Weissler, Fred B. An \(L^{\infty}\) blow-up estimate for a nonlinear heat equation. (English) Zbl 0592.35071 Commun. Pure Appl. Math. 38, 291-295 (1985). This paper gives an elegant proof to the following theorem: Suppose for some \(\rho\), \(\tau >0\) the function u(x,t), defined on \(\Gamma =\{(x,t):| x| <\rho,0<t<\tau \}\) satisfies (a) \(u\in C^ 1(\Gamma)\), and u has continuous 2nd order x-derivatives in \(\Gamma\) ; (b) \(u\geq 0\), \(u_ t\geq 0\) in \(\Gamma\) ; (c) for each \(t\in (0,\tau)\), u(.,t) is radially symmetric and nonincreasing as a function of \(| x|;\) (d) for each \(t\in (0,\tau)\), \(u_ t(.,t)\) achieves its maximum at \(x=0;\) (e) u satisfies \(u_ t=\Delta u+u^ p\) in \(\Gamma\), \(p>1;\) (f) u(0,t)\(\to \infty\) as \(t\to \tau.\) Assume also that \(n\leq 2\) or \(p<(n+2)/(n-2)\). Then there exists a constant \(c>0\) such that u(x,t)\(\leq c(\tau -t)^{-1/(p-1)}\) for all (x,t)\(\in \Gamma.\) The author points out that such an estimate is the main hypothesis in a paper of Y. Giga and R. V. Kohn in the same volume [ibid. 38, 297-319 (1985; Zbl 0585.35051)]. The references include four items. Reviewer: J.E.Bouillet Cited in 62 Documents MSC: 35K55 Nonlinear parabolic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35L99 Hyperbolic equations and hyperbolic systems 35K05 Heat equation Keywords:blow-up solutions; nonlinear heat equation; estimate Citations:Zbl 0585.35051 PDFBibTeX XMLCite \textit{F. B. Weissler}, Commun. Pure Appl. Math. 38, 291--295 (1985; Zbl 0592.35071) Full Text: DOI References: [1] Giga, Comm. Pur Appl. Math. 38 pp 297– (1985) [2] Haraux, Ind. Univ. Math. Jnl. 31 pp 167– (1982) [3] Joseph, Arch. Rat. Mech. Anal. 49 pp 241– (1973) [4] Weissler, Ind. Univ. Math. Jnl. 29 pp 79– (1980) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.