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

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.