Blow-up set for a superlinear heat equation and pointedness of the initial data

We study the blow-up problem for a superlinear heat equation\begin{equation}\label{eq:P}\tag{P}\left\{\begin{array}{ll}\partial_t u = \epsilon \Delta u + f(u), x\in\Omega, \,\,\, t>0, \\u(x,t)=0,         x\in\partial\Omega, \,\,\, t>0, \\u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0), x\in\Omega,\end{array}\right.\end{equation}where $\partial_t=\partial/\partial t$, $\epsilon>0$ is a sufficiently small constant, $N\ge 1$, $\Omega\subset {\bf R}^N$ is a domain,$\varphi\in C^2(\Omega)\cap C(\overline{\Omega})$ is a nonnegative bounded function,and $f$ is a positive convex function in $(0,\infty)$.In [10], the author of this paper and Ishige characterized the location of the blow-up setfor problem (p) with $f(u)=u^p$ ($p>1$)with the aid of the invariance of the equation under some scale transformation for the solution,which played an important role in their argument.However, due to the lack of such scale invariance for problem (p),we can not apply their argument directly to problem (p).In this paper we introduce a new transformation for the solution of problem (p),which is a generalization of the scale transformation introduced in [10],and generalize the argument of [10].In particular, we show the relationship between the blow-up set for problem (p)and pointedness of the initial function under suitable assumptions on $f$.