Advances in Differential Equations

Universal blow-up rates for a semilinear heat equation and applications

Júlia Matos and Philippe Souplet

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Abstract

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.

Article information

Source
Adv. Differential Equations Volume 8, Number 5 (2003), 615-639.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926843

Mathematical Reviews number (MathSciNet)
MR1972493

Zentralblatt MATH identifier
1028.35065

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Citation

Matos, Júlia; Souplet, Philippe. Universal blow-up rates for a semilinear heat equation and applications. Adv. Differential Equations 8 (2003), no. 5, 615--639. https://projecteuclid.org/euclid.ade/1355926843.


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