Improving a method of constructing finite time blow-up solutions and its an application

We in this paper improve a method for establishing the finite time blow-up solutions on the parabolic or pseudoparabolic equations. And we apply it to restudy the finite time blow-up and upper bound for the blow-up time on the initial boundary value problem of ut-Delta ut-Delta u=|u|p-1u$$ {u}_t-\Delta {u}_t-\Delta u equal to {\left vertical bar u\right vertical bar} circumflex {p-1}u $$. Specifically, we prove that I(u0)<0$$ I\left({u}_0\right)<0 $$ is a sufficient condition of the existence of finite time blow-up solutions and 2(p-1)-1||u0||H012(p-1)|| backward difference u0||22-2(p+1)J(u0)$$ \frac{2{\left(p-1\right)} circumflex {-1}{\left\Vert {u}_0\right\Vert}_{H_0 circumflex 1} circumflex 2}{\left(p-1\right){\left\Vert \nabla {u}_0\right\Vert}_2 circumflex 2-2\left(p+1\right)J\left({u}_0\right)} $$ is a new upper bound for the blow-up time, which generalizes the results of previous studies in the variational sense. Moreover, we estimate the upper blow-up rate of these solutions, too.