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

Abstract

We consider the higher order diffusion Schrödinger equation with a time nonlocal nonlinearity i∂tu - (-Δℍ)mu = λ/Γ(α) ∫t0 (t - s)α-1 |u(s)|pds, posed in (η, t) ∈ ℍ x (0, +∞), supplemented with an initial data u(η, 0) = ƒ(η), where m > 1, p > 1, < α < 1, and Δℍ is the Laplacian operator on the (2N + 1)-dimensional Heisenberg group 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.