Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds

Article type: Research Article

Authors: Punzo, Fabio

Affiliations: Dipartimento di Matematica ‘G. Castelnuovo’, Università di Roma ‘La Sapienza’, Piazzale A. Moro 5, I-00185 Roma, Italia. E-mail: [email protected]

Abstract: We study, on weighted Riemannian model manifolds, well posedness of the Cauchy problem for a class of quasilinear parabolic equations with a coefficient which can be singular at infinity. We establish uniqueness or non-uniqueness of bounded solutions, under suitable assumptions on the behavior at infinity of the singular coefficient and on the Green function for the weighted Laplace–Beltrami operator.

Keywords: singular non-linear parabolic equations, weighted Riemannian manifolds, weighted Laplace–Beltrami operator, well-posedness, sub-supersolutions

DOI: 10.3233/ASY-2012-1098

Journal: Asymptotic Analysis, vol. 79, no. 3-4, pp. 273-301, 2012