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

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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.

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Notes

  1. These conditions can be relaxed, see below, but they allow to simplify the presentation.

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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).

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Gabriele Grillo & Matteo Muratori

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Juan Luis Vázquez

Authors
  1. Gabriele Grillo
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  2. Matteo Muratori
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  3. Juan Luis Vázquez
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Correspondence to Gabriele Grillo.

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Communicated by Y. Giga.

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Grillo, G., Muratori, M. & Vázquez, J.L. The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case. Math. Ann. 373, 119–153 (2019). https://doi.org/10.1007/s00208-018-1680-1

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  • DOI: https://doi.org/10.1007/s00208-018-1680-1

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