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

• Main Authors: Ahmed Alsaedi, Bashir Ahmad, Mokhtar Kirane, Aberrazak Nabti
• Format: Article
• Language: English
• Published: Texas State University 2020-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: schrodinger equation; heisenberg group; life span; riemann-liouville fractional integrals and derivatives
• Online Access: http://ejde.math.txstate.edu/Volumes/2020/02/abstr.html

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.