## 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

### Keywords:

blow-up solutions; nonlinear heat equation; estimate

Zbl 0585.35051
Full Text:

### 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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