Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

We investigate the initial boundary value problem of the pseudo-parabolic equation ut-/△ut-/△u=φuu+|u|p-1u,whereφu is the Newtonian potential,which was studied by Zhu et al.(Appl.Math.Comput.,329(2018)38-51),and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels.We in this note determine the upper and lower bounds for the blow-up time.While estimating the upper bound of blow-up time,we also find a sufficient condition of the solution blowingup in finite time at arbitrary initial energy level.Moreover,we also refine the upper bounds for the blow-up time under the negative initial energy.