Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power non-linearity. We prove that the critical exponent is the Fujita exponent pFuj(Q) = 1 + 2/Q, where Q is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for p>pFuj(Q) in an exponential weighted energy space. On the other hand, a blow-up result for 1 < p ≤ pFuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.