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

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.

Keywords

• schrodinger equation
• heisenberg group
• life span
• riemann-liouville fractional integrals and derivatives