Asymptotic behaviour of the porous media equation in an exterior domain

 Access to full text

 Full (PDF)

1. [Ar] D.G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (A. Fasano and M. Primicerio, eds) Lecture Notes in Mathematics1224, Springer-Verlag, New York, 1986, pp. 1-46. Zbl0626.76097MR877986
2. [ACP] D.G. Aronson - M.G. Crandall - L.A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal.6 n. 10 (1982), 1001-1022. Zbl0518.35050MR678053
3. [AGV] D.G. Aronson - O. Gil - J.L. Vazquez, Limit behaviour of focusing solutions to nonlinear diffusions, Comm. Partial Differential Equations23 n. 1, 2 (1998), 307-332. Zbl0895.35055MR1608532
4. [AG] D.G. Aronson - J. Graveleau, A selfsimilar solution to the focusing problem for the porous medium equation, Euro. J. Appl. Math.4 (1992), 65-81. Zbl0780.35079MR1208420
5. [AtP] F.V. Atkinson - L.A. Peletier, Similarity profiles of flows through porous media, Arch. Rational Mech. Anal.42 (1971), 369-379. Zbl0249.35043MR334666
6. [Ba] G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mekh.16 (1952), 67-78 (in Russian). Zbl0049.41902MR46217
7. [Ba2] G.I. Barenblatt, A class of exact solutions of the planar one-dimensional problem of non-stationary gas filtration in a porous medium, Prikl. Mat. Mekh.17 n. 6 (1953), 739-742 (in Russian). Zbl0053.46103MR64556
8. [BV] G.I. Barenblatt - J.L. Vazquez, On a new free boundary problem for unsteady flows in porous media, Euro. J. Appl. Math.9 (1998), 37-54. Zbl0931.76091MR1617007
9. [Be] J. Bear, "Dynamics of Fluids in Porous Media", Elsevier, New York, 1971. 
10. [dB] E. Di Benedetto, Continuity of weak solutions to a general porous media equation, Indiana Univ. Math. J.32 (1983), 83-118. Zbl0526.35042MR684758
11. [Bn] PH. Bénilan, "Equations d'évolution dans un espace de Banach quelconque et applications", Thèse Orsay, 1972. 
12. [BC] PH. Bénilan - M.G. Crandall, The continuous dependence on ϕ of solutions of ut - Δϕ(u) = 0, Indiana Univ. Math. J.30 (1981), 161-177. Zbl0482.35012
13. [BHV] F. Bernis - J. Hulshof - J.L. Vázquez, A very singular solution for the dual porous media equation and the asymptotic behaviour of general solutions, J. Reine Angew. Math.435 (1993), 1-31. Zbl0756.35038MR1203909
14. [Br] M. Bertsch, A class of degenerate diffusion equations with a singular nonlinear term, Nonlinear Anal.7 n. 1 (1983), 117-127. Zbl0509.35047MR687037
15. [C] J. Carr, "Applications of Centre Manifold Theory", Springer-Verlag, New York, 1981. Zbl0464.58001MR635782
16. [GV] V.A. Galaktionov - J.L. Vázquez, Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach, J. Funct. Anal.100 (1991), 435-462. Zbl0755.35010MR1125235
17. [Gi] B.H. Gilding, On a class of similarity solutions of the porous media equation III, J. Math. Anal. Appl.77 (1980), 381-402. Zbl0454.35053MR593221
18. [GP1] B.H. Gilding - L.A. Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl.55 (1976), 351-364. Zbl0356.35049MR436751
19. [GP2] B.H. Gilding - L.A. Peletier, On a class of similarity solutions of the porous media equation II, J. Math. Anal. Appl.57 (1977), 522-538. Zbl0365.35029MR436752
20. [G4] J. Goncerzewicz - H. Marcinkowska - W. Okrasinski - K. Tabisz, On the percolation of water from a cylindrical reservoir into the surrounding soil, Zastosow. Mat.16 (1978), 249-261. Zbl0403.76078MR517791
21. [GO] J. Goncerzewicz - W. Okrasinski, A lower estimate of the interface of some nonlinear diffusion problems, Extracta Math.9 (1994), 95-100. Zbl0978.35103MR1325289
22. [H] J. Hulshof, Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl.157 n. 1 (1991), 75-111. Zbl0777.35034MR1109445
23. [HV] J. Hulshof - J.L. Vázquez, The dipole solution for the porous medium equation in several space dimensions, Annali Scuola Norm. Sup. Pisa Cl. Sc. 20 (1993), 193-217. Zbl0832.35082MR1233636
24. [J] C.W. Jones, On reducible nonlinear differential equations occurring in mechanics, Proc. Roy. Soc. A217 (1953), 327-343. Zbl0052.31303MR54129
25. [LOT] A.A. Lacey - J.R. Ockendon - A.B. Tayler, "Waiting-time" solutions of a nonlinear diffusion equation, SIAM J. Appl. Math.42 (1982), 1252-1264. Zbl0531.76093MR678215
26. [M] M. Muskat, "The Flow of Homogeneous Fluids Through Porous Media", McGraw-Hill, New York, 1937. Zbl63.1368.03JFM63.1368.03
27. [O] W. Okrasinski, Asymptotic behaviour of the free boundary of some nonlinear diffusion problems, Math. Models Methods Appl. Sci.19 (1996), 375-380. Zbl0856.35139MR1376511
28. [P] L.A. Peletier, Asymptotic behavior of solutions of the porous media equation, SIAM J. Appl. Math.21 n. 4 (1971), 542-551. Zbl0229.35010MR304894
29. [PK] P. Ya. Polubarinova-Kochina, On a nonlinear differential equation encountered in the theory of infiltration, Dokl. Akad. Nauk UZSSR63 n. 6 (1948), 623-627. 
30. [QV] F. Quirós - J.L. Vázquez, Asymptotic convergence of the Stefan problem to Hele-Shaw, Trans. Amer. Math. Soc., to appear. Zbl0956.35136MR1804510
31. [QV2] F. Quirós - J.L. Vázquez, Asymptotic behaviour of the porous media, Hele-Shaw and Stefan equations in an exterior domain, in preparation. 
32. [S4] A.A. Samarskii - V.A. Galaktionov - S.P. Kurdyumov - A.P. Mikhailov, "Blow-up in Problems for Quasilinear Parabolic Equations", Walter de Gruyter, Berlin, 1995. Zbl1020.35001MR1330922
33. [V1] J.L. Vázquez, Symétrisation pour ut = Δϕ (u) et applications, C. R. Acad. Sci. Paris295 (1982), 71-74. Zbl0501.35015
34. [V2] J.L. Vázquez, An introduction to the mathematical theory of the porous medium equation, in "Shape Optimization and Free Boundaries" (M.C. Delfour, eds.) Mathematical and Physical Sciences, Series C, vol. 380, Kluwer Ac. Publ., Dordrecht, Boston and Leiden, 1992, pp. 347-389. Zbl0765.76086MR1260981
35. [VV] J.L. Vázquez - L. Véron, Different kinds of singular solutions of nonlinear parabolic equations, in "Nonlinear Problems in Applied Mathematics", volume in honor of Ivar Stakgold on his 70th birthday (T. S. Angell et al., eds.) SIAM, Philadelphia, 1996, pp. 240-249. Zbl0886.35078
36. [Z] W.P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc.271 n. 2 (1982), 733-748. Zbl0506.35053MR654859

1. Razvan Gabriel Iagar, Juan Luis Vázquez, Asymptotic analysis of the $p$-laplacian flow in an exterior domain