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Zbl 1191.35074
Zhang, Zhengce; Hu, Bei
Gradient blowup rate for a semilinear parabolic equation. (English)
[J] Discrete Contin. Dyn. Syst. 26, No. 2, 767-779 (2010). ISSN 1078-0947; ISSN 1553-5231/e

The paper is devoted to the initial boundary value problem for the heat equation with a nonlinear gradient source term: $u_t = u_{xx} + x^m|u_x|^p$, $0<x<1$, $t>0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions remain bounded. It is shown that the spatial derivative of solutions is globally bounded in the case $p\le m+2$ while blowup occurs at the boundary when $p > m+2$. The authors prove that, with an additional restriction on $p$ and the assumption on the initial data so that the solution is monotonically increasing both in time and in space, both the lower and upper bound of the blowup rate are $(T-t)^{-(m+1)/(p-m-2)}$.
[Vyacheslav I. Maksimov (Ekaterinburg)]

MSC 2000:: *35B44
35K58
35K20 Second order parabolic equations, boundary value problems

Keywords: blowup rate; nonlinear gradient source; one space dimension

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