Porous medium equations on manifolds with critical negative curvature: unbounded initial data. Issue 10 (27th July 2019)

• Record Type: Journal Article
• Authors: Muratori, Matteo; Punzo, Fabio
• Is Part Of: Applicable analysis. Volume 98:Issue 10(2019)
• Page Start: 1756
• Page End: 1772
• Publication Date: 2019-07-27
• Subjects: Porous medium equation -- Riemannian manifolds -- blow-up -- unbounded data -- negative curvature; 35R01 -- 35K65 -- 35A01 -- 35A02 -- 35B44; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1441996
• Languages: English
• ISSNs: 0003-6811

Abstract

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.