Universal blow-up rates for a semilinear heat equation and applications. (English) Zbl 1028.35065

The authors consider positive solutions of the semilinear heat equation \[ u_t=\Delta u+u^p,\quad \text{ in} (0,T)\times\mathbb R^N, \] with \(p>1\) if \(N=1\) or 2 and \(1<p<(N+2)/(N-2)\) if \(N=3.\) It is shown that the blow-up rate of all radially decreasing solutions of the problem satisfies a universal global a priori estimate. Namely, \[ \|u(t)\|_\infty\leq C(T-t)^{-1/(p-1)},\quad \varepsilon T<t<T,\;\forall \varepsilon\in(0,1). \] This estimate implies that all global positive radially decreasing solutions decay at least like \(t^{-1/(p-1)}.\) Also, as a consequence of the above estimate, a parabolic Liouville theorem for the considered problem is derived. For some equations of the form \(u_t=\Delta u+f(u,\nabla u)\) results on blow-up rates and a priori estimates of global solutions are obtained.


35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
47D06 One-parameter semigroups and linear evolution equations
35K15 Initial value problems for second-order parabolic equations