Waiting time phenomena forced by critical boundary conditions in classical diffusion problems
Authors:
A. Fasano, A. Mancini, M. Primicerio and B. Zaltzman
Journal:
Quart. Appl. Math. 69 (2011), 105-122
MSC (2000):
Primary 35K60
DOI:
https://doi.org/10.1090/S0033-569X-2010-01205-0
Published electronically:
December 29, 2010
MathSciNet review:
2807980
Full-text PDF Free Access
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Abstract: This paper revisits some very classical initial-boundary value problems for parabolic equations, providing simple examples in which the occurrence of flux discontinuities at the boundary when the unknown function reaches some critical value may give rise to a waiting time phenomenon. A physical interpretation could be a modification of the surface of the considered body taking place at the mentioned critical value, affecting the way the body interacts with the surroundings. The waiting time, whose length (finite or infinite) is a priori unknown allows the system to evolve gradually through the critical state. Some numerical simulations are also presented.
References
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References
- D.G. Aronson, The porous media equation in “nonlinear diffusion problems”, Springer Lecture Notes in Mathematics, no. 1224, 1986, pp. 1–46, A. Fasano, M. Primicerio, eds. MR 877986 (88a:35130)
- G. Astarita and G. Sarti, A class of mathematical models for sorption of swelling solvents in glassy polymers, Polym. Eng. Sci. 18 (1978), 388–395.
- G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Prikl. Math. Meh. 16 (1952), 67–78. MR 0046217 (13:700a)
- J.R. Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley, Menlo Park, 1984. MR 747979 (86b:35073)
- H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids, Clarendon Press, Oxford, 1959. MR 0022294 (9:188a)
- A. Fasano and A. Mancini, A phenomenon of waiting time in phase change problems driven by radiative heat transfer, Math. Meth. Appl. Sci. 32 (2009), 1105–1117. MR 2523565
- A. Fasano, G.H. Meyer, and M. Primicerio, On a problem in the polymer industry: theoretical and numerical investigation of swelling, SIAM J. Math. Anal. 17 (1986), 945–960. MR 846399 (87h:35355)
- A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, 1964. MR 0181836 (31:6062)
- M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, J. de Math. 10 (1913), no. 6, 105–148.
- W.H. Green and G.A. Ampt, Studies in soil physics. The flow of air and water through soils, J. Agric. Sci. 4 (1911), 1–24.
- O.A. Ladyzenskaja, N.A. Solonnikov, and N.N. Uraltseva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monogr., vol. 23, American Mathematical Society, Providence, 1968.
- A. Mikelic and M. Primicerio, Homogenization of the heat equation for a domain with a network of pipes with a well-mixed fluid, Ann. Mat. Pura Appl. 166 (1994), no. 4, 227–251. MR 1313806 (96f:35167)
- M. Ughi, Teoremi di esistenza per problemi al contorno di iv ev tipo per una equazione parabolica lineare, Riv. Mat. Univ. Parma 5 (1979), 591–606.
- P. Willmott, J.N. Dewynne, and S.D. Howison, Option pricing: Mathematical models and computation, Financial Press, Oxford, 1993.
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A. Fasano
Affiliation:
Dip. di Matematica “U.Dini” - Universitá degli Studi di Firenze, viale Morgagni, 67a - 50134 Firenze, Italy
Email:
fasano@math.unifi.it
A. Mancini
Affiliation:
Dip. di Matematica “U.Dini” - Universitá degli Studi di Firenze, viale Morgagni, 67a - 50134 Firenze, Italy
Email:
mancini@math.unifi.it
M. Primicerio
Affiliation:
Dip. di Matematica “U.Dini” - Universitá degli Studi di Firenze, viale Morgagni, 67a - 50134 Firenze, Italy
Email:
primicerio@math.unifi.it
B. Zaltzman
Affiliation:
DSEEP, Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede-Boker Campus, 84990, Israel
Email:
boris@bgumail.bgu.ac.il
Received by editor(s):
July 21, 2009
Published electronically:
December 29, 2010
Article copyright:
© Copyright 2010
Brown University
