The blow-up of solutions of the Cauchy problem \[ u_t= u_{xx}+|u|^{p-1}u\quad\text{in }\mathbb{R}\times (0,\infty),\quad u(x,0)= u_0(x)\quad\text{in }\mathbb{R}, \] is studied. For each nonnegative integer \(k\), let \(\Sigma_k\) be the set of functions on \(\mathbb{R}\) which change \(\text{sign }k\) times and satisfy some decay condition. It is shown that for \(p_k=1+2/(k+1)\), any solution with \(u_0\in\Sigma_k\) blows up in finite time if \(1<p\leq p_k\), but there exists a global solution with \(u_0\in\Sigma_k\) if \(p>p_k\).