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Mathematische Annalen

, Volume 373, Issue 1–2, pp 119–153 | Cite as

The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

  • Gabriele GrilloEmail author
  • Matteo Muratori
  • Juan Luis Vázquez
Article
  • 56 Downloads

Abstract

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan–Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the porous medium equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan–Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted porous medium equations in the Euclidean space having special weights.

Mathematics Subject Classification

Primary 35R01 Secondary 35K65 58J35 35A01 35A02 35B44 

Notes

Acknowledgements

J.L.V. was supported by Spanish Project MTM2014-52240-P. G.G. was partially supported by the PRIN Project “Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni” (Italy). M.M. was partially supported by the GNAMPA Project “Equazioni diffusive non-lineari in contesti non-Euclidei e disuguaglianze funzionali associate” (Italy). Both G.G. and M.M. have also been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM, Italy).

References

  1. 1.
    Arnaudon, M., Thalmaier, A.: Random Walks, Boundaries and Spectra. Brownian Motion and Negative Curvature, vol. 64, pp. 143–161. Birkhäuser/Springer, Basel (2011)Google Scholar
  2. 2.
    Bénilan, P., Crandall, M.G.: The continuous dependence on \(\varphi \) of solutions of \(u_t - \Delta \varphi (u) = 0\). Indiana Univ. Math. J. 30, 161–177 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bénilan, P., Crandall, M.G., Pierre, M.: Solutions of the porous medium equation in \({\mathbb{R}}^N\) under optimal conditions on initial values. Indiana Univ. Math. J. 33, 51–87 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonforte, M., Vázquez, J.L.: Global positivity estimates and Harnack inequalities for the fast diffusion equation. J. Funct. Anal. 240, 399–428 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brezis, H., Kamin, S.: Sublinear elliptic equations in \({\mathbb{R}}^n\). Manuscr. Math. 74, 87–106 (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    DiBenedetto, E., Kwong, Y., Vespri, V.: Local space-analyticity of solutions of certain singular parabolic equations. Indiana Univ. Math. J. 40, 741–765 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eidus, D., Kamin, S.: The filtration equation in a class of functions decreasing at infinity. Proc. Am. Math. Soc. 120, 825–830 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Greene, R.E., Wu, H.: Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Mathematics, 699. Springer, Berlin (1979)CrossRefGoogle Scholar
  9. 9.
    Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grillo, G., Muratori, M.: Smoothing effects for the porous medium equation on Cartan–Hadamard manifolds. Nonlinear Anal. 131, 346–362 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grillo, G., Muratori, M., Punzo, F.: Conditions at infinity for the inhomogeneous filtration equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 413–428 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grillo, G., Muratori, M., Punzo, F.: On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete Contin. Dyn. Syst. 35, 5927–5962 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grillo, G., Muratori, M., Punzo, F.: The porous medium equation with large initial data on negatively curved Riemannian manifolds. J. Math. Pures Appl. 113, 195–226 (2018).  https://doi.org/10.1016/j.matpur.2017.07.021 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grillo, G., Muratori, M., Vázquez, J.L.: The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour. Adv. Math. 314, 328–377 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ishige, K.: An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders. J. Differ. Equ. 158, 251–290 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ishige, K.: An intrinsic metric approach to uniqueness of the positive Cauchy-Neumann problem for parabolic equations. J. Math. Anal. Appl. 276, 763–790 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ishige, K., Murata, M.: Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30, 171–223 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kamin, S., Punzo, F.: Dirichlet conditions at infinity for parabolic and elliptic equations. Nonlinear Anal. 138, 156–175 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kamin, S., Reyes, G., Vázquez, J.L.: Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density. Discrete Contin. Dyn. Syst. 26, 521–549 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kersner, R., Tesei, A.: Well-posedness of initial value problems for singular parabolic equations. J. Differ. Equ. 199, 47–76 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Murata, M.: Nonuniqueness of the positive Dirichlet problem for parabolic equations in cylinders. J. Funct. Anal. 135, 456–487 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Murata, M.: Heat escape. Math. Ann. 327, 203–226 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pierre, M.: Uniqueness of the solutions of \(u_t - \Delta \varphi (u) = 0\) with initial datum a measure. Nonlinear Anal. 6, 175–187 (1982)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Punzo, F.: Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds. Asymptot. Anal. 79, 273–301 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Vázquez, J.L.: The Dirichlet problem for the porous medium equation in bounded domains. Asymptotic behavior. Monatsh. Math. 142, 81–111 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)Google Scholar
  27. 27.
    Vázquez, J.L.: Fundamental solution and long time behavior of the porous medium equation in hyperbolic space. J. Math. Pures Appl. 104, 454–484 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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