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

Article type: Research Article

Authors: Mizoguchi, Noriko

Affiliations: Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184‐8501, Japan E‐mail: mizoguti@u‐gakugei.ac.jp

Abstract: Let p>1 and Ω be a smoothly bounded domain in RN 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).

Journal: Asymptotic Analysis, vol. 35, no. 2, pp. 91-112, 2003