Porous medium equations on manifolds with critical negative curvature: unbounded initial data

ABSTRACT

We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan–Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.

KEYWORDS:

• Porous medium equation
• Riemannian manifolds
• blow-up
• unbounded data
• negative curvature

AMS Subject Classifications:

• 35R01
• 35K65
• 35A01
• 35A02
• 35B44

Acknowledgements

The authors thank the ‘Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni’ (GNAMPA) of the ‘Istituto Nazionale di Alta Matematica’ (INdAM, Italy).

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the GNAMPA Project 2017 ‘Equazioni Diffusive Non-lineari in Contesti Non-Euclidei e Disuguaglianze Funzionali Associate’. Istituto Nazionale di Alta Matematica (INdAM).