Poon, Chi-Cheung Blow-up behavior for semilinear heat equations in nonconvex domains. (English) Zbl 0990.35022 Differ. Integral Equ. 13, No. 7-9, 1111-1138 (2000). The question of the blow-up of solutions of nonlinear parabolic equations in finite time has received a lot of attention in the last 20 years. This paper focuses on the behavior of nonnegative solutions of the nonlinear heat equation \(u_t=\Delta u +u^p\) in the space-time domain \(\Omega\times(0,T)\), when \(\Omega\) is a smoothly bounded domain in \(\mathbb R^n\). Y. Giga and R. V. Kohn [Commun. Pure Appl. Math. 42, No. 6, 845-884 (1989; Zbl 0703.35020); Indiana Univ. Math. J. 36, 1-40 (1987; Zbl 0601.35052); Commun. Pure Appl. Math. 38, 297-319 (1985; Zbl 0585.35051)] studied this problem with a Dirichlet boundary condition under the additional hypothesis that \(\Omega\) is convex. They introduced a functional (in terms of the heat kernel \(K(x,y,t) = (4\pi t)^{-n/2}\exp (-|x-y|^2/(4t))\)) which they used to identify blow-up points and analyze the rate at which the solutions blow up near such a point. The current author modifies the arguments of Giga and Kohn to consider nonconvex domains and also the Neumann boundary condition. One interesting difference between the two approaches is that Poon replaces the heat kernel \(K\) by the Neumann heat kernel in the functional. Reviewer: Gary M.Lieberman (Ames) Cited in 6 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs Keywords:blow-up rate; Dirichlet boundary condition; blow-up points; Neumann boundary condition; Neumann heat kernel Citations:Zbl 0703.35020; Zbl 0601.35052; Zbl 0585.35051 PDFBibTeX XMLCite \textit{C.-C. Poon}, Differ. Integral Equ. 13, No. 7--9, 1111--1138 (2000; Zbl 0990.35022)