Numerical approach to the waiting time for the one-dimensional porous medium equation
Authors:
Tatsuyuki Nakaki and Kenji Tomoeda
Journal:
Quart. Appl. Math. 61 (2003), 601-612
MSC:
Primary 35K57; Secondary 35K65, 65M99, 76S05
DOI:
https://doi.org/10.1090/qam/2019614
MathSciNet review:
MR2019614
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Abstract: We consider the nonlinear degenerate diffusion equation. The most striking manifestation of the nonlinearity and degeneracy is an appearance of interfaces. Under some condition imposed on the initial function, the interfaces do not move on some time interval $\left [ {0,{t^*}} \right ]$. In this paper, from numerical points of view, we try to determine the value of ${t^*}$, which is called the waiting time.
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N. D. Alikakos, On the pointwise behavior of the solutions of the porous medium equation as t approaches zero or infinity, Nonlinear Anal. 9, 1095–1113 (1985).
D. G. Aronson, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math. 19, 299–307 (1970).
---, The porous medium equation, Nonlinear diffusion problems, Lecture notes in mathematics 1224 (eds. A. Fasano and M. Primicerio), Springer-Verlag (1985), 1–46.
D. G. Aronson, L. A. Caffarelli, and S. Kamin, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal. 14, 639–658 (1983).
D. G. Aronson, L. A. Caffarelli, and J. L. Vázquez, Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure and Appl. Math. 38, 375–404 (1985).
M. Bertsch and R. Dal Passo, A numerical treatment of a superdegenerate equation with applications to the porous media equation, Quart. Appl. Math. 48, 133–152 (1990).
L. A. Caffarelli and A. Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101, 1193–1218 (1979).
M. Chipot and T. Sideris, An upper bound for the waiting time for nonlinear degenerate parabolic equations, Trans. Amer. Math. Soc. 288, 423–427 (1985).
M. E. Gurtin, R. C. MacCamy, and E. A. Socolovsky, A coordinate transformation for the porous media equation that renders the free-boundary stationary, Quart. Appl. Math. 42, 345–357 (1984).
A. S. Kalashnikov, The Cauchy problem in a class of growing functions for equations of unsteady filtration type, Vest. Mosk. Univ. Ser. Mat. Mesh. 6, 17–27 (1963).
W. L. Kath and D. S. Cohen, Waiting-time behavior in a nonlinear diffusion equation, Studies in Appl. Math. 67, 79–105 (1982).
B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234, 381–415 (1977).
A. A. Lacey, J. R. Ockendon, and A. B. Tayler, "Waiting-time” solutions of a nonlinear diffusion equation, SIAM J. Appl. Math. 42, 1252–1264 (1982).
A. A. Lacey, Initial motion of the free boundary for a non-linear diffusion equation, IMA J. Appl. Math. 31, 113–119 (1983).
M. Mimura, T. Nakaki, and K. Tomoeda, A numerical approach to interface curves for some nonlinear diffusion equations, Japan J. Appl. Math. 1, 93–139 (1984).
O. A. Oleinik, A. S. Kalashnikov, and Chzou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of unsteady filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22, 667–704 (1958).
L. A. Peletier, The porous media equation, Applications of nonlinear analysis in the physical sciences (eds. H. Amann, N. Bazley, and K. Kirchgässner), Pitman (1981), 229–241.
K. Tomoeda and M. Mimura, Numerical approximations to interface curves for a porous media equation, Hiroshima Math. J. 13, 273–294 (1983).
J. L. Vázquez, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285, 717–737 (1984).
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