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In this paper, we consider the Cauchy problem for the degenerate parabolicequations on the Heisenberg groups with power law non-linearities. We obtainFujita-type critical exponents, which depend on the homogeneous dimension ofthe Heisenberg groups. The analysis includes the case of porous mediumequations. Our proof approach is based on methods of nonlinear capacityestimates specifically adapted to the nature of the Heisenberg groups. We alsouse the Kaplan eigenfunctions method in combination with the Hopf-type lemma onthe Heisenberg groups.

Fujita-type results for the degenerate parabolic equations on the Heisenberg groups