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−Δut−Δu=|u|p−1u$$ {u}_t-\Delta {u}_t-\Delta u&amp;#x0003D;{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{p-1}u $$. Specifically, we prove that I(u0)<0$$ I\left({u}_0\right)&lt;0 $$ is a sufficient condition of the existence of finite time blow‐up solutions and 2(p−1)−1‖u0‖H012(p−1)‖∇u0‖22−2(p+1)J(u0)$$ \frac{2{\left(p-1\right)}&amp;#x0005E;{-1}{\left\Vert {u}_0\right\Vert}_{H_0&amp;#x0005E;1}&amp;#x0005E;2}{\left(p-1\right){\left\Vert \nabla {u}_0\right\Vert}_2&amp;#x0005E;2-2\left(p&amp;#x0002B;1\right)J\left({u}_0\right)} $$ 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. [ABSTRACT FROM AUTHOR]