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

Abstract

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups.