Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 71, pp. 1-12.

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

Authors: Xingquan Li (Southwest Jiaotong Univ., Chengdu, China)
         Yuli Feng (Southwest Jiaotong Univ., Chengdu, China)
         Han Yang (Southwest Jiaotong Univ., Chengdu, China)

Abstract:
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.

Submitted March 24, 2025. Published July 14, 2025.
Math Subject Classifications: 35A01, 35B44, 35R03.
Key Words: Viscoelastic damped wave equation; Heisenberg group; test function method; global existence; blow up.