
12.
BLOWUP SOLUTIONS FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS
DING Juntang;LI Shengjia
Journal of Systems Science and Complexity
2005, 18 (2):
265276.
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 blowup solutions, upper bound of ``blowup time" and upper estimates of ``blowup 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.
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