Blowup rate of solutions for a semilinear heat equation with the Dirichlet boundary condition

Let p>1 and Ω be a smoothly bounded domain in R^N not necessarily convex. This paper is concerned with a Cauchy–Dirichlet problem $\def\theequation{(P)}\begin{equation}\left\{\begin{array}{l@{\quad}l}u_{t}=\Delta u+u^{p}&\mbox{in }\varOmega \times (0,\infty),\\[3pt]u(x,t)=0&\mbox{on}\ \curpartial \varOmega \times (0,\infty),\\[3pt]u(x,0)=\phi(x)\geq 0&\mbox{in }\varOmega.\end{array}\right.\end{equation}$ We show that when u blows up at t=T, it holds

|u(t)|_∞≤C(T−t)^−1/(p−1) in (0,T)

with some C>0 under some condition on the Cauchy problem for (P).