## Advances in Differential Equations

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

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

Júlia Matos and Philippe Souplet

#### 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