Nonexistence results for semi linear structural damped wave model with nonlinear memory on the Heisenberg group

In this paper we investigate the following semi-linear structurally damped wave equation with nonlinear memory
\(\left\{ \begin{array}{ll} u_{tt}-\Delta_{\mathbb{H}}u+(-\Delta_{\mathbb{H}})^{\frac{\sigma}{2}}u_{t}= \displaystyle\int_{0}^{t}(t-s)^{-\gamma}|u(s)|^{p}ds,\quad \eta\in\mathbb{R}^{2n+1},\quad t>0,\\ \cr u(\eta,0)=u_0(\eta) \quad , \quad  u_t(\eta,0)=u_1(\eta), \end{array} \right.\)
where  \(\sigma\in (0, 2),\ \gamma\in (0, 1),\ p\in \mathbb{R},\ p>1\) and \(\Delta_{\mathbb{H}} \) is the Kohn-Laplace operator on the (2n+1)-dimensional Heisenberg group \(\mathbb{H}\). We intend to apply the method of a modified test function to establish the nonexistence of global weak solutions and to overcome some difficulties as well caused by the well-known fractional Laplacian \((-\Delta)^{\frac{\sigma}{2}}.\)