DING Juntang;LI Shengjia
Journal of Systems Science and Complexity. 2005, 18(2): 265-276.
The type of problem under consideration is
$$
\left\{
\begin{array}{ll}
u_{t}=\nabla(a(u)b(x)\nabla u)+g(x,q,t)f(u)&$in$ \ D\!\times\!(0,T),\\[1mm]
\displaystyle u=0 \ $on$ \ {\it \Gamma}_1\!\times\!(0,T),~ \ \displaystyle
\frac{\partial u}{\partial n}+\sigma(x,t)u=0
\ $on$ \ {\it\Gamma}_2\!\times\!(0,T), &{\it \Gamma}_1\!\cup\!{\it \Gamma}_2\!=\!\partial\!D,\\[2mm]
u(x,0)=u_0(x)\geq0, \ \not\equiv0&$in$ \ {\overline D},
\end{array}
\right.
$$
where $D$ is a smooth bounded domain of $R^N, \ q\!=\!|\nabla\!u|^2.$ By constructing
an auxiliary function and using Hopf's maximum principles on it, existence theorems
of blow-up solutions, upper bound of ``blow-up time" and upper estimates of ``blow-up
rate" are given under suitable assumptions on $a, b, f, g, \sigma$ and initial date
$u_0(x).$ The obtained results are applied to some examples in which $a, b, f, g$ and
$\sigma$ are power functions or exponential functions.
