- Titolo:
- The porous medium equation with growing data on negatively curved Riemannian manifolds
- Quando:
- 08.05.2017 13.30 h
- Dove:
- Palazzo Campana - Torino
- Aula:
- Sala S
- Relatore:
- Dott. Matteo Muratori
- Afferenza:
- Politecnico di Milano
- Proponente:
- V. Barutello
- Locandina:

We show existence and uniqueness of very weak solutions to the Cauchy problem for the porous medium equation on complete, simply connected Riemannian manifolds with nonpositive sectional curvatures (Cartan-Hadamard), which in addition are supposed to satisfy suitable bounds from below on Ricci curvature (it can decrease at most quadratically at spatial infinity). Inspired by the seminal paper by Bénilan, Crandall and Pierre on Euclidean space, we allow the initial datum to have a power-type growth rate at infinity. Such a rate, in order to establish well-posedness, turns out to depend crucially on the curvature bounds. For instance, the associated pressure can grow at most linearly on hyperbolic space and quadratically on Euclidean space and on a class of manifolds whose Ricci curvature vanishes fast enough at infinity. The curvature conditions we require are to some extent optimal, in the sense that if they are not satisfied then uniqueness fails even for bounded initial data. Furthermore, upon assuming upper bounds on sectional curvatures that match the lower ones, we can give a sharp estimate for the maximal existence time of a solution, which shows in particular that our growth hypotheses on the initial datum cannot be improved (otherwise existence fails). Finally, at the maximal existence time, we can also provide precise pointwise blow-up results for a particular class of manifolds and initial data that fit within our framework.

- Sede:
- Palazzo Campana - Sito web
- Via:
- Via Carlo Alberto, 10
- Cap:
- 10123
- Città:
- Torino
- Provincia:
- To
- Paese:

Dipartimento di Matematica, Università degli Studi di Torino