## 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.

### MSC:

 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