We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation
$u_t= \Delta u+u^p$ in $R^N$, where $p>1+2/N$, $p(N-2)\leq N+2$. The initial data are of the form $u(x,0)=\alpha\phi(x)$, where $\phi$ is a fixed function with suitable decay at $|x|=\infty$ and $\alpha>0$ is a parameter. There exists a threshold parameter $\alpha^*$ such that the solution exists globally if and only if $\alpha\leq\alpha^*$. Our main results describe the asymptotic behavior of the solutions for $\alpha\in (0,\alpha^*]$ and in particular exhibit the difference between the behavior of sub-threshold solutions ($\alpha<\alpha^*$) and the threshold solution ($\alpha=\alpha^*$).