Abstract
In the slow diffusion case, unbounded supersolutions of the porous medium equation are of two totally different types, depending on whether the pressure is locally integrable or not. This criterion and its consequences are discussed.
Similar content being viewed by others
References
Benny Avelin and Teemu Lukkari. Lower semicontinuity of weak supersolutions to the porous medium equation. Proc. Amer. Math. Soc., 143(8):3475–3486, 2015.
Verena Bögelein, Pekka Lehtelä and Stefan Sturm. Regularity of weak solutions and supersolutions to the porous medium equation. Submitted, 2017.
Emmanuele DiBenedetto and Avner Friedman. Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math., 357:1–22, 1985.
Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri.Harnack’s Inequality for Degenerate and Singular Parabolic Equations.Springer Monographs in Mathematics. Springer, New York, 2012.
Emmanuele DiBenedetto. Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Arch. Rational Mech. Anal., 100(2):129–147, 1988.
Emmanuele DiBenedetto. Degenerate Parabolic Equations. Universitext. Springer-Verlag, New York, 1993.
Björn E. J. Dahlberg and Carlos E. Kenig. Nonnegative solutions of the porous medium equation. Comm. Partial Differential Equations, 9(5):409–437, 1984.
Panagiota Daskalopoulos and Carlos E. Kenig. Degenerate Diffusions Initial value problems and local regularity theory. EMS Tracts in Mathematics. Initial value problems and local regularity theory. EMS Tracts in Mathematics. European Mathematical Society (EMS), 2007.
Juha Kinnunen and Peter Lindqvist. Definition and properties of supersolutions to the porous medium equation. J. Reine Angew. Math., 618:135–168, 2008.
Juha Kinnunen and Peter Lindqvist. Unbounded supersolutions of some quasilinear parabolic equations: a dichotomy. Nonlinear Anal., 131:229–242, 2016.
Juha Kinnunen and Peter Lindqvist. Erratum to Definition and properties of supersolutions to the porous medium equation (J. reine angew. Math. 618 (2008), 135–168). J. Reine Angew. Math., 725:249, 2017.
Riikka Korte, Pekka Lehtelä and Stefan Sturm. Lower semicontinuous obstacles for the porous medium equation. J. Differential Equations (to appear).
Tuomo Kuusi, Peter Lindqvist, and Mikko Parviainen. Shadows of infinities. Ann. Mat. Pura Appl. (4), 195(4):1185–1206, 2016.
Pekka Lehtelä. A weak Harnack estimate for supersolutions to the porous medium equation. Differential Integral Equations, 30(11-12):879–916, 2017.
Juan Luis Vázquez. The Porous Medium Equation. Mathematical theory. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2007.
Zhuoqun Wu, Junning Zhao, Jingxue Yin, and Huilai Li. Nonlinear Diffusion Equations. World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research is supported by the Academy of Finland, the Emil Aaltonen Foundation and the Norwegian Research Council.
Rights and permissions
About this article
Cite this article
Kinnunen, J., Lehtelä, P., Lindqvist, P. et al. Supercaloric functions for the porous medium equation. J. Evol. Equ. 19, 249–270 (2019). https://doi.org/10.1007/s00028-018-0474-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-018-0474-y