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

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.