Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space

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We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space. We state sufficient conditions for uniqueness or nonuniqueness of bounded solutions, depending on the behavior of the density at infinity. Nonuniqueness relies on the prescription at infinity of suitable conditions of Dirichlet type, and possibly inhomogeneous.