Concavity of solutions of the porous medium equation

Authors:
Philippe Bénilan and Juan Luis Vázquez

Journal:
Trans. Amer. Math. Soc. **299** (1987), 81-93

MSC:
Primary 35K60; Secondary 76S05

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869400-8

MathSciNet review:
869400

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem

**[**D. C. Aronson,**A**]*Regularity properties of flows through porous media: the interface*, Arch. Rational Mech. Anal.**37**(1970), 1-10. MR**0255996 (41:656)****[**D. C. Aronson and Ph. Benilan,**AB**]*Régularité des solutions de l'équation des milieux poreux dans*, C. R. Acad. Sci. Paris**288**(1979), 103-105.**[**Ph. Bénilan and M. G. Crandall,**BC**]*The continuous dependence on**of the solutions of*, Indiana Univ. Math. J.**30**(1981), 161-177. MR**604277 (83d:35071)****[**Ph. Bénilan, M. G. Crandall and A. Pazy,**BCP**]*Evolution equations governed by accretive operators*(to appear).**[**S. H. Benton,**BT**]*The Hamilton-Jacobi equation: a global approach*, Academic Press, New York, 1977. MR**0442431 (56:813)****[**L. A. Caffarelli and A. Friedman,**CF**]*Regularity of the free-boundary for the one-dimensional flow of gas in a porous medium*, Amer. J. Math.**101**(1979), 1193-1218. MR**548877 (80k:76072)****[**M. G. Crandall and P. L. Lions,**CL**]*Viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**277**(1983), 1-42. MR**690039 (85g:35029)****[**J. L. Graveleau and P. Jamet,**GJ**]*A finite-difference approach to some degenerate nonlinear parabolic equations*, SIAM J. Appl. Math.**29**(1971), 199-223. MR**0290600 (44:7780)****[**B. F. Knerr,**K**]*The porous medium equation in one dimension*, Trans. Amer. Math. Soc.**234**(1977), 381-415. MR**0492856 (58:11917)****[**O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva,**LSU**]*Linear and quasilinear equation of parabolic type*, Transl. Math. Monos., vol. 23, Amer. Math. Soc., Providence, R.I., 1969.**[**P. L. Lions,**L**]*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Math., no. 69, Pitman, Boston, Mass., 1982. MR**667669 (84a:49038)****[**M. Ughi,**U**]*A degenerate parabolic equation modelling the spread of an epidemic*, Ann. Mat. Pura Appl. (to appear). MR**859613 (88g:35105)****[**J. L. Vazquez,**V1**]*Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium*, Trans. Amer. Math. Soc.**277**(1983), 507-527. MR**694373 (84h:35014)****[**-,**V2**]*The interfaces of one-dimensional flows in porous media*, Trans. Amer. Math. Soc.**285**(1984), 717-737. MR**752500 (85h:35229)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35K60,
76S05

Retrieve articles in all journals with MSC: 35K60, 76S05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869400-8

Keywords:
Concavity,
flow in porous media,
Trotter-Kato formula,
interfaces,
asymptotic behavior

Article copyright:
© Copyright 1987
American Mathematical Society