C R. Acad.Sci. Paris Sér. J Math. 299 (1984), 919-922.
PH. BENILAN, "Evolution Equations and Accretive Operators," Lectures Notes, University of Kenlucky, 1981.
PH. BENILAN AND M. G. CRANDALL, The continuous depcndence on of solutions of u- (u) = 0, Indialla Univ. Math, J. 30 (1981), 162-J 77.
J. I.DíAZ, Solutions with compact support for some degenerate parabolic problems, Nonlinear Anal. 3 (1971),831-847.
E. DIBENEDETTO, Continuity of weak sollltions to a general porous media equation,Indiana Univ. Math . J. 32 (1983), 83-119.
E. DIBENEDETTO, A boundary modulus of continuity for a class of singular parabolic equations, to appear.
M. I. FREIDLIN, Existence in the large of smooth solutions of' degenerate quasilinear equations, Mat. Sb. 78 (1969), 332-348; Math. USSR-Sb. 7 (1969), 323-339.
B. H. GILDING, Hölder continuity of solutions of parabolic equations, J. London Math. Soc. 13 (1976),103-106.
B. H. GILDING, A nonlincar degenerate parabolic equation, Ann Scuola Norm. Sup. Pisa 4(1977), 393-432.
B. H. GILDING, Properties of solutions of an equation in the theory of infiltration, Arch.Rational Mech. Anal. 65 (1977),203-225.
B. H. GILDING AND L. A. PELETlER, The Cauchy problem for an equation in the theory of infiltration, Arch. Rational Mech. Anal. 61 (1976), 127-140.
J. GONCERZEWICZ. On the existence of weak solutions of a boundary value problem for a certain degenerate parabolic equation,Bull.Acad.Polon.Sci.Sér.Sci.Tech.27(1979),449-454.
A. S. KALASHNIKOV, On the differential properties of generalized solitions of equations of the nonsteady-stale filtralion type, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 29 (1979),62-68.
A. S. KALASHNIKOV, On the character of the propagation of perturbation in processed described by quasilinear degenerate parabolic equations,in"Proccedings of Seminars Dedicated to I. G. Petrovskogo,pp.135-144,Otdel'nyiottisk, Moscow,1975.
R. KERSNER, Degenerate parabolic equations with general nonlinearities, Nonlinear Anal.4 (1980), 1043-1061.
R. KERSNER, Localization conditions for thermal perturbation in a scmiboundcd moving medium with absorption, Vestnik Moskov. Univ. Mat. 31 (1976), 52-58.
B. F. KNERR, The porous medium equation in one dimension, Trans. Amer. Math. Soc.234 ([977), 381--415.
S. N. KRUZHKOV, Results on the character of the regu1arity of solutions of parabolic equations and some of their applications, Mat. Zametki 6 (1969), 295•-300.
R. J. KUNZE AND D. KIRHAM, Simplified accounting for membrane indedance in capillary conductivity determinations, Soil Sci. Soc. Amer. Proc.26 1962),421-426.
O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, AND N. N. URAL'CEVA, "Linear and Quasilinear Equations of Parabolic Type," Transl. of Math. Monogr., Vol. 23, Amer.Math. Soc., Providencc, Rl, 1968.
B. J. LURCIER, On Sobolev regularizations of hyperbolic conservation laws, Comm. Partial Diflerential Eqations l0 (1985), 1-28.
T. NANBU, The Cauchy problem for quasilinear degenerate parabolic equations, Math.Rep. Kyushu Univ. 11(1977),83-96.
O. A. OLEINIK AND T. D. VENTZEL, The first boundary value problem and the Cauchy problem for the quasilinear equation of parabolic type, Mal. Sb. 41 (1957), 105-128.
O. A. OLEINIK, A. S. KALASHNIKOV, AND C. YUI-LIN, The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration, Izv. Akad. Navk.
SSSR. Soc. Mat. 22 (1958), 667-704.
L. A. PELETIER, The porous media equation, in "Applieations of Nonlinear Ana1ysis in the Physics Sciences" (H. Amman, Ed,), pp. 229-241, Pitman, London, 1981.
J. R. PHILIP, Evaporation, and moisture and heat fields in the soil, J. Meterology 14(1957),354-366.
J. R. PHILIP, The theory of infiltratation, Adv, Hydrosci. 5 (1969), 215-296.
M. E. ROSE, "Numerieal Methods for a General Class of Porous Medium Equations,"Argone National Laboratory Report, Arganne, II, 1980.
P. SACKS, "Existence and Regularity of Solutions of Inhomogeneous Porous Medium Type Equations," T.S.R. #2214, MRC, Madison, WI, 1981.
D. SWARTZENDRUBER, The flow of water in unsatured soi1s, in "F1ow Thraugh Parous Media" (R. J. M. Dewiest, Ed.), pp. 215-292, Academic Press, New York, 1969.
A. I. VOL'PERT AND S. I. HUDJAEV, Cauchy's problem for degenerate second order quasilinear parabolic equations, Mat.Sb.78 1969),374-396;Math.USSR-Sb.7 1969), 365-387.
DEQUAN WU, Uniqueness of the weak solution of quasilinear degenerate parabolic equations, Acta Math. Sinica 25 (1982),61-75.
ZHUOQUN WU, An application of the theory of nonlinear semi-groups to the Cauchy problem for quasi-linear degenerate Parabolic equation of second order, to appear.
ZHUOQUN WU AND JUNNING ZHAO, The first boundary value problem for quasilinear degenerate parabolic equations of second order in several space variables, Chinese Ann.
Math. Ser. B 4 (1983), 57-76.
W. P. ZIEMER, Interior and boundary continuity of weak solutions of degenerate parabolic equations,Trans. Amer. Math. Soc. 271, 2 (1982),733-748.
|Deposited On:||17 Sep 2012 08:45|
|Last Modified:||07 Feb 2014 09:28|
Repository Staff Only: item control page