2003 Universal blow-up rates for a semilinear heat equation and applications
Júlia Matos, Philippe Souplet
Adv. Differential Equations 8(5): 615-639 (2003). DOI: 10.57262/ade/1355926843


We consider positive solutions of the semilinear heat equation $$u_t=\Delta u+u^p,\quad \hbox{ in $(0,T)\times {\mathbb R}^N$,} \tag*{(1)} $$ with $p>1$ if $N=1$ or $2$ and $1 < p < {N+2\over N-2}$ if $N=3$. We show that the blow-up rate of all radially decreasing solutions of (1) satisfies a universal global a priori estimate. Namely, for all $\varepsilon\in (0,1)$, we prove that $$ \|u(t)\|_\infty\leq C(T-t)^{-1/(p-1)},\quad \varepsilon T < t < T, \tag*{(2)} $$ where $C=C(N, p, \varepsilon)>0$ is {\it independent of} $u$. This estimate has various applications. In particular, it implies a strong uniform decay property for global solutions of (1), which seems to have been conjectured in previous works on equation (1). Namely, all global positive radially decreasing solutions of (1) decay at least like $t^{-1/(p-1)}$. Also, as consequences of (2), we derive a parabolic Liouville theorem for (1), and for some equations of the form $u_t=\Delta u+f(u,\nabla u)$, we obtain results on blow-up rates and a priori estimates of global solutions. The proof of (2) relies on smoothing estimates in the uniformly local Lebesgue spaces $L^q_{\rho,\star}$, obtained in a previous work of the authors, together with the use of self-similar variables and suitable energy arguments.


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Júlia Matos. Philippe Souplet. "Universal blow-up rates for a semilinear heat equation and applications." Adv. Differential Equations 8 (5) 615 - 639, 2003. https://doi.org/10.57262/ade/1355926843


Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1028.35065
MathSciNet: MR1972493
Digital Object Identifier: 10.57262/ade/1355926843

Primary: 35K55
Secondary: 35B40 , 47D06

Rights: Copyright © 2003 Khayyam Publishing, Inc.


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Vol.8 • No. 5 • 2003
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