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 $$^{c}\mathbf{D}_{0\mid t}^β\left( u \right) +\left(-Δ_{\mathbb{H}} \right)^{m} \left( u \right) = \frac{1}{Γ(α)}\int_{0}^{t}\left( t-ω\right)^{α-1}\vert u(ω)\vert^{p} dω,$$ posed in $ (.,t)\in\mathbb{H}\times(0,\infty) $, supplemented with an initial data $u(.,0)=u_{0}(.) $,where $ m>1 \ , \ p>1 \ , \ 0<β<1 \ , \ 0<α<1 $, and $^{c}\mathbf{D}_{0\mid t}^β $ denotes the caputo fractional derivative of order $ β$, and $Δ_{\mathbb{H}}$ is the Laplacian operator on the $ (2N+1) $-dimensional Heisenberg group $\mathbb{H} $.Then, we prove a blow up result for its solutions.
math.GM↗