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 $$ u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}\bigl(| \nabla u|^{2q}\nabla u\bigr)=u^{p} $$ 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 $J(u_{0})<0$ .

Abstract licence: CC BY-NC-ND

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