Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation u t − △ u t − △ u − div ( | ∇ u | 2 q ∇ u ) = u p which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016 ), where the blow-up phenomena and the lifespan for the initial energy J ( u 0 ) < 0 were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J ( u 0 ) < 0 .