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

Abstract: We consider the fractional order integral equation with a time nonlocal nonlinearity $$^{c}\mathbf{D}_{0\mid t}^{\beta}\left( u \right) +\left(-\Delta_{\mathbb{H}} \right)^{m} \left( u \right) = \frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left( t-\omega\right)^{\alpha-1}\vert u(\omega)\vert^{p} d\omega,$$ posed in $ (.,t)\in\mathbb{H}\times(0,\infty) $, supplemented with an initial data $u(.,0)=u_{0}(.) $,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 $\Delta_{\mathbb{H}}$ is the Laplacian operator on the $ (2N+1) $-dimensional Heisenberg group $\mathbb{H} $.Then, we prove a blow up result for its solutions.