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An \(L^{\infty}\) blow-up estimate for a nonlinear heat equation. (English) Zbl 0592.35071

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

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

Citations:

Zbl 0585.35051
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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)
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