Support properties of solutions to nonlinear parabolic equationswith variable density in the hyperbolic space

Fabio Punzo

doi:10.3934/dcdss.2012.5.657

Article Contents

2012, Volume 5, Issue 3: 657-670. Doi: 10.3934/dcdss.2012.5.657

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Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space

Fabio Punzo^1,

1.
Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro 5, I-00185 Roma

Received: May 31, 2010

Revised: July 31, 2010

Published: May 2012

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Abstract

We consider the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space, assuming that the initial datum has compact support. We provide simple conditions, involving the behaviour of the density at infinity, so that the support of every nonnegative solution is not compact at some positive time, or it remains compact for any positive time. These results extend to the case of the hyperbolic space those given in [8] for the Cauchy problem in $\mathbb{R}^n$.

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Mathematics Subject Classification: Primary: 35K61, 35K67, 35B99; Secondary: 35B40, 35B51.

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