Abstract
We show the existence of solutions to the fast diffusion equation with a general finite and positive Borel measure as the right hand side source term.
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Akagi G.: Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces. J. Differ. Equ. 231(1), 32–56 (2006)
Alt H.W., Luckhaus S.: Quasilinear elliptic–parabolic differential equations. Math. Z. 183(3), 311–341 (1983)
Barenblatt G.I.: On self-similar motions of a compressible fluid in a porous medium. Akad. Nauk SSSR. Prikl. Mat. Meh. 16, 679–698 (1952)
Bénilan P., Boccardo L., Gallouët T., Gariepy R., Pierre M., Vázquez J.L.: An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(2), 241–273 (1995)
Blanchard D., Murat F.: Renormalised solutions of nonlinear parabolic problems with L 1 data: existence and uniqueness. Proc. Roy. Soc. Edinburgh Sect. A 127(6), 1137–1152 (1997)
Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vázquez J.L.: Asymptotics of the fast diffusion equation via entropy estimates. Arch. Ration. Mech. Anal. 191(2), 347–385 (2009)
Boccardo L., Dall’Aglio A., Gallouët T., Orsina L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147(1), 237–258 (1997)
Boccardo L., Gallouët T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87(1), 149–169 (1989)
Bögelein V., Duzaar F., Mingione G.: Degenerate problems with irregular obstacles. J. Reine Angew. Math. 650, 107–160 (2011)
Brézis H., Friedman A.: Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. (9) 62(1), 73–97 (1983)
Chasseigne E., Vazquez J.L.: Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities. Arch. Ration. Mech. Anal. 164(2), 133–187 (2002)
Dahlberg B.E.J., Kenig C.E.: Nonnegative solutions of the porous medium equation. Comm. Partial Differ. Equ. 9(5), 409–437 (1984)
Dahlberg B.E.J., Kenig C.E.: Nonnegative solutions to fast diffusions. Rev. Mat. Iberoamericana 4(1), 11–29 (1988)
Daskalopoulos, P., Kenig, C.E.: Degenerate diffusions. EMS tracts in mathematics, vol. 1. European Mathematical Society (EMS), Zürich (2007)
Herrero M.A., Pierre M.: The Cauchy problem for u t = Δu m when 0 < m < 1. Trans. Am. Math. Soc. 291(1), 145–158 (1985)
Kilpeläinen T., Malý J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19(4), 591–613 (1992)
Kilpeläinen T., Xu X.: On the uniqueness problem for quasilinear elliptic equations involving measures. Rev. Mat. Iberoamericana 12(2), 461–475 (1996)
Kilpeläinen T., Zhong X.: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc. 130(6), 1681–1688 (2002) (electronic)
Kinnunen J., Lindqvist P.: Definition and properties of supersolutions to the porous medium equation. J. Reine Angew. Math. 618, 135–168 (2008)
Korte R., Kuusi T., Siljander J.: Obstacle problem for nonlinear parabolic equations. J. Differ. Equ. 246(9), 3668–3680 (2009)
Lukkari T.: The porous medium equation with measure data. J. Evol. Equ. 10(3), 711–729 (2010)
Mikkonen P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Diss. 104, 1–71 (1996)
Naumann, J.: Einführung in die Theorie parabolischer Variationsungleichungen, volume 64 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. BSB B. G. Teubner Verlagsgesellschaft, Leipzig. With English, French and Russian summaries (1984)
Pierre M.: Uniqueness of the solutions of u t −Δ φ (u) = 0 with initial datum a measure. Nonlinear Anal. 6(2), 175–187 (1982)
Prignet A.: Existence and uniqueness of “entropy” solutions of parabolic problems with L 1 data. Nonlinear Anal. 28(12), 1943–1954 (1997)
Simon J.: Compact sets in the space L p (0, T; B). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)
Vázquez J.L.: Smoothing and decay estimates for nonlinear diffusion equations. Oxford lecture series in mathematics and its applications, vol. 33. Oxford University Press, Oxford (2006)
Vázquez J.L.: The porous medium equation. Oxford mathematical monographs. The Clarendon Press/Oxford University Press, Oxford (2007)
Zel′dovič, Ya. B., Kompaneec, A.S.: On the theory of propagation of heat with the heat conductivity depending upon the temperature. In: Collection in honor of the seventieth birthday of academician A. F. Ioffe, pp. 61–71. Izdat. Akad. Nauk SSSR, Moscow (1950)
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Lukkari, T. The fast diffusion equation with measure data. Nonlinear Differ. Equ. Appl. 19, 329–343 (2012). https://doi.org/10.1007/s00030-011-0131-4
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DOI: https://doi.org/10.1007/s00030-011-0131-4