Nonexistence results of global solutions for fractional order integral equations on the Heisenberg group

We consider the fractional order integral equation with a time nonlocal nonlinearity $${}^c{\mathbf{D}}_{0\mid t}^{\beta }\left(u\right)+{(-{\mathrm{\Delta }}_{\mathbb{H}})}^m\left(u\right)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-\omega )}^{\alpha -1}\mathrm{\mid }u(\omega ){\mathrm{\mid }}^{p}d\omega ,$$ posed in $(\mathrm{.},t)\in \mathbb{H}\times (0,\mathrm{\infty })$, supplemented with an initial data $u(\mathrm{.},0)=u_0(\mathrm{.})$,where $m>1,p>1,0<\beta <1,0<\alpha <1$, and ${}^c{\mathbf{D}}_{0\mid t}^{\beta }$ denotes the caputo fractional derivative of order $\beta$, and ${\mathrm{\Delta }}_{\mathbb{H}}$ is the Laplacian operator on the $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$.Then, we prove a blow up result for its solutions.