Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 02, pp. 1-10.

Title: Lifespan of solutions of a fractional  evolution equation with
       higher order diffusion on the Heisenberg group

Authors: Ahmed Alsaedi (King Abdulaziz Univ., Jeddah, Saudi Arabia)
         Bashir Ahmad  (King Abdulaziz Univ., Jeddah, Saudi Arabia)
         Mokhtar Kirane (Univ. de La Rochelle, La Rochelle, France)
         Aberrazak Nabti (King Abdulaziz Univ., Jeddah, Saudi Arabia)

Abstract:
 We consider the higher order diffusion Schrodinger equation with a time nonlocal
 nonlinearity
 $$
 i\partial_tu-(-\Delta_{\mathbb{H}})^mu
 =\frac{\lambda}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1} | u(s)|^{p}\,ds,
 $$
 posed in $(\eta, t) \in \mathbb{H}\times(0,+\infty)$, supplemented with an initial
 data $u(\eta,0)=f(\eta)$, where $m>1,\,p>1,\,0<\alpha<1$, 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. Furthermore, we give an upper
 bound estimate of the life span of blow up solutions.
 
Submitted June 8, 2019. Published January 07, 2020.
Math Subject Classifications: 35Q55, 35B44, 26A33, 35B30.
Key Words: Schrodinger equation; Heisenberg group; life span;
           Riemann-Liouville fractional integrals and derivatives.