  We consider the viscous $n$-dimensional Camassa-Holm equations, with
$n=2,3,4$ in the whole space. We establish existence and regularity of the
solutions and study the large time behavior of the solutions in several Sobolev
spaces. We first show that if the data is only in $L^2$ then the solution
decays without a rate and that this is the best that can be expected for data
in $L^2$. For solutions with data in $H^m\cap L^1$ we obtain decay at an
algebraic rate which is optimal in the sense that it coincides with the rate of
the underlying linear part.
