Initial blow-up rates and universal bounds for nonlinear heat equations

We establish a universal upper bound on the initial blow-up rate for all positive classical solutions of the Dirichlet problem for the nonlinear heat equation ut=(delta)u+u^p on (0,T)×(omega), where p>1 and (omega) is a smoothly bounded domain in R^N. Namely, we show that ||u(t)||=< C(p,(omega),T) t^-(alpha) on (0,T/2] for some (alpha)=(alpha)(N,p)>0. This is proved for all subcritical p (i.e., p< (N+2)/(N-2)) if N=<4 (and under a stronger assumption on p if N>=5). As a consequence, we improve the known results on universal bounds for global solutions. Furthermore, if p<(N+3)/(N+1), then we may take (alpha)=(N+1)/2 and we show that this value of (alpha) is optimal. Interestingly, the rate can be faster than the previously known, maximal initial blow-up rate of the Cauchy problem. Applications to universal blow-up estimates at t=T are given. The Neumann problem is also considered and we obtain the estimate on the initial rate for all subcritical p up to dimension N=6.