We in this paper improve a method of establishing the existence of
finite time blow-up solutions, and then apply it to study the finite
time blow-up, the blow-up time and the blow-up rate of the weak
solutions on the initial boundary problem of u_t -
\Delta u_{t} - \Delta u_{t} =
|u|^{p - 1}u. By applying this improved method,
we prove that I(u_{0}) < 0 is a sufficient condition of the
existence of the finite time blow-up solutions and
\frac{2(p -
1)^{-1}\|u_{0}\|_{H_{0}^{1}}^{2}}{(p
- 1) \|\nabla
u_{0}\|_{2}^{2} - 2(p +
1)J(u_{0})} is an upper bound for the blow-up time, which generalize
the blow-up results of the predecessors in the sense of the variation.
Moreover, we estimate the upper blow-up rate of the blow-up solutions,
too.
Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation