Global existence and blow-up for the viscoelastic damped wave equation on the Heisenberg group

The purpose of this article is to study the Cauchy problem for the viscoelastic damped wave equation on the Heisenberg group. We first prove the global existence of small data solutions for \(p\in[2,Q/(Q-4)]\) if \(n=2,3\), \(p>2\) if \(n=1\) using the contraction principle. Then, a blow-up result is obtained by using the test function method under certain integral sign assumptions for the Cauchy data when \(1< p\leq1+2/(Q-1)\), where \(Q=2n+2\) is the homogeneous dimension of the Heisenberg group. Moreover, we obtain the upper bound for the lifespan of the solution by employing a revisited test function method. For more information see https://ejde.math.txstate.edu/Volumes/2025/71/abstr.html