We consider the Dirichlet problem for the equation
$$
 u_t =\Delta u +\lambda e ^u
$$
on  the unit ball in $R^N$, $3 \le N\le 9$.  Given two equilibria $\phi^-$, $\phi^+$, we give a necessary and sufficient condition for  the existence of $L^1$-connections  from $\phi^-$ to  $\phi^+$. By an $L^1$-connection we mean a function $u(\cdot,t)$ which is a classical solution  on the interval $(-\infty,T)$, for some $T\in \mathbb R$, and blows up at $t=T$, but continues to exist in the space $L^1$ for $t\in[ T,\infty)$ and converges to $\phi^\pm$ (in a suitable sense) as $t\to\pm\infty$. In a  preliminary analysis, we give a detailed description of the classical connections between the equilibria.